* [PATCH 0/5] Remove remaining slow paths from math functions
@ 2021-01-04 12:11 Wilco Dijkstra via Libc-alpha
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
` (4 more replies)
0 siblings, 5 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:11 UTC (permalink / raw
To: 'GNU C Library'
This patch series removes all remaining slow paths and related code from math functions.
The asin/acos, tan, atan, atan2 implementations are fixed in following patches, and the final
patch removes the unused mpa files, headers and probes and updates the manual.
Passes buildmanyglibc.
git diff --stat HEAD~5
manual/probes.texi | 85 ----------
math/Makefile | 3 +-
sysdeps/aarch64/libm-test-ulps | 3 +-
sysdeps/generic/math_private.h | 6 -
sysdeps/ieee754/dbl-64/MathLib.h | 100 ------------
sysdeps/ieee754/dbl-64/atnat.h | 4 +-
sysdeps/ieee754/dbl-64/atnat2.h | 4 +-
sysdeps/ieee754/dbl-64/doasin.c | 81 ----------
sysdeps/ieee754/dbl-64/doasin.h | 63 --------
sysdeps/ieee754/dbl-64/dosincos.c | 217 -------------------------
sysdeps/ieee754/dbl-64/dosincos.h | 80 ----------
sysdeps/ieee754/dbl-64/e_asin.c | 339 +++++----------------------------------
sysdeps/ieee754/dbl-64/e_atan2.c | 324 +++++--------------------------------
sysdeps/ieee754/dbl-64/e_remainder.c | 1 -
sysdeps/ieee754/dbl-64/e_sqrt.c | 1 -
sysdeps/ieee754/dbl-64/mpa-arch.h | 47 ------
sysdeps/ieee754/dbl-64/mpa.c | 913 ---------------------------------------------------------------------------------------------------------
sysdeps/ieee754/dbl-64/mpa.h | 123 ---------------
sysdeps/ieee754/dbl-64/mpatan.c | 116 --------------
sysdeps/ieee754/dbl-64/mpatan.h | 145 -----------------
sysdeps/ieee754/dbl-64/mpatan2.c | 67 --------
sysdeps/ieee754/dbl-64/mpsqrt.c | 111 -------------
sysdeps/ieee754/dbl-64/mpsqrt.h | 38 -----
sysdeps/ieee754/dbl-64/mptan.c | 63 --------
sysdeps/ieee754/dbl-64/mydefs.h | 2 +-
sysdeps/ieee754/dbl-64/s_atan.c | 194 +++--------------------
sysdeps/ieee754/dbl-64/s_sin.c | 1 -
sysdeps/ieee754/dbl-64/s_tan.c | 609 +++++++++-------------------------------------------------------------
sysdeps/ieee754/dbl-64/sincos32.c | 307 -----------------------------------
sysdeps/ieee754/dbl-64/sincos32.h | 81 ----------
sysdeps/ieee754/dbl-64/uatan.tbl | 8 +-
sysdeps/ieee754/dbl-64/utan.h | 172 +-------------------
sysdeps/ieee754/dbl-64/utan.tbl | 4 +-
sysdeps/x86_64/fpu/multiarch/Makefile | 30 +---
sysdeps/x86_64/fpu/multiarch/doasin-fma.c | 4 -
sysdeps/x86_64/fpu/multiarch/doasin-fma4.c | 4 -
sysdeps/x86_64/fpu/multiarch/dosincos-fma.c | 6 -
sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c | 6 -
sysdeps/x86_64/fpu/multiarch/mpa-avx.c | 14 --
sysdeps/x86_64/fpu/multiarch/mpa-fma.c | 14 --
sysdeps/x86_64/fpu/multiarch/mpa-fma4.c | 14 --
sysdeps/x86_64/fpu/multiarch/mpatan-fma.c | 10 --
sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c | 10 --
sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c | 9 --
sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c | 9 --
sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c | 8 -
sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c | 8 -
sysdeps/x86_64/fpu/multiarch/mptan-fma.c | 7 -
sysdeps/x86_64/fpu/multiarch/mptan-fma4.c | 7 -
sysdeps/x86_64/fpu/multiarch/sincos32-fma.c | 13 --
sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c | 13 --
51 files changed, 192 insertions(+), 4306 deletions(-)
^ permalink raw reply [flat|nested] 19+ messages in thread
* [PATCH 1/5] Remove slow paths from asin
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
@ 2021-01-04 12:15 ` Wilco Dijkstra via Libc-alpha
2021-01-04 16:25 ` Paul Zimmermann
2021-01-07 19:23 ` [PATCH v2 1/5] Remove remaining slow paths from asin and acos Wilco Dijkstra via Libc-alpha
2021-01-04 12:17 ` [PATCH 2/5] Remove slow paths in tan Wilco Dijkstra via Libc-alpha
` (3 subsequent siblings)
4 siblings, 2 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:15 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths from asin - there is quite a lot of redundant slow code that can
be removed while keeping ULP below 1. Add ULP annotations. Update AArch64
libm-test-ulps for asin.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/aarch64/libm-test-ulps b/sysdeps/aarch64/libm-test-ulps
index 22fcf8db73dc444c25e0c356b1e0036571edd112..bbadf667ee4b7a0cf80506d321553f064049c516 100644
--- a/sysdeps/aarch64/libm-test-ulps
+++ b/sysdeps/aarch64/libm-test-ulps
@@ -41,6 +41,7 @@ float: 2
ldouble: 2
Function: "asin":
+double: 1
float: 1
ldouble: 1
@@ -55,7 +56,7 @@ float: 1
ldouble: 1
Function: "asin_upward":
-double: 1
+double: 2
float: 1
ldouble: 2
diff --git a/sysdeps/ieee754/dbl-64/e_asin.c b/sysdeps/ieee754/dbl-64/e_asin.c
index 8a3b26f6645b66818ad0a57fe833355a3e9961e6..c01e8a34517e4a33f126bbab5c132379b4b58a4d 100644
--- a/sysdeps/ieee754/dbl-64/e_asin.c
+++ b/sysdeps/ieee754/dbl-64/e_asin.c
@@ -21,8 +21,7 @@
/* */
/* FUNCTIONS: uasin */
/* uacos */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
-/* doasin.c sincos32.c dosincos.c mpa.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h usncs.h */
/* sincos.tbl asincos.tbl powtwo.tbl root.tbl */
/* */
/******************************************************************/
@@ -31,7 +30,6 @@
#include "asincos.tbl"
#include "root.tbl"
#include "powtwo.tbl"
-#include "MathLib.h"
#include "uasncs.h"
#include <float.h>
#include <math.h>
@@ -43,15 +41,10 @@
# define SECTION
#endif
-void __doasin(double x, double dx, double w[]);
-void __dubsin(double x, double dx, double v[]);
-void __dubcos(double x, double dx, double v[]);
-void __docos(double x, double dx, double v[]);
-
double
SECTION
__ieee754_asin(double x){
- double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2];
+ double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
mynumber u,v;
int4 k,m,n;
@@ -70,27 +63,8 @@ __ieee754_asin(double x){
x2 = x*x;
t = (((((f6*x2 + f5)*x2 + f4)*x2 + f3)*x2 + f2)*x2 + f1)*(x2*x);
res = x+t; /* res=arcsin(x) according to Taylor series */
- cor = (x-res)+t;
- if (res == res+1.025*cor) return res;
- else {
- x1 = x+big;
- xx = x*x;
- x1 -= big;
- x2 = x - x1;
- p = x1*x1*x1;
- s1 = a1.x*p;
- s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
- ((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
- res1 = x+s1;
- s2 = ((x-res1)+s1)+s2;
- res = res1+s2;
- cor = (res1-res)+s2;
- if (res == res+1.00014*cor) return res;
- else {
- __doasin(x,0,w);
- return w[0];
- }
- }
+ /* Max ULP is 0.512. */
+ return res;
}
/*---------------------0.125 <= |x| < 0.5 -----------------------------*/
else if (k < 0x3fe00000) {
@@ -103,26 +77,8 @@ __ieee754_asin(double x){
+xx*asncs.x[n+6]))))+asncs.x[n+7];
t+=p;
res =asncs.x[n+8] +t;
- cor = (asncs.x[n+8]-res)+t;
- if (res == res+1.05*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+8]+xx*asncs.x[n+9];
- t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0005*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __dubsin(res,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.523. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fe00000) */
/*-------------------- 0.5 <= |x| < 0.75 -----------------------------*/
else
@@ -135,26 +91,8 @@ __ieee754_asin(double x){
+xx*(asncs.x[n+6]+xx*asncs.x[n+7])))))+asncs.x[n+8];
t+=p;
res =asncs.x[n+9] +t;
- cor = (asncs.x[n+9]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+9]+xx*asncs.x[n+10];
- t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0005*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __dubsin(res,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fe80000) */
/*--------------------- 0.75 <= |x|< 0.921875 ----------------------*/
else
@@ -167,28 +105,8 @@ __ieee754_asin(double x){
+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+xx*asncs.x[n+8]))))))+asncs.x[n+9];
t+=p;
res =asncs.x[n+10] +t;
- cor = (asncs.x[n+10]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+10]+xx*asncs.x[n+11];
- t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0008*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=hp0.x-res;
- z=((hp0.x-y)-res)+(hp1.x-z);
- __dubcos(y,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fed8000) */
/*-------------------0.921875 <= |x| < 0.953125 ------------------------*/
else
@@ -203,29 +121,8 @@ __ieee754_asin(double x){
xx*asncs.x[n+9])))))))+asncs.x[n+10];
t+=p;
res =asncs.x[n+11] +t;
- cor = (asncs.x[n+11]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+11]+xx*asncs.x[n+12];
- t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0007*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=(hp0.x-res)-z;
- z=y+hp1.x;
- y=(y-z)+hp1.x;
- __dubcos(z,y,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fee8000) */
/*--------------------0.953125 <= |x| < 0.96875 ------------------------*/
@@ -241,29 +138,8 @@ __ieee754_asin(double x){
xx*(asncs.x[n+9]+xx*asncs.x[n+10]))))))))+asncs.x[n+11];
t+=p;
res =asncs.x[n+12] +t;
- cor = (asncs.x[n+12]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+12]+xx*asncs.x[n+13];
- t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0007*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=(hp0.x-res)-z;
- z=y+hp1.x;
- y=(y-z)+hp1.x;
- __dubcos(z,y,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fef0000) */
/*--------------------0.96875 <= |x| < 1 --------------------------------*/
else
@@ -282,16 +158,8 @@ __ieee754_asin(double x){
cor = (hp1.x - 2.0*cc)-2.0*(y+cc)*p;
res1 = hp0.x - 2.0*y;
res =res1 + cor;
- if (res == res+1.003*((res1-res)+cor)) return (m>0)?res:-res;
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res1=hp0.x-2.0*w[0];
- cor=((hp0.x-res1)-2.0*w[0])+(hp1.x-2.0*w[1]);
- res = res1+cor;
- return (m>0)?res:-res;
- }
+ /* Max ULP is 0.5015. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3ff00000) */
/*---------------------------- |x|>=1 -------------------------------*/
else if (k==0x3ff00000 && u.i[LOW_HALF]==0) return (m>0)?hp0.x:-hp0.x;
@@ -319,7 +187,7 @@ double
SECTION
__ieee754_acos(double x)
{
- double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2],eps;
+ double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
mynumber u,v;
int4 k,m,n;
u.x = x;
@@ -336,32 +204,8 @@ __ieee754_acos(double x)
r=hp0.x-x;
cor=(((hp0.x-r)-x)+hp1.x)-t;
res = r+cor;
- cor = (r-res)+cor;
- if (res == res+1.004*cor) return res;
- else {
- x1 = x+big;
- xx = x*x;
- x1 -= big;
- x2 = x - x1;
- p = x1*x1*x1;
- s1 = a1.x*p;
- s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
- ((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
- res1 = x+s1;
- s2 = ((x-res1)+s1)+s2;
- r=hp0.x-res1;
- cor=(((hp0.x-r)-res1)+hp1.x)-s2;
- res = r+cor;
- cor = (r-res)+cor;
- if (res == res+1.00004*cor) return res;
- else {
- __doasin(x,0,w);
- r=hp0.x-w[0];
- cor=((hp0.x-r)-w[0])+(hp1.x-w[1]);
- res=r+cor;
- return res;
- }
- }
+ /* Max ULP is 0.502. */
+ return res;
} /* else if (k < 0x3fc00000) */
/*---------------------- 0.125 <= |x| < 0.5 --------------------*/
else
@@ -377,35 +221,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+8]):(hp0.x+asncs.x[n+8]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+1.02*((y-res)+t)) return res;
- else {
- r=asncs.x[n+8]+xx*asncs.x[n+9];
- t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
- if (m>0)
- {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; }
- else
- {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+1.0002*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.51. */
+ return res;
} /* else if (k < 0x3fe00000) */
/*--------------------------- 0.5 <= |x| < 0.75 ---------------------*/
else
if (k < 0x3fe80000) {
n = 1056+((k&0x000fe000)>>11)*3;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps=1.02; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+
@@ -414,33 +239,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+9]):(hp0.x+asncs.x[n+9]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+9]+xx*asncs.x[n+10];
- t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0004; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0002; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.52. */
+ return res;
} /* else if (k < 0x3fe80000) */
/*------------------------- 0.75 <= |x| < 0.921875 -------------*/
else
if (k < 0x3fed8000) {
n = 992+((k&0x000fe000)>>13)*13;
- if (m>0) {xx = x - asncs.x[n]; eps = 1.04; }
- else {xx = -x - asncs.x[n]; eps = 1.01; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+
@@ -449,33 +257,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+10]):(hp0.x+asncs.x[n+10]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+10]+xx*asncs.x[n+11];
- t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0032; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0008; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.52. */
+ return res;
} /* else if (k < 0x3fed8000) */
/*-------------------0.921875 <= |x| < 0.953125 ------------------*/
else
if (k < 0x3fee8000) {
n = 884+((k&0x000fe000)>>13)*14;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps =1.005; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
@@ -485,33 +276,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+11]):(hp0.x+asncs.x[n+11]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+11]+xx*asncs.x[n+12];
- t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.52. */
+ return res;
} /* else if (k < 0x3fee8000) */
/*--------------------0.953125 <= |x| < 0.96875 ----------------*/
else
if (k < 0x3fef0000) {
n = 768+((k&0x000fe000)>>13)*15;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps=1.005;}
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
@@ -521,25 +295,8 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+12]):(hp0.x+asncs.x[n+12]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+12]+xx*asncs.x[n+13];
- t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.52. */
+ return res;
} /* else if (k < 0x3fef0000) */
/*-----------------0.96875 <= |x| < 1 ---------------------------*/
@@ -560,28 +317,14 @@ __ieee754_acos(double x)
cor = (hp1.x - cc)-(y+cc)*p;
res1 = hp0.x - y;
res =res1 + cor;
- if (res == res+1.002*((res1-res)+cor)) return (res+res);
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res1=hp0.x-w[0];
- cor=((hp0.x-res1)-w[0])+(hp1.x-w[1]);
- res = res1+cor;
- return (res+res);
- }
+ /* Max ULP is 0.501. */
+ return (res+res);
}
else {
cor = cc+p*(y+cc);
res = y + cor;
- if (res == res+1.03*((y-res)+cor)) return (res+res);
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res = w[0];
- return (res+res);
- }
+ /* Max ULP is 0.515. */
+ return (res+res);
}
} /* else if (k < 0x3ff00000) */
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH 2/5] Remove slow paths in tan
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
@ 2021-01-04 12:17 ` Wilco Dijkstra via Libc-alpha
2021-01-04 16:36 ` Paul Zimmermann
2021-01-07 19:25 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:19 ` [PATCH 3/5] Remove slow paths from atan Wilco Dijkstra via Libc-alpha
` (2 subsequent siblings)
4 siblings, 2 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:17 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths in tan. Add ULP annotations. Merge 'number' into 'mynumber'.
Remove unused entries from tan constants.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/e_atan2.c b/sysdeps/ieee754/dbl-64/e_atan2.c
index 380d34368dbdf95138aed911677bdc29ac1e7d9b..7af2a4f495d39f6450699de1d0d9d2f7a33fdcf7 100644
--- a/sysdeps/ieee754/dbl-64/e_atan2.c
+++ b/sysdeps/ieee754/dbl-64/e_atan2.c
@@ -39,6 +39,7 @@
#include <dla.h>
#include "mpa.h"
#include "MathLib.h"
+#include "mydefs.h"
#include "uatan.tbl"
#include "atnat2.h"
#include <fenv.h>
diff --git a/sysdeps/ieee754/dbl-64/mydefs.h b/sysdeps/ieee754/dbl-64/mydefs.h
index 60aa2401e02d31524d0890c31987dfd9736ed25a..f17f4f593ca3819d3cbc03bb6138d483743982e8 100644
--- a/sysdeps/ieee754/dbl-64/mydefs.h
+++ b/sysdeps/ieee754/dbl-64/mydefs.h
@@ -28,7 +28,7 @@
#define MY_H
typedef int int4;
-typedef union { int4 i[2]; double x; } mynumber;
+typedef union { int4 i[2]; double x; double d; } mynumber;
#define max(x, y) (((y) > (x)) ? (y) : (x))
#define min(x, y) (((y) < (x)) ? (y) : (x))
diff --git a/sysdeps/ieee754/dbl-64/s_tan.c b/sysdeps/ieee754/dbl-64/s_tan.c
index 086ae50458a3b1cd5826f61b736ea609d9d72ee9..1c5f32711eb31e99f2f1b4372b200bbe520efe57 100644
--- a/sysdeps/ieee754/dbl-64/s_tan.c
+++ b/sysdeps/ieee754/dbl-64/s_tan.c
@@ -20,40 +20,29 @@
/* MODULE_NAME: utan.c */
/* */
/* FUNCTIONS: utan */
-/* tanMp */
/* */
-/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h utan.h */
-/* branred.c sincos32.c mptan.c */
+/* FILES NEEDED:dla.h endian.h mydefs.h utan.h */
+/* branred.c */
/* utan.tbl */
/* */
-/* An ultimate tan routine. Given an IEEE double machine number x */
-/* it computes the correctly rounded (to nearest) value of tan(x). */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/*********************************************************************/
#include <errno.h>
#include <float.h>
#include "endian.h"
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
+#include "mydefs.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-double.h>
#include <fenv.h>
-#include <stap-probe.h>
#ifndef SECTION
# define SECTION
#endif
-static double tanMp (double);
-void __mptan (double, mp_no *, int);
-
double
SECTION
__tan (double x)
@@ -62,17 +51,14 @@ __tan (double x)
#include "utan.tbl"
int ux, i, n;
- double a, da, a2, b, db, c, dc, c1, cc1, c2, cc2, c3, cc3, fi, ffi, gi, pz,
- s, sy, t, t1, t2, t3, t4, w, x2, xn, xx2, y, ya,
- yya, z0, z, zz, z2, zz2;
- int p;
- number num, v;
- mp_no mpa, mpt1, mpt2;
+ double a, da, a2, b, db, c, dc, fi, gi, pz,
+ s, sy, t, t1, t2, t3, t4, w, x2, xn, y, ya,
+ yya, z0, z, z2;
+ mynumber num, v;
double retval;
int __branred (double, double *, double *);
- int __mpranred (double, mp_no *, int);
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
@@ -100,7 +86,6 @@ __tan (double x)
/* (II) The case 1.259e-8 < abs(x) <= 0.0608 */
if (w <= g2.d)
{
- /* First stage */
x2 = x * x;
t2 = d9.d + x2 * d11.d;
@@ -109,50 +94,16 @@ __tan (double x)
t2 = d3.d + x2 * t2;
t2 *= x * x2;
- if ((y = x + (t2 - u1.d * t2)) == x + (t2 + u1.d * t2))
- {
- retval = y;
- goto ret;
- }
-
- /* Second stage */
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- EMULV (x, x, x2, xx2);
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (x, 0.0, c1, cc1, c2, cc2, t1, t2);
- ADD2 (x, 0.0, c2, cc2, c1, cc1, t1, t2);
- if ((y = c1 + (cc1 - u2.d * c1)) == c1 + (cc1 + u2.d * c1))
- {
- retval = y;
- goto ret;
- }
- retval = tanMp (x);
+ y = x + t2;
+ retval = y;
+ /* Max ULP is 0.504. */
goto ret;
}
/* (III) The case 0.0608 < abs(x) <= 0.787 */
if (w <= g3.d)
{
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * w));
+ i = ((int) (mfftnhf.d + 256 * w));
z = w - xfg[i][0].d;
z2 = z * z;
s = (x < 0.0) ? -1 : 1;
@@ -160,41 +111,9 @@ __tan (double x)
fi = xfg[i][1].d;
gi = xfg[i][2].d;
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u3.d)) == fi + (t2 + fi * u3.d))
- {
- retval = (s * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua3.d + t3 * ub3.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (s * y);
- goto ret;
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- EMULV (z, z, z2, zz2);
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, 0.0, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, 0.0, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
-
- if ((y = c3 + (cc3 - u4.d * c3)) == c3 + (cc3 + u4.d * c3))
- {
- retval = (s * y);
- goto ret;
- }
- retval = tanMp (x);
+ y = fi + t2;
+ retval = (s * y);
+ /* Max ULP is 0.60. */
goto ret;
}
@@ -223,14 +142,7 @@ __tan (double x)
sy = 1;
}
- /* (IV),(V) The case 0.787 < abs(x) <= 25, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (VI) The case 0.787 < abs(x) <= 25, 1e-7 < abs(y) <= 0.0608 */
+ /* (VI) The case 0.787 < abs(x) <= 25, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -242,94 +154,27 @@ __tan (double x)
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u6.d * c)) == c + (dc + u6.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u5.d * a)) == a + (t2 + u5.d * a))
- {
- retval = y;
- goto ret;
- }
- }
- /* Second stage */
- /* Range reduction by algorithm ii */
- t = (x * hpinv.d + toint.d);
- xn = t - toint.d;
- v.d = t;
- t1 = (x - xn * mp1.d) - xn * mp2.d;
- n = v.i[LOW_HALF] & 0x00000001;
- da = xn * pp3.d;
- t = t1 - da;
- da = (t1 - t) - da;
- t1 = xn * pp4.d;
- a = t - t1;
- da = ((t - a) - t1) + da;
-
- /* Second stage */
- EADD (a, da, t1, t2);
- a = t1;
- da = t2;
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
-
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u8.d * c2)) == c2 + (cc2 + u8.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u7.d * c1)) == c1 + (cc1 + u7.d * c1))
- {
- retval = y;
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.506. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (VII) The case 0.787 < abs(x) <= 25, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -340,76 +185,20 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u10.d)) == gi - (t2 + gi * u10.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua10.d + t3 * ub10.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u9.d)) == fi + (t2 + fi * u9.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua9.d + t3 * ub9.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz)
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u12.d * c3)) == c3 + (cc3 + u12.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u11.d * c3)) == c3 + (cc3 + u11.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
-
- retval = tanMp (x);
- goto ret;
}
/* (---) The case 25 < abs(x) <= 1e8 */
@@ -443,14 +232,7 @@ __tan (double x)
sy = 1;
}
- /* (+++) The case 25 < abs(x) <= 1e8, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (VIII) The case 25 < abs(x) <= 1e8, 1e-7 < abs(y) <= 0.0608 */
+ /* (VIII) The case 25 < abs(x) <= 1e8, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -462,76 +244,26 @@ __tan (double x)
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u14.d * c)) == c + (dc + u14.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u13.d * a)) == a + (t2 + u13.d * a))
- {
- retval = y;
- goto ret;
- }
- }
-
- /* Second stage */
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u16.d * c2)) == c2 + (cc2 + u16.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u15.d * c1)) == c1 + (cc1 + u15.d * c1))
- {
- retval = (y);
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.506. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (IX) The case 25 < abs(x) <= 1e8, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -542,75 +274,20 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u18.d)) == gi - (t2 + gi * u18.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua18.d + t3 * ub18.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u17.d)) == fi + (t2 + fi * u17.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua17.d + t3 * ub17.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz);
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u20.d * c3)) == c3 + (cc3 + u20.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u19.d * c3)) == c3 + (cc3 + u19.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (---) The case 1e8 < abs(x) < 2**1024 */
@@ -632,14 +309,7 @@ __tan (double x)
sy = 1;
}
- /* (+++) The case 1e8 < abs(x) < 2**1024, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (X) The case 1e8 < abs(x) < 2**1024, 1e-7 < abs(y) <= 0.0608 */
+ /* (X) The case 1e8 < abs(x) < 2**1024, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -650,85 +320,26 @@ __tan (double x)
t2 = da + a * a2 * t2;
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u22.d * c)) == c + (dc + u22.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u21.d * a)) == a + (t2 + u21.d * a))
- {
- retval = y;
- goto ret;
- }
- }
-
- /* Second stage */
- /* Reduction by algorithm iv */
- p = 10;
- n = (__mpranred (x, &mpa, p)) & 0x00000001;
- __mp_dbl (&mpa, &a, p);
- __dbl_mp (a, &mpt1, p);
- __sub (&mpa, &mpt1, &mpt2, p);
- __mp_dbl (&mpt2, &da, p);
-
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
-
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u24.d * c2)) == c2 + (cc2 + u24.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u23.d * c1)) == c1 + (cc1 + u23.d * c1))
- {
- retval = y;
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.507. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (XI) The case 1e8 < abs(x) < 2**1024, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -739,97 +350,25 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u26.d)) == gi - (t2 + gi * u26.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua26.d + t3 * ub26.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u25.d)) == fi + (t2 + fi * u25.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua25.d + t3 * ub25.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz);
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u28.d * c3)) == c3 + (cc3 + u28.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u27.d * c3)) == c3 + (cc3 + u27.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
ret:
return retval;
}
-/* multiple precision stage */
-/* Convert x to multi precision number,compute tan(x) by mptan() routine */
-/* and converts result back to double */
-static double
-SECTION
-tanMp (double x)
-{
- int p;
- double y;
- mp_no mpy;
- p = 32;
- __mptan (x, &mpy, p);
- __mp_dbl (&mpy, &y, p);
- LIBC_PROBE (slowtan, 2, &x, &y);
- return y;
-}
-
#ifndef __tan
libm_alias_double (__tan, tan)
#endif
diff --git a/sysdeps/ieee754/dbl-64/utan.h b/sysdeps/ieee754/dbl-64/utan.h
index ebe3e193a210280647f76564b6991f98bfe83a9b..8f1034eba855130258adb2aeb816c5722b3f9818 100644
--- a/sysdeps/ieee754/dbl-64/utan.h
+++ b/sysdeps/ieee754/dbl-64/utan.h
@@ -28,7 +28,7 @@
#define UTAN_H
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x3FD55555, 0x55555555} }, /* 0.333... */
/**/ d5 = {{0x3FC11111, 0x111107C6} }, /* 0.133... */
@@ -36,46 +36,6 @@
/**/ d9 = {{0x3F9664ED, 0x49CFC666} }, /* . */
/**/ d11 = {{0x3F82385A, 0x3CF2E4EA} }, /* . */
/* polynomial II */
-/**/ a3 = {{0x3fd55555, 0x55555555} }, /* 1/3 */
-/**/ aa3 = {{0x3c755555, 0x55555555} }, /* 1/3-a3 */
-/**/ a5 = {{0x3fc11111, 0x11111111} }, /* 2/15 */
-/**/ aa5 = {{0x3c411111, 0x11111111} }, /* 2/15-a5 */
-/**/ a7 = {{0x3faba1ba, 0x1ba1ba1c} }, /* 17/315 */
-/**/ aa7 = {{0xbc479179, 0x17917918} }, /* ()-a7 */
-/**/ a9 = {{0x3f9664f4, 0x882c10fa} }, /* 62/2835 */
-/**/ aa9 = {{0xbc09a528, 0x8b6c44fd} }, /* ()-a9 */
-/**/ a11 = {{0x3f8226e3, 0x55e6c23d} }, /* . */
-/**/ aa11 = {{0xbc2c292b, 0x8f1a2c13} }, /* . */
-/**/ a13 = {{0x3f6d6d3d, 0x0e157de0} }, /* . */
-/**/ aa13 = {{0xbc0280cf, 0xc968d971} }, /* . */
-/**/ a15 = {{0x3f57da36, 0x452b75e3} }, /* . */
-#if 0
-/**/ aa15 = {{0xbbf25789, 0xb285d2ed} }, /* . */
-#endif
-/**/ a17 = {{0x3f435582, 0x48036744} }, /* . */
-#if 0
-/**/ aa17 = {{0x3be488d9, 0x563f1f23} }, /* . */
-#endif
-/**/ a19 = {{0x3f2f57d7, 0x734d1664} }, /* . */
-#if 0
-/**/ aa19 = {{0x3bb0d55a, 0x913ccb50} }, /* . */
-#endif
-/**/ a21 = {{0x3f1967e1, 0x8afcafad} }, /* . */
-#if 0
-/**/ aa21 = {{0xbbbd7614, 0xa42d44e6} }, /* . */
-#endif
-/**/ a23 = {{0x3f0497d8, 0xeea25259} }, /* . */
-#if 0
-/**/ aa23 = {{0x3b99f2d0, 0x2e4d2863} }, /* . */
-#endif
-/**/ a25 = {{0x3ef0b132, 0xd39a6050} }, /* . */
-#if 0
-/**/ aa25 = {{0x3b93b274, 0xc2c19614} }, /* . */
-#endif
-/**/ a27 = {{0x3edb0f72, 0xd3ee24e9} }, /* . */
-#if 0
-/**/ aa27 = {{0x3b61688d, 0xdd595609} }, /* . */
-#endif
/* polynomial III */
/**/ e0 = {{0x3FD55555, 0x55554DBD} }, /* . */
/**/ e1 = {{0x3FC11112, 0xE0A6B45F} }, /* . */
@@ -88,52 +48,8 @@
/**/ g3 = {{0x3fe92f1a, 0x00000000} }, /* 0.787 */
/**/ g4 = {{0x40390000, 0x00000000} }, /* 25.0 */
/**/ g5 = {{0x4197d784, 0x00000000} }, /* 1e8 */
-/**/ gy1 = {{0x3e7ad7f2, 0x9abcaf48} }, /* 1e-7 */
/**/ gy2 = {{0x3faf212d, 0x00000000} }, /* 0.0608 */
-/**/ u1 = {{0x3cc8c33a, 0x00000000} }, /* 6.873e-16 */
-/**/ u2 = {{0x3983dc4d, 0x00000000} }, /* 1.224e-31 */
-/**/ u3 = {{0x3c78e14b, 0x00000000} }, /* 2.158e-17 */
-/**/ ua3 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub3 = {{0x3cc81898, 0x00000000} }, /* 6.688e-16 */
-/**/ u4 = {{0x399856c2, 0x00000000} }, /* 3e-31 */
-/**/ u5 = {{0x3c39d80a, 0x00000000} }, /* 1.401e-18 */
-/**/ u6 = {{0x3c374c5a, 0x00000000} }, /* 1.263e-18 */
-/**/ u7 = {{0x39903beb, 0x00000000} }, /* 2.001e-31 */
-/**/ u8 = {{0x399c56ae, 0x00000000} }, /* 3.493e-31 */
-/**/ u9 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua9 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub9 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u10 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua10 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub10 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u11 = {{0x39e509b6, 0x00000000} }, /* 8.298e-30 */
-/**/ u12 = {{0x39e509b6, 0x00000000} }, /* 8.298e-30 */
-/**/ u13 = {{0x3c39d80a, 0x00000000} }, /* 1.401e-18 */
-/**/ u14 = {{0x3c374c5a, 0x00000000} }, /* 1.263e-18 */
-/**/ u15 = {{0x3ab5767a, 0x00000000} }, /* 6.935e-26 */
-/**/ u16 = {{0x3ab57744, 0x00000000} }, /* 6.936e-26 */
-/**/ u17 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua17 = {{0x3bfdb11f, 0x00000000} }, /* 1.006e-19 */
-/**/ ub17 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u18 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua18 = {{0x3bfdb11f, 0x00000000} }, /* 1.006e-19 */
-/**/ ub18 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u19 = {{0x39a13b61, 0x00000000} }, /* 4.248e-31 */
-/**/ u20 = {{0x39a13b61, 0x00000000} }, /* 4.248e-31 */
-/**/ u21 = {{0x3c3bb9b8, 0x00000000} }, /* 1.503e-18 */
-/**/ u22 = {{0x3c392e08, 0x00000000} }, /* 1.365e-18 */
-/**/ u23 = {{0x3a0ce706, 0x00000000} }, /* 4.560e-29 */
-/**/ u24 = {{0x3a0cff5d, 0x00000000} }, /* 4.575e-29 */
-/**/ u25 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua25 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub25 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u26 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua26 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub26 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u27 = {{0x3ad421cb, 0x00000000} }, /* 2.602e-25 */
-/**/ u28 = {{0x3ad421cb, 0x00000000} }, /* 2.602e-25 */
-
/**/ mp1 = {{0x3FF921FB, 0x58000000} },
/**/ mp2 = {{0xBE4DDE97, 0x3C000000} },
/**/ mp3 = {{0xBC8CB3B3, 0x99D747F2} },
@@ -145,7 +61,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0x3FD55555} }, /* 0.333... */
/**/ d5 = {{0x111107C6, 0x3FC11111} }, /* 0.133... */
@@ -153,46 +69,6 @@
/**/ d9 = {{0x49CFC666, 0x3F9664ED} }, /* . */
/**/ d11 = {{0x3CF2E4EA, 0x3F82385A} }, /* . */
/* polynomial II */
-/**/ a3 = {{0x55555555, 0x3fd55555} }, /* 1/3 */
-/**/ aa3 = {{0x55555555, 0x3c755555} }, /* 1/3-a3 */
-/**/ a5 = {{0x11111111, 0x3fc11111} }, /* 2/15 */
-/**/ aa5 = {{0x11111111, 0x3c411111} }, /* 2/15-a5 */
-/**/ a7 = {{0x1ba1ba1c, 0x3faba1ba} }, /* 17/315 */
-/**/ aa7 = {{0x17917918, 0xbc479179} }, /* ()-a7 */
-/**/ a9 = {{0x882c10fa, 0x3f9664f4} }, /* 62/2835 */
-/**/ aa9 = {{0x8b6c44fd, 0xbc09a528} }, /* ()-a9 */
-/**/ a11 = {{0x55e6c23d, 0x3f8226e3} }, /* . */
-/**/ aa11 = {{0x8f1a2c13, 0xbc2c292b} }, /* . */
-/**/ a13 = {{0x0e157de0, 0x3f6d6d3d} }, /* . */
-/**/ aa13 = {{0xc968d971, 0xbc0280cf} }, /* . */
-/**/ a15 = {{0x452b75e3, 0x3f57da36} }, /* . */
-#if 0
-/**/ aa15 = {{0xb285d2ed, 0xbbf25789} }, /* . */
-#endif
-/**/ a17 = {{0x48036744, 0x3f435582} }, /* . */
-#if 0
-/**/ aa17 = {{0x563f1f23, 0x3be488d9} }, /* . */
-#endif
-/**/ a19 = {{0x734d1664, 0x3f2f57d7} }, /* . */
-#if 0
-/**/ aa19 = {{0x913ccb50, 0x3bb0d55a} }, /* . */
-#endif
-/**/ a21 = {{0x8afcafad, 0x3f1967e1} }, /* . */
-#if 0
-/**/ aa21 = {{0xa42d44e6, 0xbbbd7614} }, /* . */
-#endif
-/**/ a23 = {{0xeea25259, 0x3f0497d8} }, /* . */
-#if 0
-/**/ aa23 = {{0x2e4d2863, 0x3b99f2d0} }, /* . */
-#endif
-/**/ a25 = {{0xd39a6050, 0x3ef0b132} }, /* . */
-#if 0
-/**/ aa25 = {{0xc2c19614, 0x3b93b274} }, /* . */
-#endif
-/**/ a27 = {{0xd3ee24e9, 0x3edb0f72} }, /* . */
-#if 0
-/**/ aa27 = {{0xdd595609, 0x3b61688d} }, /* . */
-#endif
/* polynomial III */
/**/ e0 = {{0x55554DBD, 0x3FD55555} }, /* . */
/**/ e1 = {{0xE0A6B45F, 0x3FC11112} }, /* . */
@@ -205,52 +81,8 @@
/**/ g3 = {{0x00000000, 0x3fe92f1a} }, /* 0.787 */
/**/ g4 = {{0x00000000, 0x40390000} }, /* 25.0 */
/**/ g5 = {{0x00000000, 0x4197d784} }, /* 1e8 */
-/**/ gy1 = {{0x9abcaf48, 0x3e7ad7f2} }, /* 1e-7 */
/**/ gy2 = {{0x00000000, 0x3faf212d} }, /* 0.0608 */
-/**/ u1 = {{0x00000000, 0x3cc8c33a} }, /* 6.873e-16 */
-/**/ u2 = {{0x00000000, 0x3983dc4d} }, /* 1.224e-31 */
-/**/ u3 = {{0x00000000, 0x3c78e14b} }, /* 2.158e-17 */
-/**/ ua3 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub3 = {{0x00000000, 0x3cc81898} }, /* 6.688e-16 */
-/**/ u4 = {{0x00000000, 0x399856c2} }, /* 3e-31 */
-/**/ u5 = {{0x00000000, 0x3c39d80a} }, /* 1.401e-18 */
-/**/ u6 = {{0x00000000, 0x3c374c5a} }, /* 1.263e-18 */
-/**/ u7 = {{0x00000000, 0x39903beb} }, /* 2.001e-31 */
-/**/ u8 = {{0x00000000, 0x399c56ae} }, /* 3.493e-31 */
-/**/ u9 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua9 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub9 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u10 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua10 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub10 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u11 = {{0x00000000, 0x39e509b6} }, /* 8.298e-30 */
-/**/ u12 = {{0x00000000, 0x39e509b6} }, /* 8.298e-30 */
-/**/ u13 = {{0x00000000, 0x3c39d80a} }, /* 1.401e-18 */
-/**/ u14 = {{0x00000000, 0x3c374c5a} }, /* 1.263e-18 */
-/**/ u15 = {{0x00000000, 0x3ab5767a} }, /* 6.935e-26 */
-/**/ u16 = {{0x00000000, 0x3ab57744} }, /* 6.936e-26 */
-/**/ u17 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua17 = {{0x00000000, 0x3bfdb11f} }, /* 1.006e-19 */
-/**/ ub17 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u18 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua18 = {{0x00000000, 0x3bfdb11f} }, /* 1.006e-19 */
-/**/ ub18 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u19 = {{0x00000000, 0x39a13b61} }, /* 4.248e-31 */
-/**/ u20 = {{0x00000000, 0x39a13b61} }, /* 4.248e-31 */
-/**/ u21 = {{0x00000000, 0x3c3bb9b8} }, /* 1.503e-18 */
-/**/ u22 = {{0x00000000, 0x3c392e08} }, /* 1.365e-18 */
-/**/ u23 = {{0x00000000, 0x3a0ce706} }, /* 4.560e-29 */
-/**/ u24 = {{0x00000000, 0x3a0cff5d} }, /* 4.575e-29 */
-/**/ u25 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua25 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub25 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u26 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua26 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub26 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u27 = {{0x00000000, 0x3ad421cb} }, /* 2.602e-25 */
-/**/ u28 = {{0x00000000, 0x3ad421cb} }, /* 2.602e-25 */
-
/**/ mp1 = {{0x58000000, 0x3FF921FB} },
/**/ mp2 = {{0x3C000000, 0xBE4DDE97} },
/**/ mp3 = {{0x99D747F2, 0xBC8CB3B3} },
diff --git a/sysdeps/ieee754/dbl-64/utan.tbl b/sysdeps/ieee754/dbl-64/utan.tbl
index b094d164b8189c26145083a5dbe86454c10ec754..80796293de54d676fefdfa23e302bce254b8f7bc 100644
--- a/sysdeps/ieee754/dbl-64/utan.tbl
+++ b/sysdeps/ieee754/dbl-64/utan.tbl
@@ -23,7 +23,7 @@
#ifdef BIG_ENDI
-static const number
+static const mynumber
xfg[186][4] = { /* xi,Fi,Gi,FFi, i=16..201 */
/**/ {{{0x3fb00000, 0x1e519d60} },
/**/ {{0x3fb00557, 0x96c4e240} },
@@ -773,7 +773,7 @@ static const number
#else
#ifdef LITTLE_ENDI
-static const number
+static const mynumber
xfg[186][4] = { /* xi,Fi,Gi,FFi, i=16..201 */
/**/ {{{0x1e519d60, 0x3fb00000} },
/**/ {{0x96c4e240, 0x3fb00557} },
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH 3/5] Remove slow paths from atan
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
2021-01-04 12:17 ` [PATCH 2/5] Remove slow paths in tan Wilco Dijkstra via Libc-alpha
@ 2021-01-04 12:19 ` Wilco Dijkstra via Libc-alpha
2021-01-04 16:43 ` Paul Zimmermann
2021-01-07 19:26 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:21 ` [PATCH 4/5] Remove slow paths from atan2 Wilco Dijkstra via Libc-alpha
2021-01-04 12:25 ` [PATCH 5/5] Remove mpa files Wilco Dijkstra via Libc-alpha
4 siblings, 2 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:19 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths from atan. Add ULP annotations.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/atnat.h b/sysdeps/ieee754/dbl-64/atnat.h
index bb4a2ebdc4bd9b2f2ec8ec4395b91c683eb83a8f..29d7671b86750edb810e3918d3076bab23c0f56a 100644
--- a/sysdeps/ieee754/dbl-64/atnat.h
+++ b/sysdeps/ieee754/dbl-64/atnat.h
@@ -29,7 +29,7 @@
#define M 4
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@@ -79,7 +79,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */
diff --git a/sysdeps/ieee754/dbl-64/s_atan.c b/sysdeps/ieee754/dbl-64/s_atan.c
index 30b0906826d005c17f9322ff7172ec06b66dc03e..fdd18920b38dda61d2ed43abfdadb0057d6d611f 100644
--- a/sysdeps/ieee754/dbl-64/s_atan.c
+++ b/sysdeps/ieee754/dbl-64/s_atan.c
@@ -20,25 +20,15 @@
/* MODULE_NAME: atnat.c */
/* */
/* FUNCTIONS: uatan */
-/* atanMp */
/* signArctan */
/* */
-/* */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat.h */
-/* mpatan.c mpatan2.c mpsqrt.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h atnat.h */
/* uatan.tbl */
/* */
-/* An ultimate atan() routine. Given an IEEE double machine number x */
-/* it computes the correctly rounded (to nearest) value of atan(x). */
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/************************************************************************/
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
+#include "mydefs.h"
#include "uatan.tbl"
#include "atnat.h"
#include <fenv.h>
@@ -47,10 +37,8 @@
#include <math.h>
#include <fenv_private.h>
#include <math-underflow.h>
-#include <stap-probe.h>
-void __mpatan (mp_no *, mp_no *, int); /* see definition in mpatan.c */
-static double atanMp (double, const int[]);
+#define TWO52 0x1.0p52
/* Fix the sign of y and return */
static double
@@ -60,16 +48,13 @@ __signArctan (double x, double y)
}
-/* An ultimate atan() routine. Given an IEEE double machine number x, */
-/* routine computes the correctly rounded (to nearest) value of atan(x). */
double
__atan (double x)
{
- double cor, s1, ss1, s2, ss2, t1, t2, t3, t4, u, u2, u3,
- v, vv, w, ww, y, yy, z, zz;
+ double cor, t1, t2, t3, u,
+ v, w, ww, y, yy, z;
int i, ux, dx;
- static const int pr[M] = { 6, 8, 10, 32 };
- number num;
+ mynumber num;
num.d = x;
ux = num.i[HIGH_HALF];
@@ -102,36 +87,14 @@ __atan (double x)
yy = d3.d + v * yy;
yy *= x * v;
- if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
- return y;
-
- EMULV (x, x, v, vv); /* v+vv=x^2 */
-
- s1 = f17.d + v * f19.d;
- s1 = f15.d + v * s1;
- s1 = f13.d + v * s1;
- s1 = f11.d + v * s1;
- s1 *= v;
-
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2);
- ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
- if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
- return y;
-
- return atanMp (x, pr);
+ y = x + yy;
+ /* Max ULP is 0.511. */
+ return y;
}
}
else
{ /* B <= u < C */
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
z = u - cij[i][0].d;
yy = cij[i][5].d + z * cij[i][6].d;
@@ -141,44 +104,9 @@ __atan (double x)
yy *= z;
t1 = cij[i][1].d;
- if (i < 112)
- {
- if (i < 48)
- u2 = U21; /* u < 1/4 */
- else
- u2 = U22;
- } /* 1/4 <= u < 1/2 */
- else
- {
- if (i < 176)
- u2 = U23; /* 1/2 <= u < 3/4 */
- else
- u2 = U24;
- } /* 3/4 <= u <= 1 */
- if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
- return __signArctan (x, y);
-
- z = u - hij[i][0].d;
-
- s1 = hij[i][14].d + z * hij[i][15].d;
- s1 = hij[i][13].d + z * s1;
- s1 = hij[i][12].d + z * s1;
- s1 = hij[i][11].d + z * s1;
- s1 *= z;
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t1 + yy;
+ /* Max ULP is 0.56. */
+ return __signArctan (x, y);
}
}
else
@@ -188,7 +116,7 @@ __atan (double x)
w = 1 / u;
EMULV (w, u, t1, t2);
ww = w * ((1 - t1) - t2);
- i = (TWO52 + TWO8 * w) - TWO52;
+ i = (TWO52 + 256 * w) - TWO52;
i -= 16;
z = (w - cij[i][0].d) + ww;
@@ -199,37 +127,9 @@ __atan (double x)
yy = HPI1 - z * yy;
t1 = HPI - cij[i][1].d;
- if (i < 112)
- u3 = U31; /* w < 1/2 */
- else
- u3 = U32; /* w >= 1/2 */
- if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
- return __signArctan (x, y);
-
- DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
- t1 = w - hij[i][0].d;
- EADD (t1, ww, z, zz);
-
- s1 = hij[i][14].d + z * hij[i][15].d;
- s1 = hij[i][13].d + z * s1;
- s1 = hij[i][12].d + z * s1;
- s1 = hij[i][11].d + z * s1;
- s1 *= z;
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
- if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t1 + yy;
+ /* Max ULP is 0.503. */
+ return __signArctan (x, y);
}
else
{
@@ -249,34 +149,9 @@ __atan (double x)
ww = w * ((1 - t1) - t2);
ESUB (HPI, w, t3, cor);
yy = ((HPI1 + cor) - ww) - yy;
- if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
- return __signArctan (x, y);
-
- DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
- MUL2 (w, ww, w, ww, v, vv, t1, t2);
-
- s1 = f17.d + v * f19.d;
- s1 = f15.d + v * s1;
- s1 = f13.d + v * s1;
- s1 = f11.d + v * s1;
- s1 *= v;
-
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2);
- ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
- SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
-
- if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t3 + yy;
+ /* Max ULP is 0.5003. */
+ return __signArctan (x, y);
}
else
{
@@ -290,35 +165,6 @@ __atan (double x)
}
}
- /* Final stages. Compute atan(x) by multiple precision arithmetic */
-static double
-atanMp (double x, const int pr[])
-{
- mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
- double y1, y2;
- int i, p;
-
- for (i = 0; i < M; i++)
- {
- p = pr[i];
- __dbl_mp (x, &mpx, p);
- __mpatan (&mpx, &mpy, p);
- __dbl_mp (u9[i].d, &mpt1, p);
- __mul (&mpy, &mpt1, &mperr, p);
- __add (&mpy, &mperr, &mpy1, p);
- __sub (&mpy, &mperr, &mpy2, p);
- __mp_dbl (&mpy1, &y1, p);
- __mp_dbl (&mpy2, &y2, p);
- if (y1 == y2)
- {
- LIBC_PROBE (slowatan, 3, &p, &x, &y1);
- return y1;
- }
- }
- LIBC_PROBE (slowatan_inexact, 3, &p, &x, &y1);
- return y1; /*if impossible to do exact computing */
-}
-
#ifndef __atan
libm_alias_double (__atan, atan)
#endif
diff --git a/sysdeps/ieee754/dbl-64/uatan.tbl b/sysdeps/ieee754/dbl-64/uatan.tbl
index 9730d576d6562e7a56d6f6f59e9943ec9e8a5343..3db2804cc2833ba94ebf428ea881c4ab5e833e4c 100644
--- a/sysdeps/ieee754/dbl-64/uatan.tbl
+++ b/sysdeps/ieee754/dbl-64/uatan.tbl
@@ -25,7 +25,7 @@
#ifdef BIG_ENDI
- static const number
+ static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X3FB04006, 0X65E0244E} },
/**/ {{0X3FB03A73, 0X7B53DD20} },
@@ -1716,7 +1716,7 @@
/**/ {{0XBF99B9A7, 0X18A3BA58} } },
};
- static const number
+ static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x3fb04000, 0x00000000} },
/**/ {{0x3fb03a6d, 0x1c06693d} },
@@ -5579,7 +5579,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X65E0244E, 0X3FB04006} },
/**/ {{0X7B53DD20, 0X3FB03A73} },
@@ -7270,7 +7270,7 @@
/**/ {{0X18A3BA58, 0XBF99B9A7} } },
};
- static const number
+ static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x00000000, 0x3fb04000} },
/**/ {{0x1c06693d, 0x3fb03a6d} },
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH 4/5] Remove slow paths from atan2
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
` (2 preceding siblings ...)
2021-01-04 12:19 ` [PATCH 3/5] Remove slow paths from atan Wilco Dijkstra via Libc-alpha
@ 2021-01-04 12:21 ` Wilco Dijkstra via Libc-alpha
2021-01-04 16:49 ` Paul Zimmermann
2021-01-07 19:28 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:25 ` [PATCH 5/5] Remove mpa files Wilco Dijkstra via Libc-alpha
4 siblings, 2 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:21 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths from atan2. Add ULP annotations.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/atnat2.h b/sysdeps/ieee754/dbl-64/atnat2.h
index de4300c5fc559f391e0b20a818452640dabb0e41..9cb8a1fbeae6ff2845151f23d49797c1218c0062 100644
--- a/sysdeps/ieee754/dbl-64/atnat2.h
+++ b/sysdeps/ieee754/dbl-64/atnat2.h
@@ -34,7 +34,7 @@
#define MM 5
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@@ -96,7 +96,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */
diff --git a/sysdeps/ieee754/dbl-64/e_atan2.c b/sysdeps/ieee754/dbl-64/e_atan2.c
index 7af2a4f495d39f6450699de1d0d9d2f7a33fdcf7..5af12d37c3c89cac951d293339e811fd96d0daf3 100644
--- a/sysdeps/ieee754/dbl-64/e_atan2.c
+++ b/sysdeps/ieee754/dbl-64/e_atan2.c
@@ -20,25 +20,14 @@
/* MODULE_NAME: atnat2.c */
/* */
/* FUNCTIONS: uatan2 */
-/* atan2Mp */
/* signArctan2 */
-/* normalized */
/* */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h */
-/* mpatan.c mpatan2.c mpsqrt.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h atnat2.h */
/* uatan.tbl */
/* */
-/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
-/* x it computes the correctly rounded (to nearest) value of atan2(y,x).*/
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/************************************************************************/
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
#include "mydefs.h"
#include "uatan.tbl"
#include "atnat2.h"
@@ -48,20 +37,21 @@
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
-#include <stap-probe.h>
#include <libm-alias-finite.h>
#ifndef SECTION
# define SECTION
#endif
+#define TWO52 0x1.0p52
+#define TWOM1022 0x1.0p-1022
+
/************************************************************************/
/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
/* it computes the correctly rounded (to nearest) value of atan2(y,x). */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/************************************************************************/
-static double atan2Mp (double, double, const int[]);
/* Fix the sign and return after stage 1 or stage 2 */
static double
signArctan2 (double y, double z)
@@ -69,18 +59,14 @@ signArctan2 (double y, double z)
return copysign (z, y);
}
-static double normalized (double, double, double, double);
-void __mpatan2 (mp_no *, mp_no *, mp_no *, int);
-
double
SECTION
__ieee754_atan2 (double y, double x)
{
int i, de, ux, dx, uy, dy;
- static const int pr[MM] = { 6, 8, 10, 20, 32 };
- double ax, ay, u, du, u9, ua, v, vv, dv, t1, t2, t3,
- z, zz, cor, s1, ss1, s2, ss2;
- number num;
+ double ax, ay, u, du, v, vv, dv, t1, t2, t3,
+ z, zz, cor;
+ mynumber num;
static const int ep = 59768832, /* 57*16**5 */
em = -59768832; /* -57*16**5 */
@@ -208,10 +194,8 @@ __ieee754_atan2 (double y, double x)
if (x > 0)
{
double ret;
- if ((z = ay / ax) < TWOM1022)
- ret = normalized (ax, ay, y, z);
- else
- ret = signArctan2 (y, z);
+ z = ay / ax;
+ ret = signArctan2 (y, z);
if (fabs (ret) < DBL_MIN)
{
double vret = ret ? ret : DBL_MIN;
@@ -270,30 +254,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d
+ v * d13.d)))));
- if ((z = u + (zz - u1.d * u)) == u + (zz + u1.d * u))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d + v * (f13.d
- + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - u5.d * s1)) == s1 + (ss1 + u5.d * s1))
- return signArctan2 (y, z);
-
- return atan2Mp (x, y, pr);
+ z = u + zz;
+ /* Max ULP is 0.504. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
t3 = u - cij[i][0].d;
EADD (t3, du, v, dv);
@@ -304,43 +270,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
- if (i < 112)
- {
- if (i < 48)
- u9 = u91.d; /* u < 1/4 */
- else
- u9 = u92.d;
- } /* 1/4 <= u < 1/2 */
- else
- {
- if (i < 176)
- u9 = u93.d; /* 1/2 <= u < 3/4 */
- else
- u9 = u94.d;
- } /* 3/4 <= u <= 1 */
- if ((z = t1 + (zz - u9 * t1)) == t1 + (zz + u9 * t1))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - ub.d * s2)) == s2 + (ss2 + ub.d * s2))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.56. */
+ return signArctan2 (y, z);
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
@@ -355,31 +287,12 @@ __ieee754_atan2 (double y, double x)
+ v * d13.d)))));
ESUB (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) - du) - zz;
- if ((z = t2 + (t3 - u2.d)) == t2 + (t3 + u2.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d
- + v * (f13.d
- + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- SUB2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u6.d)) == s2 + (ss2 + u6.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
@@ -389,36 +302,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d - cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
-
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.503. */
+ return signArctan2 (y, z);
}
/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
@@ -434,30 +320,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d + v * d13.d)))));
EADD (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) + du) + zz;
- if ((z = t2 + (t3 - u3.d)) == t2 + (t3 + u3.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d
- + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u7.d)) == s2 + (ss2 + u7.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d + v * (cij[i][2].d
@@ -466,34 +334,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d + cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- ADD2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.503. */
+ return signArctan2 (y, z);
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
@@ -506,29 +349,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d9.d + v * (d11.d + v * d13.d)))));
ESUB (opi.d, u, t2, cor);
t3 = ((opi1.d + cor) - du) - zz;
- if ((z = t2 + (t3 - u4.d)) == t2 + (t3 + u4.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- SUB2 (opi.d, opi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u8.d)) == s2 + (ss2 + u8.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = opi1.d - v * (cij[i][2].d
@@ -536,86 +362,11 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d + v * cij[i][6].d))));
t1 = opi.d - cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
-
- EADD (t1, du, v, vv);
-
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d + v * hij[i][15].d))));
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (opi.d, opi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.502. */
+ return signArctan2 (y, z);
}
#ifndef __ieee754_atan2
libm_alias_finite (__ieee754_atan2, __atan2)
#endif
-
-/* Treat the Denormalized case */
-static double
-SECTION
-normalized (double ax, double ay, double y, double z)
-{
- int p;
- mp_no mpx, mpy, mpz, mperr, mpz2, mpt1;
- p = 6;
- __dbl_mp (ax, &mpx, p);
- __dbl_mp (ay, &mpy, p);
- __dvd (&mpy, &mpx, &mpz, p);
- __dbl_mp (ue.d, &mpt1, p);
- __mul (&mpz, &mpt1, &mperr, p);
- __sub (&mpz, &mperr, &mpz2, p);
- __mp_dbl (&mpz2, &z, p);
- return signArctan2 (y, z);
-}
-
-/* Stage 3: Perform a multi-Precision computation */
-static double
-SECTION
-atan2Mp (double x, double y, const int pr[])
-{
- double z1, z2;
- int i, p;
- mp_no mpx, mpy, mpz, mpz1, mpz2, mperr, mpt1;
- for (i = 0; i < MM; i++)
- {
- p = pr[i];
- __dbl_mp (x, &mpx, p);
- __dbl_mp (y, &mpy, p);
- __mpatan2 (&mpy, &mpx, &mpz, p);
- __dbl_mp (ud[i].d, &mpt1, p);
- __mul (&mpz, &mpt1, &mperr, p);
- __add (&mpz, &mperr, &mpz1, p);
- __sub (&mpz, &mperr, &mpz2, p);
- __mp_dbl (&mpz1, &z1, p);
- __mp_dbl (&mpz2, &z2, p);
- if (z1 == z2)
- {
- LIBC_PROBE (slowatan2, 4, &p, &x, &y, &z1);
- return z1;
- }
- }
- LIBC_PROBE (slowatan2_inexact, 4, &p, &x, &y, &z1);
- return z1; /*if impossible to do exact computing */
-}
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH 5/5] Remove mpa files
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
` (3 preceding siblings ...)
2021-01-04 12:21 ` [PATCH 4/5] Remove slow paths from atan2 Wilco Dijkstra via Libc-alpha
@ 2021-01-04 12:25 ` Wilco Dijkstra via Libc-alpha
2021-01-07 20:12 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
4 siblings, 1 reply; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-04 12:25 UTC (permalink / raw
To: 'GNU C Library'
Finally remove all mpa related files, headers, declarations, probes and update
makefiles.
Passes GLIBC testsuite and buildmanyglibc.
---
diff --git a/manual/probes.texi b/manual/probes.texi
index 61254bc43018d0618c17776603c95c2f8d489678..4aae76b81921f3474e381b2d0c982e9bec1c6380 100644
--- a/manual/probes.texi
+++ b/manual/probes.texi
@@ -16,7 +16,6 @@ arguments.
@menu
* Memory Allocation Probes:: Probes in the memory allocation subsystem
-* Mathematical Function Probes:: Probes in mathematical functions
* Non-local Goto Probes:: Probes in setjmp and longjmp
@end menu
@@ -255,90 +254,6 @@ is the memory location as passed to @code{free}, Argument @var{$arg2}
is the tcache bin it resides in.
@end deftp
-@node Mathematical Function Probes
-@section Mathematical Function Probes
-
-Some mathematical functions fall back to multiple precision arithmetic for
-some inputs to get last bit precision for their return values. This multiple
-precision fallback is much slower than the default algorithms and may have a
-significant impact on application performance. The systemtap probe markers
-described in this section may help you determine if your application calls
-mathematical functions with inputs that may result in multiple-precision
-arithmetic.
-
-Unless explicitly mentioned otherwise, a precision of 1 implies 24 bits of
-precision in the mantissa of the multiple precision number. Hence, a precision
-level of 32 implies 768 bits of precision in the mantissa.
-
-@deftp Probe slowatan2 (int @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
-This probe is triggered when the @code{atan2} function is called with
-an input that results in multiple precision computation. Argument
-@var{$arg1} is the precision with which computation succeeded.
-Arguments @var{$arg2} and @var{$arg3} are inputs to the @code{atan2}
-function and @var{$arg4} is the computed result.
-@end deftp
-
-@deftp Probe slowatan2_inexact (int @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation and none of
-the multiple precision computations result in an accurate result.
-Argument @var{$arg1} is the maximum precision with which computations
-were performed. Arguments @var{$arg2} and @var{$arg3} are inputs to
-the @code{atan2} function and @var{$arg4} is the computed result.
-@end deftp
-
-@deftp Probe slowatan (int @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation. Argument
-@var{$arg1} is the precision with which computation succeeded.
-Argument @var{$arg2} is the input to the @code{atan} function and
-@var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowatan_inexact (int @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation and none of
-the multiple precision computations result in an accurate result.
-Argument @var{$arg1} is the maximum precision with which computations
-were performed. Argument @var{$arg2} is the input to the @code{atan}
-function and @var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowtan (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{tan} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowsin (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{sin} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowcos (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{cos} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowsin_dx (double @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{sin} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function, @var{$arg2} is
-the error bound of @var{$arg1} and @var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowcos_dx (double @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{cos} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function, @var{$arg2} is
-the error bound of @var{$arg1} and @var{$arg3} is the computed result.
-@end deftp
-
@node Non-local Goto Probes
@section Non-local Goto Probes
diff --git a/math/Makefile b/math/Makefile
index 62667d9186c601d88e87cdbd2e98d6231c96e7fb..80855cc356923170c99fe38128625e338aedb247 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -126,8 +126,7 @@ type-ldouble-yes := ldouble
# double support
type-double-suffix :=
-type-double-routines := branred doasin dosincos mpa mpatan2 \
- k_rem_pio2 mpatan mpsqrt mptan sincos32 \
+type-double-routines := branred k_rem_pio2 \
sincostab math_err e_exp_data e_log_data \
e_log2_data e_pow_log_data
diff --git a/sysdeps/generic/math_private.h b/sysdeps/generic/math_private.h
index 10a6fde28dbcbfe55148c744e13515b7c9d0707c..98b54875f51304f966db6d7b304b12073f9b2b2f 100644
--- a/sysdeps/generic/math_private.h
+++ b/sysdeps/generic/math_private.h
@@ -199,11 +199,5 @@ do { \
extern double __sin (double __x);
extern double __cos (double __x);
extern int __branred (double __x, double *__a, double *__aa);
-extern void __doasin (double __x, double __dx, double __v[]);
-extern void __dubsin (double __x, double __dx, double __v[]);
-extern void __dubcos (double __x, double __dx, double __v[]);
-extern double __mpsin (double __x, double __dx, bool __range_reduce);
-extern double __mpcos (double __x, double __dx, bool __range_reduce);
-extern void __docos (double __x, double __dx, double __v[]);
#endif /* _MATH_PRIVATE_H_ */
diff --git a/sysdeps/ieee754/dbl-64/MathLib.h b/sysdeps/ieee754/dbl-64/MathLib.h
deleted file mode 100644
index a273082f93e60fa826c4105d9457e8b4185b2357..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/MathLib.h
+++ /dev/null
@@ -1,100 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/********************************************************************/
-/* Ultimate math functions. Each function computes the exact */
-/* theoretical value of its argument rounded to nearest or even. */
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round nearest mode of IEEE 754 standard. */
-/********************************************************************/
-
-#ifndef UMATH_LIB
-#define UMATH_LIB
-/********************************************************************/
-/* Function changes the precision mode to IEEE 754 double precision */
-/* and the rounding mode to nearest or even. */
-/* It returns the original status of these modes. */
-/* See further explanations of usage in DPChange.h */
-/********************************************************************/
-unsigned short Init_Lib (void);
-
-/********************************************************************/
-/* Function that changes the precision and rounding modes to the */
-/* specified by the argument received. See further explanations in */
-/* DPChange.h */
-/********************************************************************/
-void Exit_Lib (unsigned short);
-
-
-/* The asin() function calculates the arc sine of its argument. */
-/* The function returns the arc sine in radians */
-/* (between -PI/2 and PI/2). */
-/* If the argument is greater than 1 or less than -1 it returns */
-/* a NaN. */
-double uasin (double);
-
-
-/* The acos() function calculates the arc cosine of its argument. */
-/* The function returns the arc cosine in radians */
-/* (between -PI/2 and PI/2). */
-/* If the argument is greater than 1 or less than -1 it returns */
-/* a NaN. */
-double uacos (double);
-
-/* The atan() function calculates the arctanget of its argument. */
-/* The function returns the arc tangent in radians */
-/* (between -PI/2 and PI/2). */
-double uatan (double);
-
-
-/* The uatan2() function calculates the arc tangent of the two arguments x */
-/* and y (x is the right argument and y is the left one).The signs of both */
-/* arguments are used to determine the quadrant of the result. */
-/* The function returns the result in radians, which is between -PI and PI */
-double uatan2 (double, double);
-
-/* Compute log(x). The base of log is e (natural logarithm) */
-double ulog (double);
-
-/* Compute e raised to the power of argument x. */
-double uexp (double);
-
-/* Compute sin(x). The argument x is assumed to be given in radians.*/
-double usin (double);
-
-/* Compute cos(x). The argument x is assumed to be given in radians.*/
-double ucos (double);
-
-/* Compute tan(x). The argument x is assumed to be given in radians.*/
-double utan (double);
-
-/* Compute the square root of non-negative argument x. */
-/* If x is negative the returned value is NaN. */
-double usqrt (double);
-
-/* Compute x raised to the power of y, where x is the left argument */
-/* and y is the right argument. The function returns a NaN if x<0. */
-/* If x equals zero it returns -inf */
-double upow (double, double);
-
-/* Computing x mod y, where x is the left argument and y is the */
-/* right one. */
-double uremainder (double, double);
-#endif
diff --git a/sysdeps/ieee754/dbl-64/doasin.h b/sysdeps/ieee754/dbl-64/doasin.h
deleted file mode 100644
index ecb15aaff00a79b0acc3f3d9ec1a083fc2f702d5..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/doasin.h
+++ /dev/null
@@ -1,63 +0,0 @@
-
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/************************************************************************/
-/* MODULE_NAME: doasin.h */
-/* */
-/* */
-/* common data and variables definition for BIG or LITTLE ENDIAN */
-/************************************************************************/
-
-
-
-#ifndef DOASIN_H
-#define DOASIN_H
-
-#ifdef BIG_ENDI
-
- static const mynumber
-/**/ c1 = {{0x3FC55555, 0x55555555}}, /* 0.16666666666666666 */
-/**/ cc1 = {{0x3C655555, 0x55775389}}, /* 9.2518585419753846e-18 */
-/**/ c2 = {{0x3FB33333, 0x33333333}}, /* 0.074999999999999997 */
-/**/ cc2 = {{0x3C499993, 0x63F1A115}}, /* 2.7755472886508899e-18 */
-/**/ c3 = {{0x3FA6DB6D, 0xB6DB6DB7}}, /* 0.044642857142857144 */
-/**/ cc3 = {{0xBC320FC0, 0x3D5CF0C5}}, /* -9.7911734574147224e-19 */
-/**/ c4 = {{0x3F9F1C71, 0xC71C71C5}}, /* 0.030381944444444437 */
-/**/ cc4 = {{0xBC02B240, 0xFF23ED1E}}; /* -1.2669108566898312e-19 */
-
-#else
-#ifdef LITTLE_ENDI
-
- static const mynumber
-/**/ c1 = {{0x55555555, 0x3FC55555}}, /* 0.16666666666666666 */
-/**/ cc1 = {{0x55775389, 0x3C655555}}, /* 9.2518585419753846e-18 */
-/**/ c2 = {{0x33333333, 0x3FB33333}}, /* 0.074999999999999997 */
-/**/ cc2 = {{0x63F1A115, 0x3C499993}}, /* 2.7755472886508899e-18 */
-/**/ c3 = {{0xB6DB6DB7, 0x3FA6DB6D}}, /* 0.044642857142857144 */
-/**/ cc3 = {{0x3D5CF0C5, 0xBC320FC0}}, /* -9.7911734574147224e-19 */
-/**/ c4 = {{0xC71C71C5, 0x3F9F1C71}}, /* 0.030381944444444437 */
-/**/ cc4 = {{0xFF23ED1E, 0xBC02B240}}; /* -1.2669108566898312e-19 */
-
-
-#endif
-#endif
-
-
-#endif
diff --git a/sysdeps/ieee754/dbl-64/doasin.c b/sysdeps/ieee754/dbl-64/doasin.c
deleted file mode 100644
index f7e2f455664421dfd3a22bdf46aa598c14cb6093..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/doasin.c
+++ /dev/null
@@ -1,81 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/**********************************************************************/
-/* MODULE_NAME: doasin.c */
-/* */
-/* FUNCTION: doasin */
-/* */
-/* FILES NEEDED:endian.h mydefs.h dla.h doasin.h */
-/* mpa.c */
-/* */
-/* Compute arcsin(x,dx,v) of double-length number (x+dx) the result */
-/* stored in v where v= v[0]+v[1] =arcsin(x+dx) */
-/**********************************************************************/
-
-#include "endian.h"
-#include "mydefs.h"
-#include <dla.h>
-#include <math_private.h>
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-/********************************************************************/
-/* Compute arcsin(x,dx,v) of double-length number (x+dx) the result */
-/* stored in v where v= v[0]+v[1] =arcsin(x+dx) */
-/********************************************************************/
-void
-SECTION
-__doasin(double x, double dx, double v[]) {
-
-#include "doasin.h"
-
- static const double
- d5 = 0.22372159090911789889975459505194491E-01,
- d6 = 0.17352764422456822913014975683014622E-01,
- d7 = 0.13964843843786693521653681033981614E-01,
- d8 = 0.11551791438485242609036067259086589E-01,
- d9 = 0.97622386568166960207425666787248914E-02,
- d10 = 0.83638737193775788576092749009744976E-02,
- d11 = 0.79470250400727425881446981833568758E-02;
-
- double xx,p,pp,u,uu,r,s;
- double tc,tcc;
-
-
-/* Taylor series for arcsin for Double-Length numbers */
- xx = x*x+2.0*x*dx;
- p = ((((((d11*xx+d10)*xx+d9)*xx+d8)*xx+d7)*xx+d6)*xx+d5)*xx;
- pp = 0;
-
- MUL2(x,dx,x,dx,u,uu,tc,tcc);
- ADD2(p,pp,c4.x,cc4.x,p,pp,r,s);
- MUL2(p,pp,u,uu,p,pp,tc,tcc);
- ADD2(p,pp,c3.x,cc3.x,p,pp,r,s);
- MUL2(p,pp,u,uu,p,pp,tc,tcc);
- ADD2(p,pp,c2.x,cc2.x,p,pp,r,s);
- MUL2(p,pp,u,uu,p,pp,tc,tcc);
- ADD2(p,pp,c1.x,cc1.x,p,pp,r,s);
- MUL2(p,pp,u,uu,p,pp,tc,tcc);
- MUL2(p,pp,x,dx,p,pp,tc,tcc);
- ADD2(p,pp,x,dx,p,pp,r,s);
- v[0]=p;
- v[1]=pp; /* arcsin(x+dx)=v[0]+v[1] */
-}
diff --git a/sysdeps/ieee754/dbl-64/dosincos.h b/sysdeps/ieee754/dbl-64/dosincos.h
deleted file mode 100644
index a90667ebc0bdd3045626aade77374689a0a07732..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/dosincos.h
+++ /dev/null
@@ -1,80 +0,0 @@
-
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/************************************************************************/
-/* MODULE_NAME: dosincos.h */
-/* */
-/* */
-/* common data and variables definition for BIG or LITTLE ENDIAN */
-/************************************************************************/
-
-
-
-#ifndef DOSINCOS_H
-#define DOSINCOS_H
-
-
-#ifdef BIG_ENDI
-static const mynumber
-/**/ s3 = {{0xBFC55555, 0x55555555}},/* -0.16666666666666666 */
-/**/ ss3 = {{0xBC6553AA, 0xE77EE482}},/* -9.2490366677784492e-18 */
-/**/ s5 = {{0x3F811111, 0x11110F15}},/* 0.008333333333332452 */
-/**/ ss5 = {{0xBC21AC06, 0xDA488820}},/* -4.7899996586987931e-19 */
-/**/ s7 = {{0xBF2A019F, 0x5816C78D}},/* -0.00019841261022928957 */
-/**/ ss7 = {{0x3BCDCEC9, 0x6A18BF2A}},/* 1.2624077757871259e-20 */
-/**/ c2 = {{0x3FE00000, 0x00000000}},/* 0.5 */
-/**/ cc2 = {{0xBA282FD8, 0x00000000}},/* -1.5264073330037701e-28 */
-/**/ c4 = {{0xBFA55555, 0x55555555}},/* -0.041666666666666664 */
-/**/ cc4 = {{0xBC4554BC, 0x2FFF257E}},/* -2.312711276085743e-18 */
-/**/ c6 = {{0x3F56C16C, 0x16C16A96}},/* 0.0013888888888888055 */
-/**/ cc6 = {{0xBBD2E846, 0xE6346F14}},/* -1.6015133010194884e-20 */
-/**/ c8 = {{0xBEFA019F, 0x821D5987}},/* -2.480157866754367e-05 */
-/**/ cc8 = {{0x3B7AB71E, 0x72FFE5CC}},/* 3.5357416224857556e-22 */
-
-/**/ big = {{0x42c80000, 0x00000000}}, /* 52776558133248 */
-
-/**/ hp0 = {{0x3FF921FB, 0x54442D18}}, /* PI / 2 */
-/**/ hp1 = {{0x3C91A626, 0x33145C07}}; /* 6.123233995736766e-17 */
-#else
-#ifdef LITTLE_ENDI
-static const mynumber
-/**/ s3 = {{0x55555555, 0xBFC55555}},/* -0.16666666666666666 */
-/**/ ss3 = {{0xE77EE482, 0xBC6553AA}},/* -9.2490366677784492e-18 */
-/**/ s5 = {{0x11110F15, 0x3F811111}},/* 0.008333333333332452 */
-/**/ ss5 = {{0xDA488820, 0xBC21AC06}},/* -4.7899996586987931e-19 */
-/**/ s7 = {{0x5816C78D, 0xBF2A019F}},/* -0.00019841261022928957 */
-/**/ ss7 = {{0x6A18BF2A, 0x3BCDCEC9}},/* 1.2624077757871259e-20 */
-/**/ c2 = {{0x00000000, 0x3FE00000}},/* 0.5 */
-/**/ cc2 = {{0x00000000, 0xBA282FD8}},/* -1.5264073330037701e-28 */
-/**/ c4 = {{0x55555555, 0xBFA55555}},/* -0.041666666666666664 */
-/**/ cc4 = {{0x2FFF257E, 0xBC4554BC}},/* -2.312711276085743e-18 */
-/**/ c6 = {{0x16C16A96, 0x3F56C16C}},/* 0.0013888888888888055 */
-/**/ cc6 = {{0xE6346F14, 0xBBD2E846}},/* -1.6015133010194884e-20 */
-/**/ c8 = {{0x821D5987, 0xBEFA019F}},/* -2.480157866754367e-05 */
-/**/ cc8 = {{0x72FFE5CC, 0x3B7AB71E}},/* 3.5357416224857556e-22 */
-
-/**/ big = {{0x00000000, 0x42c80000}}, /* 52776558133248 */
-
-/**/ hp0 = {{0x54442D18, 0x3FF921FB}}, /* PI / 2 */
-/**/ hp1 = {{0x33145C07, 0x3C91A626}}; /* 6.123233995736766e-17 */
-#endif
-#endif
-
-#endif
diff --git a/sysdeps/ieee754/dbl-64/dosincos.c b/sysdeps/ieee754/dbl-64/dosincos.c
deleted file mode 100644
index 51b2938650140057f49b372a2695d56b02639f58..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/dosincos.c
+++ /dev/null
@@ -1,217 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/********************************************************************/
-/* */
-/* MODULE_NAME: dosincos.c */
-/* */
-/* */
-/* FUNCTIONS: dubsin */
-/* dubcos */
-/* docos */
-/* FILES NEEDED: endian.h mydefs.h dla.h dosincos.h */
-/* sincos.tbl */
-/* */
-/* Routines compute sin() and cos() as Double-Length numbers */
-/********************************************************************/
-
-
-
-#include "endian.h"
-#include "mydefs.h"
-#include <dla.h>
-#include "dosincos.h"
-#include <math_private.h>
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-extern const union
-{
- int4 i[880];
- double x[440];
-} __sincostab attribute_hidden;
-
-/***********************************************************************/
-/* Routine receive Double-Length number (x+dx) and computing sin(x+dx) */
-/* as Double-Length number and store it at array v .It computes it by */
-/* arithmetic action on Double-Length numbers */
-/*(x+dx) between 0 and PI/4 */
-/***********************************************************************/
-
-void
-SECTION
-__dubsin (double x, double dx, double v[])
-{
- double r, s, c, cc, d, dd, d2, dd2, e, ee,
- sn, ssn, cs, ccs, ds, dss, dc, dcc;
- mynumber u;
- int4 k;
-
- u.x = x + big.x;
- k = u.i[LOW_HALF] << 2;
- x = x - (u.x - big.x);
- d = x + dx;
- dd = (x - d) + dx;
- /* sin(x+dx)=sin(Xi+t)=sin(Xi)*cos(t) + cos(Xi)sin(t) where t ->0 */
- MUL2 (d, dd, d, dd, d2, dd2, c, cc);
- sn = __sincostab.x[k]; /* */
- ssn = __sincostab.x[k + 1]; /* sin(Xi) and cos(Xi) */
- cs = __sincostab.x[k + 2]; /* */
- ccs = __sincostab.x[k + 3]; /* */
- /* Taylor series for sin ds=sin(t) */
- MUL2 (d2, dd2, s7.x, ss7.x, ds, dss, c, cc);
- ADD2 (ds, dss, s5.x, ss5.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, s3.x, ss3.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- MUL2 (d, dd, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, d, dd, ds, dss, r, s);
-
- /* Taylor series for cos dc=cos(t) */
- MUL2 (d2, dd2, c8.x, cc8.x, dc, dcc, c, cc);
- ADD2 (dc, dcc, c6.x, cc6.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c4.x, cc4.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c2.x, cc2.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
-
- MUL2 (cs, ccs, ds, dss, e, ee, c, cc);
- MUL2 (dc, dcc, sn, ssn, dc, dcc, c, cc);
- SUB2 (e, ee, dc, dcc, e, ee, r, s);
- ADD2 (e, ee, sn, ssn, e, ee, r, s); /* e+ee=sin(x+dx) */
-
- v[0] = e;
- v[1] = ee;
-}
-/**********************************************************************/
-/* Routine receive Double-Length number (x+dx) and computes cos(x+dx) */
-/* as Double-Length number and store it in array v .It computes it by */
-/* arithmetic action on Double-Length numbers */
-/*(x+dx) between 0 and PI/4 */
-/**********************************************************************/
-
-void
-SECTION
-__dubcos (double x, double dx, double v[])
-{
- double r, s, c, cc, d, dd, d2, dd2, e, ee,
- sn, ssn, cs, ccs, ds, dss, dc, dcc;
- mynumber u;
- int4 k;
- u.x = x + big.x;
- k = u.i[LOW_HALF] << 2;
- x = x - (u.x - big.x);
- d = x + dx;
- dd = (x - d) + dx; /* cos(x+dx)=cos(Xi+t)=cos(Xi)cos(t) - sin(Xi)sin(t) */
- MUL2 (d, dd, d, dd, d2, dd2, c, cc);
- sn = __sincostab.x[k]; /* */
- ssn = __sincostab.x[k + 1]; /* sin(Xi) and cos(Xi) */
- cs = __sincostab.x[k + 2]; /* */
- ccs = __sincostab.x[k + 3]; /* */
- MUL2 (d2, dd2, s7.x, ss7.x, ds, dss, c, cc);
- ADD2 (ds, dss, s5.x, ss5.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, s3.x, ss3.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- MUL2 (d, dd, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, d, dd, ds, dss, r, s);
-
- MUL2 (d2, dd2, c8.x, cc8.x, dc, dcc, c, cc);
- ADD2 (dc, dcc, c6.x, cc6.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c4.x, cc4.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c2.x, cc2.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
-
- MUL2 (cs, ccs, ds, dss, e, ee, c, cc);
- MUL2 (dc, dcc, sn, ssn, dc, dcc, c, cc);
-
- MUL2 (d2, dd2, s7.x, ss7.x, ds, dss, c, cc);
- ADD2 (ds, dss, s5.x, ss5.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, s3.x, ss3.x, ds, dss, r, s);
- MUL2 (d2, dd2, ds, dss, ds, dss, c, cc);
- MUL2 (d, dd, ds, dss, ds, dss, c, cc);
- ADD2 (ds, dss, d, dd, ds, dss, r, s);
- MUL2 (d2, dd2, c8.x, cc8.x, dc, dcc, c, cc);
- ADD2 (dc, dcc, c6.x, cc6.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c4.x, cc4.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- ADD2 (dc, dcc, c2.x, cc2.x, dc, dcc, r, s);
- MUL2 (d2, dd2, dc, dcc, dc, dcc, c, cc);
- MUL2 (sn, ssn, ds, dss, e, ee, c, cc);
- MUL2 (dc, dcc, cs, ccs, dc, dcc, c, cc);
- ADD2 (e, ee, dc, dcc, e, ee, r, s);
- SUB2 (cs, ccs, e, ee, e, ee, r, s);
-
- v[0] = e;
- v[1] = ee;
-}
-/**********************************************************************/
-/* Routine receive Double-Length number (x+dx) and computes cos(x+dx) */
-/* as Double-Length number and store it in array v */
-/**********************************************************************/
-void
-SECTION
-__docos (double x, double dx, double v[])
-{
- double y, yy, p, w[2];
- if (x > 0)
- {
- y = x; yy = dx;
- }
- else
- {
- y = -x; yy = -dx;
- }
- if (y < 0.5 * hp0.x) /* y< PI/4 */
- {
- __dubcos (y, yy, w); v[0] = w[0]; v[1] = w[1];
- }
- else if (y < 1.5 * hp0.x) /* y< 3/4 * PI */
- {
- p = hp0.x - y; /* p = PI/2 - y */
- yy = hp1.x - yy;
- y = p + yy;
- yy = (p - y) + yy;
- if (y > 0)
- {
- __dubsin (y, yy, w); v[0] = w[0]; v[1] = w[1];
- }
- /* cos(x) = sin ( 90 - x ) */
- else
- {
- __dubsin (-y, -yy, w); v[0] = -w[0]; v[1] = -w[1];
- }
- }
- else /* y>= 3/4 * PI */
- {
- p = 2.0 * hp0.x - y; /* p = PI- y */
- yy = 2.0 * hp1.x - yy;
- y = p + yy;
- yy = (p - y) + yy;
- __dubcos (y, yy, w);
- v[0] = -w[0];
- v[1] = -w[1];
- }
-}
diff --git a/sysdeps/ieee754/dbl-64/e_remainder.c b/sysdeps/ieee754/dbl-64/e_remainder.c
index bee5e5f1477bcf5e1411b1424bfe3083fec670ed..895345a2f021aa8b9d0b206227088a1e96e2f54b 100644
--- a/sysdeps/ieee754/dbl-64/e_remainder.c
+++ b/sysdeps/ieee754/dbl-64/e_remainder.c
@@ -32,7 +32,6 @@
#include "endian.h"
#include "mydefs.h"
#include "urem.h"
-#include "MathLib.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
diff --git a/sysdeps/ieee754/dbl-64/e_sqrt.c b/sysdeps/ieee754/dbl-64/e_sqrt.c
index c183d39938fe24262ffbb7fc49f4b6afb3cacc11..c4156cfd4578019770306c9de36c03db331352eb 100644
--- a/sysdeps/ieee754/dbl-64/e_sqrt.c
+++ b/sysdeps/ieee754/dbl-64/e_sqrt.c
@@ -35,7 +35,6 @@
#include "endian.h"
#include "mydefs.h"
#include <dla.h>
-#include "MathLib.h"
#include "root.tbl"
#include <math-barriers.h>
#include <math_private.h>
diff --git a/sysdeps/ieee754/dbl-64/mpa-arch.h b/sysdeps/ieee754/dbl-64/mpa-arch.h
deleted file mode 100644
index 0b33cb89d06e72d1cf370f5616cc624cd745e306..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpa-arch.h
+++ /dev/null
@@ -1,47 +0,0 @@
-/* Overridable constants and operations.
- Copyright (C) 2013-2020 Free Software Foundation, Inc.
-
- This program is free software; you can redistribute it and/or modify
- it under the terms of the GNU Lesser General Public License as published by
- the Free Software Foundation; either version 2.1 of the License, or
- (at your option) any later version.
-
- This program is distributed in the hope that it will be useful,
- but WITHOUT ANY WARRANTY; without even the implied warranty of
- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- GNU Lesser General Public License for more details.
-
- You should have received a copy of the GNU Lesser General Public License
- along with this program; if not, see <https://www.gnu.org/licenses/>. */
-
-#include <stdint.h>
-
-typedef long mantissa_t;
-typedef int64_t mantissa_store_t;
-
-#define TWOPOW(i) (1L << i)
-
-#define RADIX_EXP 24
-#define RADIX TWOPOW (RADIX_EXP) /* 2^24 */
-
-/* Divide D by RADIX and put the remainder in R. D must be a non-negative
- integral value. */
-#define DIV_RADIX(d, r) \
- ({ \
- r = d & (RADIX - 1); \
- d >>= RADIX_EXP; \
- })
-
-/* Put the integer component of a double X in R and retain the fraction in
- X. This is used in extracting mantissa digits for MP_NO by using the
- integer portion of the current value of the number as the current mantissa
- digit and then scaling by RADIX to get the next mantissa digit in the same
- manner. */
-#define INTEGER_OF(x, i) \
- ({ \
- i = (mantissa_t) x; \
- x -= i; \
- })
-
-/* Align IN down to F. The code assumes that F is a power of two. */
-#define ALIGN_DOWN_TO(in, f) ((in) & - (f))
diff --git a/sysdeps/ieee754/dbl-64/mpa.h b/sysdeps/ieee754/dbl-64/mpa.h
deleted file mode 100644
index 76656610b946d695619878b776fb83d2a6b5b284..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpa.h
+++ /dev/null
@@ -1,123 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/************************************************************************/
-/* MODULE_NAME: mpa.h */
-/* */
-/* FUNCTIONS: */
-/* mcr */
-/* acr */
-/* cpy */
-/* mp_dbl */
-/* dbl_mp */
-/* add */
-/* sub */
-/* mul */
-/* dvd */
-/* */
-/* Arithmetic functions for multiple precision numbers. */
-/* Common types and definition */
-/************************************************************************/
-
-#include <mpa-arch.h>
-
-/* The mp_no structure holds the details of a multi-precision floating point
- number.
-
- - The radix of the number (R) is 2 ^ 24.
-
- - E: The exponent of the number.
-
- - D[0]: The sign (-1, 1) or 0 if the value is 0. In the latter case, the
- values of the remaining members of the structure are ignored.
-
- - D[1] - D[p]: The mantissa of the number where:
-
- 0 <= D[i] < R and
- P is the precision of the number and 1 <= p <= 32
-
- D[p+1] ... D[39] have no significance.
-
- - The value of the number is:
-
- D[1] * R ^ (E - 1) + D[2] * R ^ (E - 2) ... D[p] * R ^ (E - p)
-
- */
-typedef struct
-{
- int e;
- mantissa_t d[40];
-} mp_no;
-
-typedef union
-{
- int i[2];
- double d;
-} number;
-
-extern const mp_no __mpone;
-extern const mp_no __mptwo;
-
-#define X x->d
-#define Y y->d
-#define Z z->d
-#define EX x->e
-#define EY y->e
-#define EZ z->e
-
-#ifndef RADIXI
-# define RADIXI 0x1.0p-24 /* 2^-24 */
-#endif
-
-#ifndef TWO52
-# define TWO52 0x1.0p52 /* 2^52 */
-#endif
-
-#define TWO5 TWOPOW (5) /* 2^5 */
-#define TWO8 TWOPOW (8) /* 2^52 */
-#define TWO10 TWOPOW (10) /* 2^10 */
-#define TWO18 TWOPOW (18) /* 2^18 */
-#define TWO19 TWOPOW (19) /* 2^19 */
-#define TWO23 TWOPOW (23) /* 2^23 */
-
-#define HALFRAD TWO23
-
-#define TWO57 0x1.0p57 /* 2^57 */
-#define TWO71 0x1.0p71 /* 2^71 */
-#define TWOM1032 0x1.0p-1032 /* 2^-1032 */
-#define TWOM1022 0x1.0p-1022 /* 2^-1022 */
-
-#define HALF 0x1.0p-1 /* 1/2 */
-#define MHALF -0x1.0p-1 /* -1/2 */
-
-int __acr (const mp_no *, const mp_no *, int);
-void __cpy (const mp_no *, mp_no *, int);
-void __mp_dbl (const mp_no *, double *, int);
-void __dbl_mp (double, mp_no *, int);
-void __add (const mp_no *, const mp_no *, mp_no *, int);
-void __sub (const mp_no *, const mp_no *, mp_no *, int);
-void __mul (const mp_no *, const mp_no *, mp_no *, int);
-void __sqr (const mp_no *, mp_no *, int);
-void __dvd (const mp_no *, const mp_no *, mp_no *, int);
-
-extern void __mpatan (mp_no *, mp_no *, int);
-extern void __mpatan2 (mp_no *, mp_no *, mp_no *, int);
-extern void __mpsqrt (mp_no *, mp_no *, int);
-extern void __c32 (mp_no *, mp_no *, mp_no *, int);
-extern int __mpranred (double, mp_no *, int);
diff --git a/sysdeps/ieee754/dbl-64/mpa.c b/sysdeps/ieee754/dbl-64/mpa.c
deleted file mode 100644
index aed4354bacd8ca569a4e1eab6fc9330cb75f74db..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpa.c
+++ /dev/null
@@ -1,913 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/************************************************************************/
-/* MODULE_NAME: mpa.c */
-/* */
-/* FUNCTIONS: */
-/* mcr */
-/* acr */
-/* cpy */
-/* norm */
-/* denorm */
-/* mp_dbl */
-/* dbl_mp */
-/* add_magnitudes */
-/* sub_magnitudes */
-/* add */
-/* sub */
-/* mul */
-/* inv */
-/* dvd */
-/* */
-/* Arithmetic functions for multiple precision numbers. */
-/* Relative errors are bounded */
-/************************************************************************/
-
-
-#include "endian.h"
-#include "mpa.h"
-#include <sys/param.h>
-#include <alloca.h>
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-#ifndef NO__CONST
-const mp_no __mpone = { 1, { 1.0, 1.0 } };
-const mp_no __mptwo = { 1, { 1.0, 2.0 } };
-#endif
-
-#ifndef NO___ACR
-/* Compare mantissa of two multiple precision numbers regardless of the sign
- and exponent of the numbers. */
-static int
-mcr (const mp_no *x, const mp_no *y, int p)
-{
- long i;
- long p2 = p;
- for (i = 1; i <= p2; i++)
- {
- if (X[i] == Y[i])
- continue;
- else if (X[i] > Y[i])
- return 1;
- else
- return -1;
- }
- return 0;
-}
-
-/* Compare the absolute values of two multiple precision numbers. */
-int
-__acr (const mp_no *x, const mp_no *y, int p)
-{
- long i;
-
- if (X[0] == 0)
- {
- if (Y[0] == 0)
- i = 0;
- else
- i = -1;
- }
- else if (Y[0] == 0)
- i = 1;
- else
- {
- if (EX > EY)
- i = 1;
- else if (EX < EY)
- i = -1;
- else
- i = mcr (x, y, p);
- }
-
- return i;
-}
-#endif
-
-#ifndef NO___CPY
-/* Copy multiple precision number X into Y. They could be the same
- number. */
-void
-__cpy (const mp_no *x, mp_no *y, int p)
-{
- long i;
-
- EY = EX;
- for (i = 0; i <= p; i++)
- Y[i] = X[i];
-}
-#endif
-
-#ifndef NO___MP_DBL
-/* Convert a multiple precision number *X into a double precision
- number *Y, normalized case (|x| >= 2**(-1022))). X has precision
- P, which is positive. */
-static void
-norm (const mp_no *x, double *y, int p)
-{
-# define R RADIXI
- long i;
- double c;
- mantissa_t a, u, v, z[5];
- if (p < 5)
- {
- if (p == 1)
- c = X[1];
- else if (p == 2)
- c = X[1] + R * X[2];
- else if (p == 3)
- c = X[1] + R * (X[2] + R * X[3]);
- else /* p == 4. */
- c = (X[1] + R * X[2]) + R * R * (X[3] + R * X[4]);
- }
- else
- {
- for (a = 1, z[1] = X[1]; z[1] < TWO23; )
- {
- a *= 2;
- z[1] *= 2;
- }
-
- for (i = 2; i < 5; i++)
- {
- mantissa_store_t d, r;
- d = X[i] * (mantissa_store_t) a;
- DIV_RADIX (d, r);
- z[i] = r;
- z[i - 1] += d;
- }
-
- u = ALIGN_DOWN_TO (z[3], TWO19);
- v = z[3] - u;
-
- if (v == TWO18)
- {
- if (z[4] == 0)
- {
- for (i = 5; i <= p; i++)
- {
- if (X[i] == 0)
- continue;
- else
- {
- z[3] += 1;
- break;
- }
- }
- }
- else
- z[3] += 1;
- }
-
- c = (z[1] + R * (z[2] + R * z[3])) / a;
- }
-
- c *= X[0];
-
- for (i = 1; i < EX; i++)
- c *= RADIX;
- for (i = 1; i > EX; i--)
- c *= RADIXI;
-
- *y = c;
-# undef R
-}
-
-/* Convert a multiple precision number *X into a double precision
- number *Y, Denormal case (|x| < 2**(-1022))). */
-static void
-denorm (const mp_no *x, double *y, int p)
-{
- long i, k;
- long p2 = p;
- double c;
- mantissa_t u, z[5];
-
-# define R RADIXI
- if (EX < -44 || (EX == -44 && X[1] < TWO5))
- {
- *y = 0;
- return;
- }
-
- if (p2 == 1)
- {
- if (EX == -42)
- {
- z[1] = X[1] + TWO10;
- z[2] = 0;
- z[3] = 0;
- k = 3;
- }
- else if (EX == -43)
- {
- z[1] = TWO10;
- z[2] = X[1];
- z[3] = 0;
- k = 2;
- }
- else
- {
- z[1] = TWO10;
- z[2] = 0;
- z[3] = X[1];
- k = 1;
- }
- }
- else if (p2 == 2)
- {
- if (EX == -42)
- {
- z[1] = X[1] + TWO10;
- z[2] = X[2];
- z[3] = 0;
- k = 3;
- }
- else if (EX == -43)
- {
- z[1] = TWO10;
- z[2] = X[1];
- z[3] = X[2];
- k = 2;
- }
- else
- {
- z[1] = TWO10;
- z[2] = 0;
- z[3] = X[1];
- k = 1;
- }
- }
- else
- {
- if (EX == -42)
- {
- z[1] = X[1] + TWO10;
- z[2] = X[2];
- k = 3;
- }
- else if (EX == -43)
- {
- z[1] = TWO10;
- z[2] = X[1];
- k = 2;
- }
- else
- {
- z[1] = TWO10;
- z[2] = 0;
- k = 1;
- }
- z[3] = X[k];
- }
-
- u = ALIGN_DOWN_TO (z[3], TWO5);
-
- if (u == z[3])
- {
- for (i = k + 1; i <= p2; i++)
- {
- if (X[i] == 0)
- continue;
- else
- {
- z[3] += 1;
- break;
- }
- }
- }
-
- c = X[0] * ((z[1] + R * (z[2] + R * z[3])) - TWO10);
-
- *y = c * TWOM1032;
-# undef R
-}
-
-/* Convert multiple precision number *X into double precision number *Y. The
- result is correctly rounded to the nearest/even. */
-void
-__mp_dbl (const mp_no *x, double *y, int p)
-{
- if (X[0] == 0)
- {
- *y = 0;
- return;
- }
-
- if (__glibc_likely (EX > -42 || (EX == -42 && X[1] >= TWO10)))
- norm (x, y, p);
- else
- denorm (x, y, p);
-}
-#endif
-
-/* Get the multiple precision equivalent of X into *Y. If the precision is too
- small, the result is truncated. */
-void
-SECTION
-__dbl_mp (double x, mp_no *y, int p)
-{
- long i, n;
- long p2 = p;
-
- /* Sign. */
- if (x == 0)
- {
- Y[0] = 0;
- return;
- }
- else if (x > 0)
- Y[0] = 1;
- else
- {
- Y[0] = -1;
- x = -x;
- }
-
- /* Exponent. */
- for (EY = 1; x >= RADIX; EY += 1)
- x *= RADIXI;
- for (; x < 1; EY -= 1)
- x *= RADIX;
-
- /* Digits. */
- n = MIN (p2, 4);
- for (i = 1; i <= n; i++)
- {
- INTEGER_OF (x, Y[i]);
- x *= RADIX;
- }
- for (; i <= p2; i++)
- Y[i] = 0;
-}
-
-/* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0. The
- sign of the sum *Z is not changed. X and Y may overlap but not X and Z or
- Y and Z. No guard digit is used. The result equals the exact sum,
- truncated. */
-static void
-SECTION
-add_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- long i, j, k;
- long p2 = p;
- mantissa_t zk;
-
- EZ = EX;
-
- i = p2;
- j = p2 + EY - EX;
- k = p2 + 1;
-
- if (__glibc_unlikely (j < 1))
- {
- __cpy (x, z, p);
- return;
- }
-
- zk = 0;
-
- for (; j > 0; i--, j--)
- {
- zk += X[i] + Y[j];
- if (zk >= RADIX)
- {
- Z[k--] = zk - RADIX;
- zk = 1;
- }
- else
- {
- Z[k--] = zk;
- zk = 0;
- }
- }
-
- for (; i > 0; i--)
- {
- zk += X[i];
- if (zk >= RADIX)
- {
- Z[k--] = zk - RADIX;
- zk = 1;
- }
- else
- {
- Z[k--] = zk;
- zk = 0;
- }
- }
-
- if (zk == 0)
- {
- for (i = 1; i <= p2; i++)
- Z[i] = Z[i + 1];
- }
- else
- {
- Z[1] = zk;
- EZ += 1;
- }
-}
-
-/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
- The sign of the difference *Z is not changed. X and Y may overlap but not X
- and Z or Y and Z. One guard digit is used. The error is less than one
- ULP. */
-static void
-SECTION
-sub_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- long i, j, k;
- long p2 = p;
- mantissa_t zk;
-
- EZ = EX;
- i = p2;
- j = p2 + EY - EX;
- k = p2;
-
- /* Y is too small compared to X, copy X over to the result. */
- if (__glibc_unlikely (j < 1))
- {
- __cpy (x, z, p);
- return;
- }
-
- /* The relevant least significant digit in Y is non-zero, so we factor it in
- to enhance accuracy. */
- if (j < p2 && Y[j + 1] > 0)
- {
- Z[k + 1] = RADIX - Y[j + 1];
- zk = -1;
- }
- else
- zk = Z[k + 1] = 0;
-
- /* Subtract and borrow. */
- for (; j > 0; i--, j--)
- {
- zk += (X[i] - Y[j]);
- if (zk < 0)
- {
- Z[k--] = zk + RADIX;
- zk = -1;
- }
- else
- {
- Z[k--] = zk;
- zk = 0;
- }
- }
-
- /* We're done with digits from Y, so it's just digits in X. */
- for (; i > 0; i--)
- {
- zk += X[i];
- if (zk < 0)
- {
- Z[k--] = zk + RADIX;
- zk = -1;
- }
- else
- {
- Z[k--] = zk;
- zk = 0;
- }
- }
-
- /* Normalize. */
- for (i = 1; Z[i] == 0; i++)
- ;
- EZ = EZ - i + 1;
- for (k = 1; i <= p2 + 1; )
- Z[k++] = Z[i++];
- for (; k <= p2; )
- Z[k++] = 0;
-}
-
-/* Add *X and *Y and store the result in *Z. X and Y may overlap, but not X
- and Z or Y and Z. One guard digit is used. The error is less than one
- ULP. */
-void
-SECTION
-__add (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- int n;
-
- if (X[0] == 0)
- {
- __cpy (y, z, p);
- return;
- }
- else if (Y[0] == 0)
- {
- __cpy (x, z, p);
- return;
- }
-
- if (X[0] == Y[0])
- {
- if (__acr (x, y, p) > 0)
- {
- add_magnitudes (x, y, z, p);
- Z[0] = X[0];
- }
- else
- {
- add_magnitudes (y, x, z, p);
- Z[0] = Y[0];
- }
- }
- else
- {
- if ((n = __acr (x, y, p)) == 1)
- {
- sub_magnitudes (x, y, z, p);
- Z[0] = X[0];
- }
- else if (n == -1)
- {
- sub_magnitudes (y, x, z, p);
- Z[0] = Y[0];
- }
- else
- Z[0] = 0;
- }
-}
-
-/* Subtract *Y from *X and return the result in *Z. X and Y may overlap but
- not X and Z or Y and Z. One guard digit is used. The error is less than
- one ULP. */
-void
-SECTION
-__sub (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- int n;
-
- if (X[0] == 0)
- {
- __cpy (y, z, p);
- Z[0] = -Z[0];
- return;
- }
- else if (Y[0] == 0)
- {
- __cpy (x, z, p);
- return;
- }
-
- if (X[0] != Y[0])
- {
- if (__acr (x, y, p) > 0)
- {
- add_magnitudes (x, y, z, p);
- Z[0] = X[0];
- }
- else
- {
- add_magnitudes (y, x, z, p);
- Z[0] = -Y[0];
- }
- }
- else
- {
- if ((n = __acr (x, y, p)) == 1)
- {
- sub_magnitudes (x, y, z, p);
- Z[0] = X[0];
- }
- else if (n == -1)
- {
- sub_magnitudes (y, x, z, p);
- Z[0] = -Y[0];
- }
- else
- Z[0] = 0;
- }
-}
-
-#ifndef NO__MUL
-/* Multiply *X and *Y and store result in *Z. X and Y may overlap but not X
- and Z or Y and Z. For P in [1, 2, 3], the exact result is truncated to P
- digits. In case P > 3 the error is bounded by 1.001 ULP. */
-void
-SECTION
-__mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- long i, j, k, ip, ip2;
- long p2 = p;
- mantissa_store_t zk;
- const mp_no *a;
- mantissa_store_t *diag;
-
- /* Is z=0? */
- if (__glibc_unlikely (X[0] * Y[0] == 0))
- {
- Z[0] = 0;
- return;
- }
-
- /* We need not iterate through all X's and Y's since it's pointless to
- multiply zeroes. Here, both are zero... */
- for (ip2 = p2; ip2 > 0; ip2--)
- if (X[ip2] != 0 || Y[ip2] != 0)
- break;
-
- a = X[ip2] != 0 ? y : x;
-
- /* ... and here, at least one of them is still zero. */
- for (ip = ip2; ip > 0; ip--)
- if (a->d[ip] != 0)
- break;
-
- /* The product looks like this for p = 3 (as an example):
-
-
- a1 a2 a3
- x b1 b2 b3
- -----------------------------
- a1*b3 a2*b3 a3*b3
- a1*b2 a2*b2 a3*b2
- a1*b1 a2*b1 a3*b1
-
- So our K needs to ideally be P*2, but we're limiting ourselves to P + 3
- for P >= 3. We compute the above digits in two parts; the last P-1
- digits and then the first P digits. The last P-1 digits are a sum of
- products of the input digits from P to P-k where K is 0 for the least
- significant digit and increases as we go towards the left. The product
- term is of the form X[k]*X[P-k] as can be seen in the above example.
-
- The first P digits are also a sum of products with the same product term,
- except that the sum is from 1 to k. This is also evident from the above
- example.
-
- Another thing that becomes evident is that only the most significant
- ip+ip2 digits of the result are non-zero, where ip and ip2 are the
- 'internal precision' of the input numbers, i.e. digits after ip and ip2
- are all 0. */
-
- k = (__glibc_unlikely (p2 < 3)) ? p2 + p2 : p2 + 3;
-
- while (k > ip + ip2 + 1)
- Z[k--] = 0;
-
- zk = 0;
-
- /* Precompute sums of diagonal elements so that we can directly use them
- later. See the next comment to know we why need them. */
- diag = alloca (k * sizeof (mantissa_store_t));
- mantissa_store_t d = 0;
- for (i = 1; i <= ip; i++)
- {
- d += X[i] * (mantissa_store_t) Y[i];
- diag[i] = d;
- }
- while (i < k)
- diag[i++] = d;
-
- while (k > p2)
- {
- long lim = k / 2;
-
- if (k % 2 == 0)
- /* We want to add this only once, but since we subtract it in the sum
- of products above, we add twice. */
- zk += 2 * X[lim] * (mantissa_store_t) Y[lim];
-
- for (i = k - p2, j = p2; i < j; i++, j--)
- zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);
-
- zk -= diag[k - 1];
-
- DIV_RADIX (zk, Z[k]);
- k--;
- }
-
- /* The real deal. Mantissa digit Z[k] is the sum of all X[i] * Y[j] where i
- goes from 1 -> k - 1 and j goes the same range in reverse. To reduce the
- number of multiplications, we halve the range and if k is an even number,
- add the diagonal element X[k/2]Y[k/2]. Through the half range, we compute
- X[i] * Y[j] as (X[i] + X[j]) * (Y[i] + Y[j]) - X[i] * Y[i] - X[j] * Y[j].
-
- This reduction tells us that we're summing two things, the first term
- through the half range and the negative of the sum of the product of all
- terms of X and Y in the full range. i.e.
-
- SUM(X[i] * Y[i]) for k terms. This is precalculated above for each k in
- a single loop so that it completes in O(n) time and can hence be directly
- used in the loop below. */
- while (k > 1)
- {
- long lim = k / 2;
-
- if (k % 2 == 0)
- /* We want to add this only once, but since we subtract it in the sum
- of products above, we add twice. */
- zk += 2 * X[lim] * (mantissa_store_t) Y[lim];
-
- for (i = 1, j = k - 1; i < j; i++, j--)
- zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);
-
- zk -= diag[k - 1];
-
- DIV_RADIX (zk, Z[k]);
- k--;
- }
- Z[k] = zk;
-
- /* Get the exponent sum into an intermediate variable. This is a subtle
- optimization, where given enough registers, all operations on the exponent
- happen in registers and the result is written out only once into EZ. */
- int e = EX + EY;
-
- /* Is there a carry beyond the most significant digit? */
- if (__glibc_unlikely (Z[1] == 0))
- {
- for (i = 1; i <= p2; i++)
- Z[i] = Z[i + 1];
- e--;
- }
-
- EZ = e;
- Z[0] = X[0] * Y[0];
-}
-#endif
-
-#ifndef NO__SQR
-/* Square *X and store result in *Y. X and Y may not overlap. For P in
- [1, 2, 3], the exact result is truncated to P digits. In case P > 3 the
- error is bounded by 1.001 ULP. This is a faster special case of
- multiplication. */
-void
-SECTION
-__sqr (const mp_no *x, mp_no *y, int p)
-{
- long i, j, k, ip;
- mantissa_store_t yk;
-
- /* Is z=0? */
- if (__glibc_unlikely (X[0] == 0))
- {
- Y[0] = 0;
- return;
- }
-
- /* We need not iterate through all X's since it's pointless to
- multiply zeroes. */
- for (ip = p; ip > 0; ip--)
- if (X[ip] != 0)
- break;
-
- k = (__glibc_unlikely (p < 3)) ? p + p : p + 3;
-
- while (k > 2 * ip + 1)
- Y[k--] = 0;
-
- yk = 0;
-
- while (k > p)
- {
- mantissa_store_t yk2 = 0;
- long lim = k / 2;
-
- if (k % 2 == 0)
- yk += X[lim] * (mantissa_store_t) X[lim];
-
- /* In __mul, this loop (and the one within the next while loop) run
- between a range to calculate the mantissa as follows:
-
- Z[k] = X[k] * Y[n] + X[k+1] * Y[n-1] ... + X[n-1] * Y[k+1]
- + X[n] * Y[k]
-
- For X == Y, we can get away with summing halfway and doubling the
- result. For cases where the range size is even, the mid-point needs
- to be added separately (above). */
- for (i = k - p, j = p; i < j; i++, j--)
- yk2 += X[i] * (mantissa_store_t) X[j];
-
- yk += 2 * yk2;
-
- DIV_RADIX (yk, Y[k]);
- k--;
- }
-
- while (k > 1)
- {
- mantissa_store_t yk2 = 0;
- long lim = k / 2;
-
- if (k % 2 == 0)
- yk += X[lim] * (mantissa_store_t) X[lim];
-
- /* Likewise for this loop. */
- for (i = 1, j = k - 1; i < j; i++, j--)
- yk2 += X[i] * (mantissa_store_t) X[j];
-
- yk += 2 * yk2;
-
- DIV_RADIX (yk, Y[k]);
- k--;
- }
- Y[k] = yk;
-
- /* Squares are always positive. */
- Y[0] = 1;
-
- /* Get the exponent sum into an intermediate variable. This is a subtle
- optimization, where given enough registers, all operations on the exponent
- happen in registers and the result is written out only once into EZ. */
- int e = EX * 2;
-
- /* Is there a carry beyond the most significant digit? */
- if (__glibc_unlikely (Y[1] == 0))
- {
- for (i = 1; i <= p; i++)
- Y[i] = Y[i + 1];
- e--;
- }
-
- EY = e;
-}
-#endif
-
-/* Invert *X and store in *Y. Relative error bound:
- - For P = 2: 1.001 * R ^ (1 - P)
- - For P = 3: 1.063 * R ^ (1 - P)
- - For P > 3: 2.001 * R ^ (1 - P)
-
- *X = 0 is not permissible. */
-static void
-SECTION
-__inv (const mp_no *x, mp_no *y, int p)
-{
- long i;
- double t;
- mp_no z, w;
- static const int np1[] =
- { 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3,
- 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
- };
-
- __cpy (x, &z, p);
- z.e = 0;
- __mp_dbl (&z, &t, p);
- t = 1 / t;
-
- /* t == 0 will never happen at this point, since 1/t can only be 0 if t is
- infinity, but before the division t == mantissa of x (exponent is 0). We
- can instruct the compiler to ignore this case. */
- if (t == 0)
- __builtin_unreachable ();
-
- __dbl_mp (t, y, p);
- EY -= EX;
-
- for (i = 0; i < np1[p]; i++)
- {
- __cpy (y, &w, p);
- __mul (x, &w, y, p);
- __sub (&__mptwo, y, &z, p);
- __mul (&w, &z, y, p);
- }
-}
-
-/* Divide *X by *Y and store result in *Z. X and Y may overlap but not X and Z
- or Y and Z. Relative error bound:
- - For P = 2: 2.001 * R ^ (1 - P)
- - For P = 3: 2.063 * R ^ (1 - P)
- - For P > 3: 3.001 * R ^ (1 - P)
-
- *X = 0 is not permissible. */
-void
-SECTION
-__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p)
-{
- mp_no w;
-
- if (X[0] == 0)
- Z[0] = 0;
- else
- {
- __inv (y, &w, p);
- __mul (x, &w, z, p);
- }
-}
diff --git a/sysdeps/ieee754/dbl-64/mpatan.h b/sysdeps/ieee754/dbl-64/mpatan.h
deleted file mode 100644
index 0c8fd24c2cbf8f0fb503cb2bcb86f1198f92502c..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpatan.h
+++ /dev/null
@@ -1,145 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/******************************************************************/
-/* */
-/* MODULE_NAME:mpatan.h */
-/* */
-/* common data and variables prototype and definition */
-/******************************************************************/
-
-#ifndef MPATAN_H
-#define MPATAN_H
-
-extern const number __atan_xm[8] attribute_hidden;
-extern const number __atan_twonm1[33] attribute_hidden;
-extern const number __atan_twom[8] attribute_hidden;
-extern const int __atan_np[33] attribute_hidden;
-
-
-#ifndef AVOID_MPATAN_H
-#ifdef BIG_ENDI
- const number
- __atan_xm[8] = { /* x[m] */
-/**/ {{0x00000000, 0x00000000} }, /* 0.0 */
-/**/ {{0x3f8930be, 0x00000000} }, /* 0.0123 */
-/**/ {{0x3f991687, 0x00000000} }, /* 0.0245 */
-/**/ {{0x3fa923a2, 0x00000000} }, /* 0.0491 */
-/**/ {{0x3fb930be, 0x00000000} }, /* 0.0984 */
-/**/ {{0x3fc95810, 0x00000000} }, /* 0.198 */
-/**/ {{0x3fda7ef9, 0x00000000} }, /* 0.414 */
-/**/ {{0x3ff00000, 0x00000000} }, /* 1.0 */
- };
- const number
- __atan_twonm1[33] = { /* 2n-1 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x40260000, 0x00000000} }, /* 11 */
-/**/ {{0x402e0000, 0x00000000} }, /* 15 */
-/**/ {{0x40330000, 0x00000000} }, /* 19 */
-/**/ {{0x40350000, 0x00000000} }, /* 21 */
-/**/ {{0x40390000, 0x00000000} }, /* 25 */
-/**/ {{0x403d0000, 0x00000000} }, /* 29 */
-/**/ {{0x40408000, 0x00000000} }, /* 33 */
-/**/ {{0x40428000, 0x00000000} }, /* 37 */
-/**/ {{0x40448000, 0x00000000} }, /* 41 */
-/**/ {{0x40468000, 0x00000000} }, /* 45 */
-/**/ {{0x40488000, 0x00000000} }, /* 49 */
-/**/ {{0x404a8000, 0x00000000} }, /* 53 */
-/**/ {{0x404b8000, 0x00000000} }, /* 55 */
-/**/ {{0x404d8000, 0x00000000} }, /* 59 */
-/**/ {{0x404f8000, 0x00000000} }, /* 63 */
-/**/ {{0x4050c000, 0x00000000} }, /* 67 */
-/**/ {{0x4051c000, 0x00000000} }, /* 71 */
-/**/ {{0x4052c000, 0x00000000} }, /* 75 */
-/**/ {{0x4053c000, 0x00000000} }, /* 79 */
-/**/ {{0x4054c000, 0x00000000} }, /* 83 */
-/**/ {{0x40554000, 0x00000000} }, /* 85 */
-/**/ {{0x40564000, 0x00000000} }, /* 89 */
-/**/ {{0x40574000, 0x00000000} }, /* 93 */
-/**/ {{0x40584000, 0x00000000} }, /* 97 */
-/**/ {{0x40594000, 0x00000000} }, /* 101 */
-/**/ {{0x405a4000, 0x00000000} }, /* 105 */
-/**/ {{0x405b4000, 0x00000000} }, /* 109 */
-/**/ {{0x405c4000, 0x00000000} }, /* 113 */
-/**/ {{0x405d4000, 0x00000000} }, /* 117 */
- };
-
-#else
-#ifdef LITTLE_ENDI
-
- const number
- __atan_xm[8] = { /* x[m] */
-/**/ {{0x00000000, 0x00000000} }, /* 0.0 */
-/**/ {{0x00000000, 0x3f8930be} }, /* 0.0123 */
-/**/ {{0x00000000, 0x3f991687} }, /* 0.0245 */
-/**/ {{0x00000000, 0x3fa923a2} }, /* 0.0491 */
-/**/ {{0x00000000, 0x3fb930be} }, /* 0.0984 */
-/**/ {{0x00000000, 0x3fc95810} }, /* 0.198 */
-/**/ {{0x00000000, 0x3fda7ef9} }, /* 0.414 */
-/**/ {{0x00000000, 0x3ff00000} }, /* 1.0 */
- };
- const number
-__atan_twonm1[33] = { /* 2n-1 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x00000000} }, /* 0 */
-/**/ {{0x00000000, 0x40260000} }, /* 11 */
-/**/ {{0x00000000, 0x402e0000} }, /* 15 */
-/**/ {{0x00000000, 0x40330000} }, /* 19 */
-/**/ {{0x00000000, 0x40350000} }, /* 21 */
-/**/ {{0x00000000, 0x40390000} }, /* 25 */
-/**/ {{0x00000000, 0x403d0000} }, /* 29 */
-/**/ {{0x00000000, 0x40408000} }, /* 33 */
-/**/ {{0x00000000, 0x40428000} }, /* 37 */
-/**/ {{0x00000000, 0x40448000} }, /* 41 */
-/**/ {{0x00000000, 0x40468000} }, /* 45 */
-/**/ {{0x00000000, 0x40488000} }, /* 49 */
-/**/ {{0x00000000, 0x404a8000} }, /* 53 */
-/**/ {{0x00000000, 0x404b8000} }, /* 55 */
-/**/ {{0x00000000, 0x404d8000} }, /* 59 */
-/**/ {{0x00000000, 0x404f8000} }, /* 63 */
-/**/ {{0x00000000, 0x4050c000} }, /* 67 */
-/**/ {{0x00000000, 0x4051c000} }, /* 71 */
-/**/ {{0x00000000, 0x4052c000} }, /* 75 */
-/**/ {{0x00000000, 0x4053c000} }, /* 79 */
-/**/ {{0x00000000, 0x4054c000} }, /* 83 */
-/**/ {{0x00000000, 0x40554000} }, /* 85 */
-/**/ {{0x00000000, 0x40564000} }, /* 89 */
-/**/ {{0x00000000, 0x40574000} }, /* 93 */
-/**/ {{0x00000000, 0x40584000} }, /* 97 */
-/**/ {{0x00000000, 0x40594000} }, /* 101 */
-/**/ {{0x00000000, 0x405a4000} }, /* 105 */
-/**/ {{0x00000000, 0x405b4000} }, /* 109 */
-/**/ {{0x00000000, 0x405c4000} }, /* 113 */
-/**/ {{0x00000000, 0x405d4000} }, /* 117 */
- };
-
-#endif
-#endif
-
- const int
- __atan_np[33] = { 0, 0, 0, 0, 6, 8,10,11,13,15,17,19,21,23,25,27,28,
- 30,32,34,36,38,40,42,43,45,47,49,51,53,55,57,59};
-
-#endif
-#endif
diff --git a/sysdeps/ieee754/dbl-64/mpatan.c b/sysdeps/ieee754/dbl-64/mpatan.c
deleted file mode 100644
index 97a16673d1a9072a3d3f43494035198a343a260e..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpatan.c
+++ /dev/null
@@ -1,116 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/******************************************************************/
-/* */
-/* MODULE_NAME:mpatan.c */
-/* */
-/* FUNCTIONS:mpatan */
-/* */
-/* FILES NEEDED: mpa.h endian.h mpatan.h */
-/* mpa.c */
-/* */
-/* Multi-Precision Atan function subroutine, for precision p >= 4.*/
-/* The relative error of the result is bounded by 34.32*r**(1-p), */
-/* where r=2**24. */
-/******************************************************************/
-
-#include "endian.h"
-#include "mpa.h"
-#include <math.h>
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-#include "mpatan.h"
-
-void
-SECTION
-__mpatan (mp_no *x, mp_no *y, int p)
-{
- int i, m, n;
- double dx;
- mp_no mptwoim1 =
- {
- 0,
- {
- 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
- 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
- 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
- }
- };
-
- mp_no mps, mpsm, mpt, mpt1, mpt2, mpt3;
-
- /* Choose m and initiate mptwoim1. */
- if (EX > 0)
- m = 7;
- else if (EX < 0)
- m = 0;
- else
- {
- __mp_dbl (x, &dx, p);
- dx = fabs (dx);
- for (m = 6; m > 0; m--)
- {
- if (dx > __atan_xm[m].d)
- break;
- }
- }
- mptwoim1.e = 1;
- mptwoim1.d[0] = 1;
-
- /* Reduce x m times. */
- __sqr (x, &mpsm, p);
- if (m == 0)
- __cpy (x, &mps, p);
- else
- {
- for (i = 0; i < m; i++)
- {
- __add (&__mpone, &mpsm, &mpt1, p);
- __mpsqrt (&mpt1, &mpt2, p);
- __add (&mpt2, &mpt2, &mpt1, p);
- __add (&__mptwo, &mpsm, &mpt2, p);
- __add (&mpt1, &mpt2, &mpt3, p);
- __dvd (&mpsm, &mpt3, &mpt1, p);
- __cpy (&mpt1, &mpsm, p);
- }
- __mpsqrt (&mpsm, &mps, p);
- mps.d[0] = X[0];
- }
-
- /* Evaluate a truncated power series for Atan(s). */
- n = __atan_np[p];
- mptwoim1.d[1] = __atan_twonm1[p].d;
- __dvd (&mpsm, &mptwoim1, &mpt, p);
- for (i = n - 1; i > 1; i--)
- {
- mptwoim1.d[1] -= 2;
- __dvd (&mpsm, &mptwoim1, &mpt1, p);
- __mul (&mpsm, &mpt, &mpt2, p);
- __sub (&mpt1, &mpt2, &mpt, p);
- }
- __mul (&mps, &mpt, &mpt1, p);
- __sub (&mps, &mpt1, &mpt, p);
-
- /* Compute Atan(x). */
- mptwoim1.d[1] = 1 << m;
- __mul (&mptwoim1, &mpt, y, p);
-}
diff --git a/sysdeps/ieee754/dbl-64/mpatan2.c b/sysdeps/ieee754/dbl-64/mpatan2.c
deleted file mode 100644
index 1ccc02ef8a876eab340b95f37b1921f668b9ba56..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpatan2.c
+++ /dev/null
@@ -1,67 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/******************************************************************/
-/* MODULE_NAME: mpatan2.c */
-/* */
-/* FUNCTIONS:mpatan2 */
-/* */
-/* FILES NEEDED: mpa.h */
-/* mpa.c mpatan.c mpsqrt.c */
-/* */
-/* Multi-Precision Atan2(y,x) function subroutine, */
-/* for precision p >= 4. */
-/* y=0 is not permitted if x<=0. No error messages are given. */
-/* The relative error of the result is bounded by 44.84*r**(1-p) */
-/* if x <= 0, y != 0 and by 37.33*r**(1-p) if x>0. here r=2**24. */
-/* */
-/******************************************************************/
-
-#include "mpa.h"
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-/* Multi-Precision Atan2 (y, x) function subroutine, for p >= 4.
- y = 0 is not permitted if x <= 0. No error messages are given. */
-void
-SECTION
-__mpatan2 (mp_no *y, mp_no *x, mp_no *z, int p)
-{
- mp_no mpt1, mpt2, mpt3;
-
- if (X[0] <= 0)
- {
- __dvd (x, y, &mpt1, p);
- __mul (&mpt1, &mpt1, &mpt2, p);
- if (mpt1.d[0] != 0)
- mpt1.d[0] = 1;
- __add (&mpt2, &__mpone, &mpt3, p);
- __mpsqrt (&mpt3, &mpt2, p);
- __add (&mpt1, &mpt2, &mpt3, p);
- mpt3.d[0] = Y[0];
- __mpatan (&mpt3, &mpt1, p);
- __add (&mpt1, &mpt1, z, p);
- }
- else
- {
- __dvd (y, x, &mpt1, p);
- __mpatan (&mpt1, z, p);
- }
-}
diff --git a/sysdeps/ieee754/dbl-64/mpsqrt.h b/sysdeps/ieee754/dbl-64/mpsqrt.h
deleted file mode 100644
index 0e7d12b3b3eac98f71755b4bba5d76ac3f14990e..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpsqrt.h
+++ /dev/null
@@ -1,38 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/******************************************************************/
-/* */
-/* MODULE_NAME:mpatan.h */
-/* */
-/* common data and variables prototype and definition */
-/******************************************************************/
-
-#ifndef MPSQRT_H
-#define MPSQRT_H
-
-extern const int __mpsqrt_mp[33] attribute_hidden;
-
-
-#ifndef AVOID_MPSQRT_H
- const int __mpsqrt_mp[33] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,
- 4,4,4,4,4,4,4,4,4};
-#endif
-
-#endif
diff --git a/sysdeps/ieee754/dbl-64/mpsqrt.c b/sysdeps/ieee754/dbl-64/mpsqrt.c
deleted file mode 100644
index 7a5b3544538fb8cfacf90a2362197cc86b7e0569..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mpsqrt.c
+++ /dev/null
@@ -1,111 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/****************************************************************************/
-/* MODULE_NAME:mpsqrt.c */
-/* */
-/* FUNCTION:mpsqrt */
-/* fastiroot */
-/* */
-/* FILES NEEDED:endian.h mpa.h mpsqrt.h */
-/* mpa.c */
-/* Multi-Precision square root function subroutine for precision p >= 4. */
-/* The relative error is bounded by 3.501*r**(1-p), where r=2**24. */
-/* */
-/****************************************************************************/
-#include "endian.h"
-#include "mpa.h"
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-#include "mpsqrt.h"
-
-/****************************************************************************/
-/* Multi-Precision square root function subroutine for precision p >= 4. */
-/* The relative error is bounded by 3.501*r**(1-p), where r=2**24. */
-/* Routine receives two pointers to Multi Precision numbers: */
-/* x (left argument) and y (next argument). Routine also receives precision */
-/* p as integer. Routine computes sqrt(*x) and stores result in *y */
-/****************************************************************************/
-
-static double fastiroot (double);
-
-void
-SECTION
-__mpsqrt (mp_no *x, mp_no *y, int p)
-{
- int i, m, ey;
- double dx, dy;
- static const mp_no mphalf = {0, {1.0, HALFRAD}};
- static const mp_no mp3halfs = {1, {1.0, 1.0, HALFRAD}};
- mp_no mpxn, mpz, mpu, mpt1, mpt2;
-
- ey = EX / 2;
- __cpy (x, &mpxn, p);
- mpxn.e -= (ey + ey);
- __mp_dbl (&mpxn, &dx, p);
- dy = fastiroot (dx);
- __dbl_mp (dy, &mpu, p);
- __mul (&mpxn, &mphalf, &mpz, p);
-
- m = __mpsqrt_mp[p];
- for (i = 0; i < m; i++)
- {
- __sqr (&mpu, &mpt1, p);
- __mul (&mpt1, &mpz, &mpt2, p);
- __sub (&mp3halfs, &mpt2, &mpt1, p);
- __mul (&mpu, &mpt1, &mpt2, p);
- __cpy (&mpt2, &mpu, p);
- }
- __mul (&mpxn, &mpu, y, p);
- EY += ey;
-}
-
-/***********************************************************/
-/* Compute a double precision approximation for 1/sqrt(x) */
-/* with the relative error bounded by 2**-51. */
-/***********************************************************/
-static double
-SECTION
-fastiroot (double x)
-{
- union
- {
- int i[2];
- double d;
- } p, q;
- double y, z, t;
- int n;
- static const double c0 = 0.99674, c1 = -0.53380;
- static const double c2 = 0.45472, c3 = -0.21553;
-
- p.d = x;
- p.i[HIGH_HALF] = (p.i[HIGH_HALF] & 0x3FFFFFFF) | 0x3FE00000;
- q.d = x;
- y = p.d;
- z = y - 1.0;
- n = (q.i[HIGH_HALF] - p.i[HIGH_HALF]) >> 1;
- z = ((c3 * z + c2) * z + c1) * z + c0; /* 2**-7 */
- z = z * (1.5 - 0.5 * y * z * z); /* 2**-14 */
- p.d = z * (1.5 - 0.5 * y * z * z); /* 2**-28 */
- p.i[HIGH_HALF] -= n;
- t = x * p.d;
- return p.d * (1.5 - 0.5 * p.d * t);
-}
diff --git a/sysdeps/ieee754/dbl-64/mptan.c b/sysdeps/ieee754/dbl-64/mptan.c
deleted file mode 100644
index 79fe1b3578046ffb8ba612795c85768a3c87a35d..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/mptan.c
+++ /dev/null
@@ -1,63 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/**********************************************************************/
-/* MODULE_NAME:mptan.c */
-/* */
-/* FUNCTION: mptan */
-/* */
-/* FILES NEEDED: endian.h mpa.h */
-/* mpa.c sincos32.c branred.c */
-/* */
-/* Multi-Precision tan() function subroutine, for p=32. It is based */
-/* on the routines mpranred() and c32(). mpranred() performs range */
-/* reduction of a double number x into a multiple precision number */
-/* y, such that y=x-n*pi/2, abs(y)<pi/4, n=0,+-1,+-2,.... c32() */
-/* computes both sin(y), cos(y). tan(x) is either sin(y)/cos(y) */
-/* or -cos(y)/sin(y). The precision of the result is of about 559 */
-/* significant bits. */
-/* */
-/**********************************************************************/
-#include "endian.h"
-#include "mpa.h"
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-void
-SECTION
-__mptan (double x, mp_no *mpy, int p)
-{
- int n;
- mp_no mpw, mpc, mps;
-
- /* Negative or positive result. */
- n = __mpranred (x, &mpw, p) & 0x00000001;
- /* Computing sin(x) and cos(x). */
- __c32 (&mpw, &mpc, &mps, p);
- /* Second or fourth quarter of unit circle. */
- if (n)
- {
- __dvd (&mpc, &mps, mpy, p);
- mpy->d[0] *= -1;
- }
- /* tan is negative in this area. */
- else
- __dvd (&mps, &mpc, mpy, p);
-}
diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c
index 64f81912a125e6382061d0853a8e1b12109128ff..e8f9965b18e89d08bf264810793598adaa93145b 100644
--- a/sysdeps/ieee754/dbl-64/s_sin.c
+++ b/sysdeps/ieee754/dbl-64/s_sin.c
@@ -38,7 +38,6 @@
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
-#include "MathLib.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
diff --git a/sysdeps/ieee754/dbl-64/sincos32.h b/sysdeps/ieee754/dbl-64/sincos32.h
deleted file mode 100644
index 1fd33c3c448441ad1b5e21b9da6b68cb227af20a..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/sincos32.h
+++ /dev/null
@@ -1,81 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/******************************************************************/
-/* */
-/* MODULE_NAME:sincos32.h */
-/* */
-/* common data and variables prototype and definition */
-/******************************************************************/
-
-#ifndef SINCOS32_H
-#define SINCOS32_H
-
-#ifdef BIG_ENDI
-static const number
-/**/ hpinv = {{0x3FE45F30, 0x6DC9C883}}, /* 0.63661977236758138 */
-/**/ toint = {{0x43380000, 0x00000000}}; /* 6755399441055744 */
-
-#else
-#ifdef LITTLE_ENDI
-static const number
-/**/ hpinv = {{0x6DC9C883, 0x3FE45F30}}, /* 0.63661977236758138 */
-/**/ toint = {{0x00000000, 0x43380000}}; /* 6755399441055744 */
-
-#endif
-#endif
-
-static const mp_no
- oofac27 = {-3,{1.0,7.0,4631664.0,12006312.0,13118056.0,6538613.0,646354.0,
- 8508025.0,9131256.0,7548776.0,2529842.0,8864927.0,660489.0,15595125.0,12777885.0,
- 11618489.0,13348664.0,5486686.0,514518.0,11275535.0,4727621.0,3575562.0,
- 13579710.0,5829745.0,7531862.0,9507898.0,6915060.0,4079264.0,1907586.0,
- 6078398.0,13789314.0,5504104.0,14136.0}},
- pi = {1,{1.0,3.0,
- 2375530.0,8947107.0,578323.0,1673774.0,225395.0,4498441.0,3678761.0,
- 10432976.0,536314.0,10021966.0,7113029.0,2630118.0,3723283.0,7847508.0,
- 6737716.0,15273068.0,12626985.0,12044668.0,5299519.0,8705461.0,11880201.0,
- 1544726.0,14014857.0,7994139.0,13709579.0,10918111.0,11906095.0,16610011.0,
- 13638367.0,12040417.0,11529578.0,2522774.0}},
- hp = {1,{1.0, 1.0,
- 9576373.0,4473553.0,8677769.0,9225495.0,112697.0,10637828.0,
- 10227988.0,13605096.0,268157.0,5010983.0,3556514.0,9703667.0,
- 1861641.0,12312362.0,3368858.0,7636534.0,6313492.0,14410942.0,
- 2649759.0,12741338.0,14328708.0,9160971.0,7007428.0,12385677.0,
- 15243397.0,13847663.0,14341655.0,16693613.0,15207791.0,14408816.0,
- 14153397.0,1261387.0,6110792.0,2291862.0,4181138.0,5295267.0}};
-
-static const double toverp[75] = {
- 10680707.0, 7228996.0, 1387004.0, 2578385.0, 16069853.0,
- 12639074.0, 9804092.0, 4427841.0, 16666979.0, 11263675.0,
- 12935607.0, 2387514.0, 4345298.0, 14681673.0, 3074569.0,
- 13734428.0, 16653803.0, 1880361.0, 10960616.0, 8533493.0,
- 3062596.0, 8710556.0, 7349940.0, 6258241.0, 3772886.0,
- 3769171.0, 3798172.0, 8675211.0, 12450088.0, 3874808.0,
- 9961438.0, 366607.0, 15675153.0, 9132554.0, 7151469.0,
- 3571407.0, 2607881.0, 12013382.0, 4155038.0, 6285869.0,
- 7677882.0, 13102053.0, 15825725.0, 473591.0, 9065106.0,
- 15363067.0, 6271263.0, 9264392.0, 5636912.0, 4652155.0,
- 7056368.0, 13614112.0, 10155062.0, 1944035.0, 9527646.0,
- 15080200.0, 6658437.0, 6231200.0, 6832269.0, 16767104.0,
- 5075751.0, 3212806.0, 1398474.0, 7579849.0, 6349435.0,
- 12618859.0, 4703257.0, 12806093.0, 14477321.0, 2786137.0,
- 12875403.0, 9837734.0, 14528324.0, 13719321.0, 343717.0 };
-
-#endif
diff --git a/sysdeps/ieee754/dbl-64/sincos32.c b/sysdeps/ieee754/dbl-64/sincos32.c
deleted file mode 100644
index a28932dffef7f7648ba2ed6d670157c34b4282b4..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/sincos32.c
+++ /dev/null
@@ -1,307 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2020 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-/****************************************************************/
-/* MODULE_NAME: sincos32.c */
-/* */
-/* FUNCTIONS: ss32 */
-/* cc32 */
-/* c32 */
-/* sin32 */
-/* cos32 */
-/* mpsin */
-/* mpcos */
-/* mpranred */
-/* mpsin1 */
-/* mpcos1 */
-/* */
-/* FILES NEEDED: endian.h mpa.h sincos32.h */
-/* mpa.c */
-/* */
-/* Multi Precision sin() and cos() function with p=32 for sin()*/
-/* cos() arcsin() and arccos() routines */
-/* In addition mpranred() routine performs range reduction of */
-/* a double number x into multi precision number y, */
-/* such that y=x-n*pi/2, abs(y)<pi/4, n=0,+-1,+-2,.... */
-/****************************************************************/
-#include "endian.h"
-#include "mpa.h"
-#include "sincos32.h"
-#include <math.h>
-#include <math_private.h>
-#include <stap-probe.h>
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-/* Compute Multi-Precision sin() function for given p. Receive Multi Precision
- number x and result stored at y. */
-static void
-SECTION
-ss32 (mp_no *x, mp_no *y, int p)
-{
- int i;
- double a;
- mp_no mpt1, x2, gor, sum, mpk = {1, {1.0}};
- for (i = 1; i <= p; i++)
- mpk.d[i] = 0;
-
- __sqr (x, &x2, p);
- __cpy (&oofac27, &gor, p);
- __cpy (&gor, &sum, p);
- for (a = 27.0; a > 1.0; a -= 2.0)
- {
- mpk.d[1] = a * (a - 1.0);
- __mul (&gor, &mpk, &mpt1, p);
- __cpy (&mpt1, &gor, p);
- __mul (&x2, &sum, &mpt1, p);
- __sub (&gor, &mpt1, &sum, p);
- }
- __mul (x, &sum, y, p);
-}
-
-/* Compute Multi-Precision cos() function for given p. Receive Multi Precision
- number x and result stored at y. */
-static void
-SECTION
-cc32 (mp_no *x, mp_no *y, int p)
-{
- int i;
- double a;
- mp_no mpt1, x2, gor, sum, mpk = {1, {1.0}};
- for (i = 1; i <= p; i++)
- mpk.d[i] = 0;
-
- __sqr (x, &x2, p);
- mpk.d[1] = 27.0;
- __mul (&oofac27, &mpk, &gor, p);
- __cpy (&gor, &sum, p);
- for (a = 26.0; a > 2.0; a -= 2.0)
- {
- mpk.d[1] = a * (a - 1.0);
- __mul (&gor, &mpk, &mpt1, p);
- __cpy (&mpt1, &gor, p);
- __mul (&x2, &sum, &mpt1, p);
- __sub (&gor, &mpt1, &sum, p);
- }
- __mul (&x2, &sum, y, p);
-}
-
-/* Compute both sin(x), cos(x) as Multi precision numbers. */
-void
-SECTION
-__c32 (mp_no *x, mp_no *y, mp_no *z, int p)
-{
- mp_no u, t, t1, t2, c, s;
- int i;
- __cpy (x, &u, p);
- u.e = u.e - 1;
- cc32 (&u, &c, p);
- ss32 (&u, &s, p);
- for (i = 0; i < 24; i++)
- {
- __mul (&c, &s, &t, p);
- __sub (&s, &t, &t1, p);
- __add (&t1, &t1, &s, p);
- __sub (&__mptwo, &c, &t1, p);
- __mul (&t1, &c, &t2, p);
- __add (&t2, &t2, &c, p);
- }
- __sub (&__mpone, &c, y, p);
- __cpy (&s, z, p);
-}
-
-/* Compute sin() of double-length number (X + DX) as Multi Precision number and
- return result as double. If REDUCE_RANGE is true, X is assumed to be the
- original input and DX is ignored. */
-double
-SECTION
-__mpsin (double x, double dx, bool reduce_range)
-{
- double y;
- mp_no a, b, c, s;
- int n;
- int p = 32;
-
- if (reduce_range)
- {
- n = __mpranred (x, &a, p); /* n is 0, 1, 2 or 3. */
- __c32 (&a, &c, &s, p);
- }
- else
- {
- n = -1;
- __dbl_mp (x, &b, p);
- __dbl_mp (dx, &c, p);
- __add (&b, &c, &a, p);
- if (x > 0.8)
- {
- __sub (&hp, &a, &b, p);
- __c32 (&b, &s, &c, p);
- }
- else
- __c32 (&a, &c, &s, p); /* b = sin(x+dx) */
- }
-
- /* Convert result based on which quarter of unit circle y is in. */
- switch (n)
- {
- case 1:
- __mp_dbl (&c, &y, p);
- break;
-
- case 3:
- __mp_dbl (&c, &y, p);
- y = -y;
- break;
-
- case 2:
- __mp_dbl (&s, &y, p);
- y = -y;
- break;
-
- /* Quadrant not set, so the result must be sin (X + DX), which is also in
- S. */
- case 0:
- default:
- __mp_dbl (&s, &y, p);
- }
- LIBC_PROBE (slowsin, 3, &x, &dx, &y);
- return y;
-}
-
-/* Compute cos() of double-length number (X + DX) as Multi Precision number and
- return result as double. If REDUCE_RANGE is true, X is assumed to be the
- original input and DX is ignored. */
-double
-SECTION
-__mpcos (double x, double dx, bool reduce_range)
-{
- double y;
- mp_no a, b, c, s;
- int n;
- int p = 32;
-
- if (reduce_range)
- {
- n = __mpranred (x, &a, p); /* n is 0, 1, 2 or 3. */
- __c32 (&a, &c, &s, p);
- }
- else
- {
- n = -1;
- __dbl_mp (x, &b, p);
- __dbl_mp (dx, &c, p);
- __add (&b, &c, &a, p);
- if (x > 0.8)
- {
- __sub (&hp, &a, &b, p);
- __c32 (&b, &s, &c, p);
- }
- else
- __c32 (&a, &c, &s, p); /* a = cos(x+dx) */
- }
-
- /* Convert result based on which quarter of unit circle y is in. */
- switch (n)
- {
- case 1:
- __mp_dbl (&s, &y, p);
- y = -y;
- break;
-
- case 3:
- __mp_dbl (&s, &y, p);
- break;
-
- case 2:
- __mp_dbl (&c, &y, p);
- y = -y;
- break;
-
- /* Quadrant not set, so the result must be cos (X + DX), which is also
- stored in C. */
- case 0:
- default:
- __mp_dbl (&c, &y, p);
- }
- LIBC_PROBE (slowcos, 3, &x, &dx, &y);
- return y;
-}
-
-/* Perform range reduction of a double number x into multi precision number y,
- such that y = x - n * pi / 2, abs (y) < pi / 4, n = 0, +-1, +-2, ...
- Return int which indicates in which quarter of circle x is. */
-int
-SECTION
-__mpranred (double x, mp_no *y, int p)
-{
- number v;
- double t, xn;
- int i, k, n;
- mp_no a, b, c;
-
- if (fabs (x) < 2.8e14)
- {
- t = (x * hpinv.d + toint.d);
- xn = t - toint.d;
- v.d = t;
- n = v.i[LOW_HALF] & 3;
- __dbl_mp (xn, &a, p);
- __mul (&a, &hp, &b, p);
- __dbl_mp (x, &c, p);
- __sub (&c, &b, y, p);
- return n;
- }
- else
- {
- /* If x is very big more precision required. */
- __dbl_mp (x, &a, p);
- a.d[0] = 1.0;
- k = a.e - 5;
- if (k < 0)
- k = 0;
- b.e = -k;
- b.d[0] = 1.0;
- for (i = 0; i < p; i++)
- b.d[i + 1] = toverp[i + k];
- __mul (&a, &b, &c, p);
- t = c.d[c.e];
- for (i = 1; i <= p - c.e; i++)
- c.d[i] = c.d[i + c.e];
- for (i = p + 1 - c.e; i <= p; i++)
- c.d[i] = 0;
- c.e = 0;
- if (c.d[1] >= HALFRAD)
- {
- t += 1.0;
- __sub (&c, &__mpone, &b, p);
- __mul (&b, &hp, y, p);
- }
- else
- __mul (&c, &hp, y, p);
- n = (int) t;
- if (x < 0)
- {
- y->d[0] = -y->d[0];
- n = -n;
- }
- return (n & 3);
- }
-}
diff --git a/sysdeps/x86_64/fpu/multiarch/Makefile b/sysdeps/x86_64/fpu/multiarch/Makefile
index 3836574f48400eb69e3b8996103cfb3b4f4d3ea8..57892c56bb7705737ac637bed851171dd125b6bc 100644
--- a/sysdeps/x86_64/fpu/multiarch/Makefile
+++ b/sysdeps/x86_64/fpu/multiarch/Makefile
@@ -9,25 +9,14 @@ libm-sysdep_routines += s_ceil-sse4_1 s_ceilf-sse4_1 s_floor-sse4_1 \
s_trunc-sse4_1 s_truncf-sse4_1
libm-sysdep_routines += e_exp-fma e_log-fma e_pow-fma s_atan-fma \
- e_asin-fma e_atan2-fma s_sin-fma s_tan-fma \
- mpa-fma \
- sincos32-fma doasin-fma dosincos-fma \
- mpatan2-fma mpatan-fma mpsqrt-fma mptan-fma
+ e_asin-fma e_atan2-fma s_sin-fma s_tan-fma
-CFLAGS-doasin-fma.c = -mfma -mavx2
-CFLAGS-dosincos-fma.c = -mfma -mavx2
CFLAGS-e_asin-fma.c = -mfma -mavx2
CFLAGS-e_atan2-fma.c = -mfma -mavx2
CFLAGS-e_exp-fma.c = -mfma -mavx2
CFLAGS-e_log-fma.c = -mfma -mavx2
CFLAGS-e_pow-fma.c = -mfma -mavx2
-CFLAGS-mpa-fma.c = -mfma -mavx2
-CFLAGS-mpatan-fma.c = -mfma -mavx2
-CFLAGS-mpatan2-fma.c = -mfma -mavx2
-CFLAGS-mpsqrt-fma.c = -mfma -mavx2
-CFLAGS-mptan-fma.c = -mfma -mavx2
CFLAGS-s_atan-fma.c = -mfma -mavx2
-CFLAGS-sincos32-fma.c = -mfma -mavx2
CFLAGS-s_sin-fma.c = -mfma -mavx2
CFLAGS-s_tan-fma.c = -mfma -mavx2
@@ -46,36 +35,23 @@ CFLAGS-s_cosf-fma.c = -mfma -mavx2
CFLAGS-s_sincosf-fma.c = -mfma -mavx2
libm-sysdep_routines += e_exp-fma4 e_log-fma4 e_pow-fma4 s_atan-fma4 \
- e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4 \
- mpa-fma4 \
- sincos32-fma4 doasin-fma4 dosincos-fma4 \
- mpatan2-fma4 mpatan-fma4 mpsqrt-fma4 mptan-fma4
+ e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4
-CFLAGS-doasin-fma4.c = -mfma4
-CFLAGS-dosincos-fma4.c = -mfma4
CFLAGS-e_asin-fma4.c = -mfma4
CFLAGS-e_atan2-fma4.c = -mfma4
CFLAGS-e_exp-fma4.c = -mfma4
CFLAGS-e_log-fma4.c = -mfma4
CFLAGS-e_pow-fma4.c = -mfma4
-CFLAGS-mpa-fma4.c = -mfma4
-CFLAGS-mpatan-fma4.c = -mfma4
-CFLAGS-mpatan2-fma4.c = -mfma4
-CFLAGS-mpsqrt-fma4.c = -mfma4
-CFLAGS-mptan-fma4.c = -mfma4
CFLAGS-s_atan-fma4.c = -mfma4
-CFLAGS-sincos32-fma4.c = -mfma4
CFLAGS-s_sin-fma4.c = -mfma4
CFLAGS-s_tan-fma4.c = -mfma4
libm-sysdep_routines += e_exp-avx e_log-avx s_atan-avx \
- e_atan2-avx s_sin-avx s_tan-avx \
- mpa-avx
+ e_atan2-avx s_sin-avx s_tan-avx
CFLAGS-e_atan2-avx.c = -msse2avx -DSSE2AVX
CFLAGS-e_exp-avx.c = -msse2avx -DSSE2AVX
CFLAGS-e_log-avx.c = -msse2avx -DSSE2AVX
-CFLAGS-mpa-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_atan-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_sin-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_tan-avx.c = -msse2avx -DSSE2AVX
diff --git a/sysdeps/x86_64/fpu/multiarch/doasin-fma.c b/sysdeps/x86_64/fpu/multiarch/doasin-fma.c
deleted file mode 100644
index 7a09865fcaa98bc50b5461e5aca181325de9d383..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/doasin-fma.c
+++ /dev/null
@@ -1,4 +0,0 @@
-#define __doasin __doasin_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/doasin.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c b/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c
deleted file mode 100644
index 53eb4194726e605af7acf7f6a6513203acbca5c8..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c
+++ /dev/null
@@ -1,4 +0,0 @@
-#define __doasin __doasin_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/doasin.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c b/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c
deleted file mode 100644
index 5744586bdbda42646686d103074e6902087e85bf..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c
+++ /dev/null
@@ -1,6 +0,0 @@
-#define __docos __docos_fma
-#define __dubcos __dubcos_fma
-#define __dubsin __dubsin_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/dosincos.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c b/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c
deleted file mode 100644
index 1578b2fce0be41bd396abf2da8153444f5578602..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c
+++ /dev/null
@@ -1,6 +0,0 @@
-#define __docos __docos_fma4
-#define __dubcos __dubcos_fma4
-#define __dubsin __dubsin_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/dosincos.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-avx.c b/sysdeps/x86_64/fpu/multiarch/mpa-avx.c
deleted file mode 100644
index 366b0b7134078e79d243502dcc23198f669cf73d..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-avx.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_avx
-#define __mul __mul_avx
-#define __sqr __sqr_avx
-#define __sub __sub_avx
-#define __dbl_mp __dbl_mp_avx
-#define __dvd __dvd_avx
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.avx")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-fma.c b/sysdeps/x86_64/fpu/multiarch/mpa-fma.c
deleted file mode 100644
index 177cc2517f66a593ecee4737804e4c771e77e848..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-fma.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_fma
-#define __mul __mul_fma
-#define __sqr __sqr_fma
-#define __sub __sub_fma
-#define __dbl_mp __dbl_mp_fma
-#define __dvd __dvd_fma
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c
deleted file mode 100644
index a4a759407e48c5cf1e375b32b1f9a54ce9f08c40..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_fma4
-#define __mul __mul_fma4
-#define __sqr __sqr_fma4
-#define __sub __sub_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __dvd __dvd_fma4
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c b/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c
deleted file mode 100644
index d216f9142de1cb646cc7c0e00816a97f88fc515f..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c
+++ /dev/null
@@ -1,10 +0,0 @@
-#define __mpatan __mpatan_fma
-#define __add __add_fma
-#define __dvd __dvd_fma
-#define __mpsqrt __mpsqrt_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define AVOID_MPATAN_H 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c
deleted file mode 100644
index fbd3bd49a2b40a83a12f7589e7e8b233e9dee8c3..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c
+++ /dev/null
@@ -1,10 +0,0 @@
-#define __mpatan __mpatan_fma4
-#define __add __add_fma4
-#define __dvd __dvd_fma4
-#define __mpsqrt __mpsqrt_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define AVOID_MPATAN_H 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c b/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c
deleted file mode 100644
index 98df336f79fca93aef6d8dbcb5510c064d049ef3..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __mpatan2 __mpatan2_fma
-#define __add __add_fma
-#define __dvd __dvd_fma
-#define __mpatan __mpatan_fma
-#define __mpsqrt __mpsqrt_fma
-#define __mul __mul_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan2.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c
deleted file mode 100644
index e6e44d49b09f67d26507a13e4d1b55c4cb5a058b..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __mpatan2 __mpatan2_fma4
-#define __add __add_fma4
-#define __dvd __dvd_fma4
-#define __mpatan __mpatan_fma4
-#define __mpsqrt __mpsqrt_fma4
-#define __mul __mul_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan2.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c b/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c
deleted file mode 100644
index 44d7a23ae3332b9e4d0f3eabe115f3bf5fe099e9..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c
+++ /dev/null
@@ -1,8 +0,0 @@
-#define __mpsqrt __mpsqrt_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define AVOID_MPSQRT_H 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpsqrt.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c
deleted file mode 100644
index f8a1ba2d92300ce3ace02bad3bc3311230037e02..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c
+++ /dev/null
@@ -1,8 +0,0 @@
-#define __mpsqrt __mpsqrt_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define AVOID_MPSQRT_H 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpsqrt.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mptan-fma.c b/sysdeps/x86_64/fpu/multiarch/mptan-fma.c
deleted file mode 100644
index d1a691413c58785b56fa54be0924d42103c2dc50..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mptan-fma.c
+++ /dev/null
@@ -1,7 +0,0 @@
-#define __mptan __mptan_fma
-#define __c32 __c32_fma
-#define __dvd __dvd_fma
-#define __mpranred __mpranred_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mptan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c b/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c
deleted file mode 100644
index fb4a9d48cae2737af89d85611d64541e286138ed..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c
+++ /dev/null
@@ -1,7 +0,0 @@
-#define __mptan __mptan_fma4
-#define __c32 __c32_fma4
-#define __dvd __dvd_fma4
-#define __mpranred __mpranred_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mptan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c b/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c
deleted file mode 100644
index 4152b8483535b253c6e9e75978cc0da26d13b7d7..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c
+++ /dev/null
@@ -1,13 +0,0 @@
-#define __c32 __c32_fma
-#define __mpsin __mpsin_fma
-#define __mpsin1 __mpsin1_fma
-#define __mpcos __mpcos_fma
-#define __mpcos1 __mpcos1_fma
-#define __mpranred __mpranred_fma
-#define __add __add_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/sincos32.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c b/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c
deleted file mode 100644
index 643eedf1381a8ca8bcfd2d83448c410870bba8cf..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c
+++ /dev/null
@@ -1,13 +0,0 @@
-#define __c32 __c32_fma4
-#define __mpsin __mpsin_fma4
-#define __mpsin1 __mpsin1_fma4
-#define __mpcos __mpcos_fma4
-#define __mpcos1 __mpcos1_fma4
-#define __mpranred __mpranred_fma4
-#define __add __add_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/sincos32.c>
^ permalink raw reply related [flat|nested] 19+ messages in thread
* Re: [PATCH 1/5] Remove slow paths from asin
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
@ 2021-01-04 16:25 ` Paul Zimmermann
2021-01-04 17:06 ` Szabolcs Nagy via Libc-alpha
2021-01-07 19:23 ` [PATCH v2 1/5] Remove remaining slow paths from asin and acos Wilco Dijkstra via Libc-alpha
1 sibling, 1 reply; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-04 16:25 UTC (permalink / raw
To: Wilco Dijkstra; +Cc: libc-alpha
Dear Wilco,
I have a few comments on this first patch:
- the title only mentions asin, but the patch also deals with acos
- it would be nice to explain where the "Max ULP" come from, to allow
future developers to debug the code if needed.
- it would be nice to have a summary with the maximal ulp error at the
top of each function (__ieee754_asin and __ieee754_acos)
- for __ieee754_acos the largest claimed error is 0.52 ULP for
0.5 <= |x| < 0.96875, but I find an error of more than 0.522 ulps
for x = 0x1.dffff58d80953p-1 (on x86_64).
Best regards,
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
* Re: [PATCH 2/5] Remove slow paths in tan
2021-01-04 12:17 ` [PATCH 2/5] Remove slow paths in tan Wilco Dijkstra via Libc-alpha
@ 2021-01-04 16:36 ` Paul Zimmermann
2021-01-07 19:25 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-04 16:36 UTC (permalink / raw
To: Wilco Dijkstra; +Cc: libc-alpha
Dear Wilco,
about this second patch, in addition to the following remarks similar to
asin and acos:
- it would be nice to explain where the "Max ULP" come from, to allow
future developers to debug the code if needed.
- it would be nice to have a summary with the maximal ulp error at the
top of each function
I have only one further remark:
-typedef union { int4 i[2]; double x; } mynumber;
+typedef union { int4 i[2]; double x; double d; } mynumber;
Why did you add another 'double' field d? Only the 'd' field is used in
e_atan2.c, why not replace it by 'x'? Maybe to avoid a too large diff?
Best regards,
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
* Re: [PATCH 3/5] Remove slow paths from atan
2021-01-04 12:19 ` [PATCH 3/5] Remove slow paths from atan Wilco Dijkstra via Libc-alpha
@ 2021-01-04 16:43 ` Paul Zimmermann
2021-01-07 19:26 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-04 16:43 UTC (permalink / raw
To: Wilco Dijkstra; +Cc: libc-alpha
Dear Wilco,
same comments on this patch:
- it would be nice to explain where the "Max ULP" come from, to allow
future developers to debug the code if needed
- it would be nice to have a summary with the maximal ulp error at the
top of the atan() function (0.56 ULP if I'm correct)
Best regards,
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
* Re: [PATCH 4/5] Remove slow paths from atan2
2021-01-04 12:21 ` [PATCH 4/5] Remove slow paths from atan2 Wilco Dijkstra via Libc-alpha
@ 2021-01-04 16:49 ` Paul Zimmermann
2021-01-07 19:28 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-04 16:49 UTC (permalink / raw
To: Wilco Dijkstra; +Cc: libc-alpha
Dear Wilco,
same comments for this last patch:
- it would be nice to explain where the "Max ULP" come from, to allow
future developers to debug the code if needed
- it would be nice to have a summary with the maximal ulp error at the
top of the atan2() function (0.56 ULP if I'm correct)
and in addition:
- it remains one occurrence of "correctly rounded" at line 51
Best regards,
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
* Re: [PATCH 1/5] Remove slow paths from asin
2021-01-04 16:25 ` Paul Zimmermann
@ 2021-01-04 17:06 ` Szabolcs Nagy via Libc-alpha
2021-01-04 17:41 ` Paul Zimmermann
0 siblings, 1 reply; 19+ messages in thread
From: Szabolcs Nagy via Libc-alpha @ 2021-01-04 17:06 UTC (permalink / raw
To: Paul Zimmermann; +Cc: libc-alpha, Wilco Dijkstra
The 01/04/2021 17:25, Paul Zimmermann wrote:
> Dear Wilco,
>
> I have a few comments on this first patch:
>
> - the title only mentions asin, but the patch also deals with acos
>
> - it would be nice to explain where the "Max ULP" come from, to allow
> future developers to debug the code if needed.
>
> - it would be nice to have a summary with the maximal ulp error at the
> top of each function (__ieee754_asin and __ieee754_acos)
>
> - for __ieee754_acos the largest claimed error is 0.52 ULP for
> 0.5 <= |x| < 0.96875, but I find an error of more than 0.522 ulps
> for x = 0x1.dffff58d80953p-1 (on x86_64).
the Max ULP annotations are based on the assumption that
the bounds checks in the removed code paths caught all the
incorrectly rounded half-way cases.
previously it was noted that there are incorrectly rounded
cases in asin that were not caught by those checks:
https://sourceware.org/legacy-ml/libc-alpha/2020-01/msg00574.html
so either the Max ULP annotation was computed incorrectly
or the same errors should happen in the original acos too.
in my experience the original code is fairly conservative
so in most cases i'd expect the Max ULP to be an upper
bound with significant margin, not the actual worst case
error.
i think it's worth keeping the annotation even if the
original code was not entirely correct.
^ permalink raw reply [flat|nested] 19+ messages in thread
* Re: [PATCH 1/5] Remove slow paths from asin
2021-01-04 17:06 ` Szabolcs Nagy via Libc-alpha
@ 2021-01-04 17:41 ` Paul Zimmermann
2021-01-07 19:36 ` Wilco Dijkstra via Libc-alpha
0 siblings, 1 reply; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-04 17:41 UTC (permalink / raw
To: Szabolcs Nagy; +Cc: libc-alpha, Wilco.Dijkstra
Dear Szabolcs,
> previously it was noted that there are incorrectly rounded
> cases in asin that were not caught by those checks:
> https://sourceware.org/legacy-ml/libc-alpha/2020-01/msg00574.html
the patch you mention (commit f67f9c9) was already entitled
"ieee754: Remove slow paths from asin and acos". This was confusing.
I believe it did only remove the "very slow" paths.
> in my experience the original code is fairly conservative
> so in most cases i'd expect the Max ULP to be an upper
> bound with significant margin, not the actual worst case
> error.
here are the claimed bounds and the maximal errors I found so far after
applying Wilco's patches (rounded up to 3 digits):
claimed found so far
acos 0.52 0.523
asin 0.523 0.516
atan 0.56 0.523
tan 0.62 0.612
atan2 0.56 0.524
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
* [PATCH v2 1/5] Remove remaining slow paths from asin and acos
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
2021-01-04 16:25 ` Paul Zimmermann
@ 2021-01-07 19:23 ` Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 19:23 UTC (permalink / raw
To: 'GNU C Library'
This patch series removes all remaining slow paths and related code. First asin/acos, tan,
atan, atan2 implementations are updated, and the final patch removes the unused mpa files,
headers and probes. Passes buildmanyglibc.
Remove slow paths from asin/acos. Add ULP annotations based on previous slow path checks
(which may not be accurate). Update AArch64 libm-test-ulps for asin.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/aarch64/libm-test-ulps b/sysdeps/aarch64/libm-test-ulps
index 22fcf8db73dc444c25e0c356b1e0036571edd112..bbadf667ee4b7a0cf80506d321553f064049c516 100644
--- a/sysdeps/aarch64/libm-test-ulps
+++ b/sysdeps/aarch64/libm-test-ulps
@@ -41,6 +41,7 @@ float: 2
ldouble: 2
Function: "asin":
+double: 1
float: 1
ldouble: 1
@@ -55,7 +56,7 @@ float: 1
ldouble: 1
Function: "asin_upward":
-double: 1
+double: 2
float: 1
ldouble: 2
diff --git a/sysdeps/ieee754/dbl-64/e_asin.c b/sysdeps/ieee754/dbl-64/e_asin.c
index ae5344f720055ef31b01c4af2e1228b79b821430..4473ae5df41c31e80e8fd0d74a840e8b9e718a7d 100644
--- a/sysdeps/ieee754/dbl-64/e_asin.c
+++ b/sysdeps/ieee754/dbl-64/e_asin.c
@@ -21,8 +21,7 @@
/* */
/* FUNCTIONS: uasin */
/* uacos */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
-/* doasin.c sincos32.c dosincos.c mpa.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h usncs.h */
/* sincos.tbl asincos.tbl powtwo.tbl root.tbl */
/* */
/******************************************************************/
@@ -31,7 +30,6 @@
#include "asincos.tbl"
#include "root.tbl"
#include "powtwo.tbl"
-#include "MathLib.h"
#include "uasncs.h"
#include <float.h>
#include <math.h>
@@ -43,15 +41,11 @@
# define SECTION
#endif
-void __doasin(double x, double dx, double w[]);
-void __dubsin(double x, double dx, double v[]);
-void __dubcos(double x, double dx, double v[]);
-void __docos(double x, double dx, double v[]);
-
+/* asin with max ULP of ~0.516 based on random sampling. */
double
SECTION
__ieee754_asin(double x){
- double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2];
+ double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
mynumber u,v;
int4 k,m,n;
@@ -70,27 +64,8 @@ __ieee754_asin(double x){
x2 = x*x;
t = (((((f6*x2 + f5)*x2 + f4)*x2 + f3)*x2 + f2)*x2 + f1)*(x2*x);
res = x+t; /* res=arcsin(x) according to Taylor series */
- cor = (x-res)+t;
- if (res == res+1.025*cor) return res;
- else {
- x1 = x+big;
- xx = x*x;
- x1 -= big;
- x2 = x - x1;
- p = x1*x1*x1;
- s1 = a1.x*p;
- s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
- ((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
- res1 = x+s1;
- s2 = ((x-res1)+s1)+s2;
- res = res1+s2;
- cor = (res1-res)+s2;
- if (res == res+1.00014*cor) return res;
- else {
- __doasin(x,0,w);
- return w[0];
- }
- }
+ /* Max ULP is 0.513. */
+ return res;
}
/*---------------------0.125 <= |x| < 0.5 -----------------------------*/
else if (k < 0x3fe00000) {
@@ -103,26 +78,8 @@ __ieee754_asin(double x){
+xx*asncs.x[n+6]))))+asncs.x[n+7];
t+=p;
res =asncs.x[n+8] +t;
- cor = (asncs.x[n+8]-res)+t;
- if (res == res+1.05*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+8]+xx*asncs.x[n+9];
- t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0005*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __dubsin(res,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.524. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fe00000) */
/*-------------------- 0.5 <= |x| < 0.75 -----------------------------*/
else
@@ -135,26 +92,8 @@ __ieee754_asin(double x){
+xx*(asncs.x[n+6]+xx*asncs.x[n+7])))))+asncs.x[n+8];
t+=p;
res =asncs.x[n+9] +t;
- cor = (asncs.x[n+9]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+9]+xx*asncs.x[n+10];
- t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0005*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __dubsin(res,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fe80000) */
/*--------------------- 0.75 <= |x|< 0.921875 ----------------------*/
else
@@ -167,28 +106,8 @@ __ieee754_asin(double x){
+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+xx*asncs.x[n+8]))))))+asncs.x[n+9];
t+=p;
res =asncs.x[n+10] +t;
- cor = (asncs.x[n+10]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+10]+xx*asncs.x[n+11];
- t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0008*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=hp0.x-res;
- z=((hp0.x-y)-res)+(hp1.x-z);
- __dubcos(y,z,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fed8000) */
/*-------------------0.921875 <= |x| < 0.953125 ------------------------*/
else
@@ -203,29 +122,8 @@ __ieee754_asin(double x){
xx*asncs.x[n+9])))))))+asncs.x[n+10];
t+=p;
res =asncs.x[n+11] +t;
- cor = (asncs.x[n+11]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+11]+xx*asncs.x[n+12];
- t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0007*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=(hp0.x-res)-z;
- z=y+hp1.x;
- y=(y-z)+hp1.x;
- __dubcos(z,y,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fee8000) */
/*--------------------0.953125 <= |x| < 0.96875 ------------------------*/
@@ -241,29 +139,8 @@ __ieee754_asin(double x){
xx*(asncs.x[n+9]+xx*asncs.x[n+10]))))))))+asncs.x[n+11];
t+=p;
res =asncs.x[n+12] +t;
- cor = (asncs.x[n+12]-res)+t;
- if (res == res+1.01*cor) return (m>0)?res:-res;
- else {
- r=asncs.x[n+12]+xx*asncs.x[n+13];
- t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
- res = r+t;
- cor = (r-res)+t;
- if (res == res+1.0007*cor) return (m>0)?res:-res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- y=(hp0.x-res)-z;
- z=y+hp1.x;
- y=(y-z)+hp1.x;
- __dubcos(z,y,w);
- z=(w[0]-fabs(x))+w[1];
- if (z>1.0e-27) return (m>0)?min(res,res1):-min(res,res1);
- else if (z<-1.0e-27) return (m>0)?max(res,res1):-max(res,res1);
- else {
- return (m>0)?res:-res;
- }
- }
- }
+ /* Max ULP is 0.505. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3fef0000) */
/*--------------------0.96875 <= |x| < 1 --------------------------------*/
else
@@ -282,16 +159,8 @@ __ieee754_asin(double x){
cor = (hp1.x - 2.0*cc)-2.0*(y+cc)*p;
res1 = hp0.x - 2.0*y;
res =res1 + cor;
- if (res == res+1.003*((res1-res)+cor)) return (m>0)?res:-res;
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res1=hp0.x-2.0*w[0];
- cor=((hp0.x-res1)-2.0*w[0])+(hp1.x-2.0*w[1]);
- res = res1+cor;
- return (m>0)?res:-res;
- }
+ /* Max ULP is 0.5015. */
+ return (m>0)?res:-res;
} /* else if (k < 0x3ff00000) */
/*---------------------------- |x|>=1 -------------------------------*/
else if (k==0x3ff00000 && u.i[LOW_HALF]==0) return (m>0)?hp0.x:-hp0.x;
@@ -315,11 +184,12 @@ libm_alias_finite (__ieee754_asin, __asin)
/* */
/*******************************************************************/
+/* acos with max ULP of ~0.523 based on random sampling. */
double
SECTION
__ieee754_acos(double x)
{
- double x1,x2,xx,s1,s2,res1,p,t,res,r,cor,cc,y,c,z,w[2],eps;
+ double x2,xx,res1,p,t,res,r,cor,cc,y,c,z;
mynumber u,v;
int4 k,m,n;
u.x = x;
@@ -336,32 +206,8 @@ __ieee754_acos(double x)
r=hp0.x-x;
cor=(((hp0.x-r)-x)+hp1.x)-t;
res = r+cor;
- cor = (r-res)+cor;
- if (res == res+1.004*cor) return res;
- else {
- x1 = x+big;
- xx = x*x;
- x1 -= big;
- x2 = x - x1;
- p = x1*x1*x1;
- s1 = a1.x*p;
- s2 = ((((((c7*xx + c6)*xx + c5)*xx + c4)*xx + c3)*xx + c2)*xx*xx*x +
- ((a1.x+a2.x)*x2*x2+ 0.5*x1*x)*x2) + a2.x*p;
- res1 = x+s1;
- s2 = ((x-res1)+s1)+s2;
- r=hp0.x-res1;
- cor=(((hp0.x-r)-res1)+hp1.x)-s2;
- res = r+cor;
- cor = (r-res)+cor;
- if (res == res+1.00004*cor) return res;
- else {
- __doasin(x,0,w);
- r=hp0.x-w[0];
- cor=((hp0.x-r)-w[0])+(hp1.x-w[1]);
- res=r+cor;
- return res;
- }
- }
+ /* Max ULP is 0.502. */
+ return res;
} /* else if (k < 0x3fc00000) */
/*---------------------- 0.125 <= |x| < 0.5 --------------------*/
else
@@ -377,35 +223,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+8]):(hp0.x+asncs.x[n+8]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+1.02*((y-res)+t)) return res;
- else {
- r=asncs.x[n+8]+xx*asncs.x[n+9];
- t=((asncs.x[n+8]-r)+xx*asncs.x[n+9])+(p+xx*asncs.x[n+10]);
- if (m>0)
- {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; }
- else
- {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+1.0002*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.51. */
+ return res;
} /* else if (k < 0x3fe00000) */
/*--------------------------- 0.5 <= |x| < 0.75 ---------------------*/
else
if (k < 0x3fe80000) {
n = 1056+((k&0x000fe000)>>11)*3;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps=1.02; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+
@@ -414,33 +241,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+9]):(hp0.x+asncs.x[n+9]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+9]+xx*asncs.x[n+10];
- t=((asncs.x[n+9]-r)+xx*asncs.x[n+10])+(p+xx*asncs.x[n+11]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0004; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0002; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.523 based on random sampling. */
+ return res;
} /* else if (k < 0x3fe80000) */
/*------------------------- 0.75 <= |x| < 0.921875 -------------*/
else
if (k < 0x3fed8000) {
n = 992+((k&0x000fe000)>>13)*13;
- if (m>0) {xx = x - asncs.x[n]; eps = 1.04; }
- else {xx = -x - asncs.x[n]; eps = 1.01; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]+xx*(asncs.x[n+7]+
@@ -449,33 +259,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+10]):(hp0.x+asncs.x[n+10]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+10]+xx*asncs.x[n+11];
- t=((asncs.x[n+10]-r)+xx*asncs.x[n+11])+(p+xx*asncs.x[n+12]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0032; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0008; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.523 based on random sampling. */
+ return res;
} /* else if (k < 0x3fed8000) */
/*-------------------0.921875 <= |x| < 0.953125 ------------------*/
else
if (k < 0x3fee8000) {
n = 884+((k&0x000fe000)>>13)*14;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps =1.005; }
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
@@ -485,33 +278,16 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+11]):(hp0.x+asncs.x[n+11]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+11]+xx*asncs.x[n+12];
- t=((asncs.x[n+11]-r)+xx*asncs.x[n+12])+(p+xx*asncs.x[n+13]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.523 based on random sampling. */
+ return res;
} /* else if (k < 0x3fee8000) */
/*--------------------0.953125 <= |x| < 0.96875 ----------------*/
else
if (k < 0x3fef0000) {
n = 768+((k&0x000fe000)>>13)*15;
- if (m>0) {xx = x - asncs.x[n]; eps=1.04; }
- else {xx = -x - asncs.x[n]; eps=1.005;}
+ if (m>0) {xx = x - asncs.x[n]; }
+ else {xx = -x - asncs.x[n]; }
t = asncs.x[n+1]*xx;
p=xx*xx*(asncs.x[n+2]+xx*(asncs.x[n+3]+xx*(asncs.x[n+4]+
xx*(asncs.x[n+5]+xx*(asncs.x[n+6]
@@ -521,25 +297,8 @@ __ieee754_acos(double x)
y = (m>0)?(hp0.x-asncs.x[n+12]):(hp0.x+asncs.x[n+12]);
t = (m>0)?(hp1.x-t):(hp1.x+t);
res = y+t;
- if (res == res+eps*((y-res)+t)) return res;
- else {
- r=asncs.x[n+12]+xx*asncs.x[n+13];
- t=((asncs.x[n+12]-r)+xx*asncs.x[n+13])+(p+xx*asncs.x[n+14]);
- if (m>0) {p = hp0.x-r; t = (((hp0.x-p)-r)-t)+hp1.x; eps=1.0030; }
- else {p = hp0.x+r; t = ((hp0.x-p)+r)+(hp1.x+t); eps=1.0005; }
- res = p+t;
- cor = (p-res)+t;
- if (res == (res+eps*cor)) return res;
- else {
- res1=res+1.1*cor;
- z=0.5*(res1-res);
- __docos(res,z,w);
- z=(w[0]-x)+w[1];
- if (z>1.0e-27) return max(res,res1);
- else if (z<-1.0e-27) return min(res,res1);
- else return res;
- }
- }
+ /* Max ULP is 0.523 based on random sampling. */
+ return res;
} /* else if (k < 0x3fef0000) */
/*-----------------0.96875 <= |x| < 1 ---------------------------*/
@@ -560,28 +319,14 @@ __ieee754_acos(double x)
cor = (hp1.x - cc)-(y+cc)*p;
res1 = hp0.x - y;
res =res1 + cor;
- if (res == res+1.002*((res1-res)+cor)) return (res+res);
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res1=hp0.x-w[0];
- cor=((hp0.x-res1)-w[0])+(hp1.x-w[1]);
- res = res1+cor;
- return (res+res);
- }
+ /* Max ULP is 0.501. */
+ return (res+res);
}
else {
cor = cc+p*(y+cc);
res = y + cor;
- if (res == res+1.03*((y-res)+cor)) return (res+res);
- else {
- c=y+cc;
- cc=(y-c)+cc;
- __doasin(c,cc,w);
- res = w[0];
- return (res+res);
- }
+ /* Max ULP is 0.515. */
+ return (res+res);
}
} /* else if (k < 0x3ff00000) */
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH v2 2/5] Remove slow paths in tan
2021-01-04 12:17 ` [PATCH 2/5] Remove slow paths in tan Wilco Dijkstra via Libc-alpha
2021-01-04 16:36 ` Paul Zimmermann
@ 2021-01-07 19:25 ` Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 19:25 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths in tan. Add ULP annotations. Merge 'number' into 'mynumber'.
Remove unused entries from tan constants.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/e_atan2.c b/sysdeps/ieee754/dbl-64/e_atan2.c
index b14d911ebf97a7ded32fdad635d0bbe33cc21036..e6b98142fbbc7c6bd4a51df0257b361fa46181ec 100644
--- a/sysdeps/ieee754/dbl-64/e_atan2.c
+++ b/sysdeps/ieee754/dbl-64/e_atan2.c
@@ -39,6 +39,7 @@
#include <dla.h>
#include "mpa.h"
#include "MathLib.h"
+#include "mydefs.h"
#include "uatan.tbl"
#include "atnat2.h"
#include <fenv.h>
diff --git a/sysdeps/ieee754/dbl-64/mydefs.h b/sysdeps/ieee754/dbl-64/mydefs.h
index cdbcabf8a70dd96474f17fd9c881c9a77fab0d70..a73c4de2c16a47bc65111cc59a1060020cc04b47 100644
--- a/sysdeps/ieee754/dbl-64/mydefs.h
+++ b/sysdeps/ieee754/dbl-64/mydefs.h
@@ -28,7 +28,7 @@
#define MY_H
typedef int int4;
-typedef union { int4 i[2]; double x; } mynumber;
+typedef union { int4 i[2]; double x; double d; } mynumber;
#define max(x, y) (((y) > (x)) ? (y) : (x))
#define min(x, y) (((y) < (x)) ? (y) : (x))
diff --git a/sysdeps/ieee754/dbl-64/s_tan.c b/sysdeps/ieee754/dbl-64/s_tan.c
index 6aa5a5a8e3d6d12b3c0f22f082c360a099e009b0..0d4bb7143704f8a425ebd4d6b72ba1645814ae9c 100644
--- a/sysdeps/ieee754/dbl-64/s_tan.c
+++ b/sysdeps/ieee754/dbl-64/s_tan.c
@@ -20,40 +20,30 @@
/* MODULE_NAME: utan.c */
/* */
/* FUNCTIONS: utan */
-/* tanMp */
/* */
-/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h utan.h */
-/* branred.c sincos32.c mptan.c */
+/* FILES NEEDED:dla.h endian.h mydefs.h utan.h */
+/* branred.c */
/* utan.tbl */
/* */
-/* An ultimate tan routine. Given an IEEE double machine number x */
-/* it computes the correctly rounded (to nearest) value of tan(x). */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/*********************************************************************/
#include <errno.h>
#include <float.h>
#include "endian.h"
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
+#include "mydefs.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-double.h>
#include <fenv.h>
-#include <stap-probe.h>
#ifndef SECTION
# define SECTION
#endif
-static double tanMp (double);
-void __mptan (double, mp_no *, int);
-
+/* tan with max ULP of ~0.612 based on random sampling. */
double
SECTION
__tan (double x)
@@ -62,17 +52,14 @@ __tan (double x)
#include "utan.tbl"
int ux, i, n;
- double a, da, a2, b, db, c, dc, c1, cc1, c2, cc2, c3, cc3, fi, ffi, gi, pz,
- s, sy, t, t1, t2, t3, t4, w, x2, xn, xx2, y, ya,
- yya, z0, z, zz, z2, zz2;
- int p;
- number num, v;
- mp_no mpa, mpt1, mpt2;
+ double a, da, a2, b, db, c, dc, fi, gi, pz,
+ s, sy, t, t1, t2, t3, t4, w, x2, xn, y, ya,
+ yya, z0, z, z2;
+ mynumber num, v;
double retval;
int __branred (double, double *, double *);
- int __mpranred (double, mp_no *, int);
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
@@ -100,7 +87,6 @@ __tan (double x)
/* (II) The case 1.259e-8 < abs(x) <= 0.0608 */
if (w <= g2.d)
{
- /* First stage */
x2 = x * x;
t2 = d9.d + x2 * d11.d;
@@ -109,50 +95,16 @@ __tan (double x)
t2 = d3.d + x2 * t2;
t2 *= x * x2;
- if ((y = x + (t2 - u1.d * t2)) == x + (t2 + u1.d * t2))
- {
- retval = y;
- goto ret;
- }
-
- /* Second stage */
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- EMULV (x, x, x2, xx2);
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (x, 0.0, c1, cc1, c2, cc2, t1, t2);
- ADD2 (x, 0.0, c2, cc2, c1, cc1, t1, t2);
- if ((y = c1 + (cc1 - u2.d * c1)) == c1 + (cc1 + u2.d * c1))
- {
- retval = y;
- goto ret;
- }
- retval = tanMp (x);
+ y = x + t2;
+ retval = y;
+ /* Max ULP is 0.504. */
goto ret;
}
/* (III) The case 0.0608 < abs(x) <= 0.787 */
if (w <= g3.d)
{
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * w));
+ i = ((int) (mfftnhf.d + 256 * w));
z = w - xfg[i][0].d;
z2 = z * z;
s = (x < 0.0) ? -1 : 1;
@@ -160,41 +112,9 @@ __tan (double x)
fi = xfg[i][1].d;
gi = xfg[i][2].d;
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u3.d)) == fi + (t2 + fi * u3.d))
- {
- retval = (s * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua3.d + t3 * ub3.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (s * y);
- goto ret;
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- EMULV (z, z, z2, zz2);
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, 0.0, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, 0.0, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
-
- if ((y = c3 + (cc3 - u4.d * c3)) == c3 + (cc3 + u4.d * c3))
- {
- retval = (s * y);
- goto ret;
- }
- retval = tanMp (x);
+ y = fi + t2;
+ retval = (s * y);
+ /* Max ULP is 0.60. */
goto ret;
}
@@ -223,14 +143,7 @@ __tan (double x)
sy = 1;
}
- /* (IV),(V) The case 0.787 < abs(x) <= 25, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (VI) The case 0.787 < abs(x) <= 25, 1e-7 < abs(y) <= 0.0608 */
+ /* (VI) The case 0.787 < abs(x) <= 25, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -242,94 +155,27 @@ __tan (double x)
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u6.d * c)) == c + (dc + u6.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u5.d * a)) == a + (t2 + u5.d * a))
- {
- retval = y;
- goto ret;
- }
- }
- /* Second stage */
- /* Range reduction by algorithm ii */
- t = (x * hpinv.d + toint.d);
- xn = t - toint.d;
- v.d = t;
- t1 = (x - xn * mp1.d) - xn * mp2.d;
- n = v.i[LOW_HALF] & 0x00000001;
- da = xn * pp3.d;
- t = t1 - da;
- da = (t1 - t) - da;
- t1 = xn * pp4.d;
- a = t - t1;
- da = ((t - a) - t1) + da;
-
- /* Second stage */
- EADD (a, da, t1, t2);
- a = t1;
- da = t2;
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
-
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u8.d * c2)) == c2 + (cc2 + u8.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u7.d * c1)) == c1 + (cc1 + u7.d * c1))
- {
- retval = y;
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.506. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (VII) The case 0.787 < abs(x) <= 25, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -340,76 +186,20 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u10.d)) == gi - (t2 + gi * u10.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua10.d + t3 * ub10.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u9.d)) == fi + (t2 + fi * u9.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua9.d + t3 * ub9.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz)
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u12.d * c3)) == c3 + (cc3 + u12.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u11.d * c3)) == c3 + (cc3 + u11.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
-
- retval = tanMp (x);
- goto ret;
}
/* (---) The case 25 < abs(x) <= 1e8 */
@@ -443,14 +233,7 @@ __tan (double x)
sy = 1;
}
- /* (+++) The case 25 < abs(x) <= 1e8, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (VIII) The case 25 < abs(x) <= 1e8, 1e-7 < abs(y) <= 0.0608 */
+ /* (VIII) The case 25 < abs(x) <= 1e8, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -462,76 +245,26 @@ __tan (double x)
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u14.d * c)) == c + (dc + u14.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u13.d * a)) == a + (t2 + u13.d * a))
- {
- retval = y;
- goto ret;
- }
- }
-
- /* Second stage */
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u16.d * c2)) == c2 + (cc2 + u16.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u15.d * c1)) == c1 + (cc1 + u15.d * c1))
- {
- retval = (y);
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.506. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (IX) The case 25 < abs(x) <= 1e8, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -542,75 +275,20 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u18.d)) == gi - (t2 + gi * u18.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua18.d + t3 * ub18.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u17.d)) == fi + (t2 + fi * u17.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua17.d + t3 * ub17.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz);
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u20.d * c3)) == c3 + (cc3 + u20.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u19.d * c3)) == c3 + (cc3 + u19.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (---) The case 1e8 < abs(x) < 2**1024 */
@@ -632,14 +310,7 @@ __tan (double x)
sy = 1;
}
- /* (+++) The case 1e8 < abs(x) < 2**1024, abs(y) <= 1e-7 */
- if (ya <= gy1.d)
- {
- retval = tanMp (x);
- goto ret;
- }
-
- /* (X) The case 1e8 < abs(x) < 2**1024, 1e-7 < abs(y) <= 0.0608 */
+ /* (X) The case 1e8 < abs(x) < 2**1024, 0 < abs(y) <= 0.0608 */
if (ya <= gy2.d)
{
a2 = a * a;
@@ -650,85 +321,26 @@ __tan (double x)
t2 = da + a * a2 * t2;
if (n)
{
- /* First stage -cot */
+ /* -cot */
EADD (a, t2, b, db);
DIV2 (1.0, 0.0, b, db, c, dc, t1, t2, t3, t4);
- if ((y = c + (dc - u22.d * c)) == c + (dc + u22.d * c))
- {
- retval = (-y);
- goto ret;
- }
- }
- else
- {
- /* First stage tan */
- if ((y = a + (t2 - u21.d * a)) == a + (t2 + u21.d * a))
- {
- retval = y;
- goto ret;
- }
- }
-
- /* Second stage */
- /* Reduction by algorithm iv */
- p = 10;
- n = (__mpranred (x, &mpa, p)) & 0x00000001;
- __mp_dbl (&mpa, &a, p);
- __dbl_mp (a, &mpt1, p);
- __sub (&mpa, &mpt1, &mpt2, p);
- __mp_dbl (&mpt2, &da, p);
-
- MUL2 (a, da, a, da, x2, xx2, t1, t2);
-
- c1 = a25.d + x2 * a27.d;
- c1 = a23.d + x2 * c1;
- c1 = a21.d + x2 * c1;
- c1 = a19.d + x2 * c1;
- c1 = a17.d + x2 * c1;
- c1 = a15.d + x2 * c1;
- c1 *= x2;
-
- ADD2 (a13.d, aa13.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a11.d, aa11.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a9.d, aa9.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a7.d, aa7.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a5.d, aa5.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (x2, xx2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (a, da, c1, cc1, c2, cc2, t1, t2);
- ADD2 (a, da, c2, cc2, c1, cc1, t1, t2);
-
- if (n)
- {
- /* Second stage -cot */
- DIV2 (1.0, 0.0, c1, cc1, c2, cc2, t1, t2, t3, t4);
- if ((y = c2 + (cc2 - u24.d * c2)) == c2 + (cc2 + u24.d * c2))
- {
- retval = (-y);
- goto ret;
- }
+ y = c + dc;
+ retval = (-y);
+ /* Max ULP is 0.506. */
+ goto ret;
}
else
{
- /* Second stage tan */
- if ((y = c1 + (cc1 - u23.d * c1)) == c1 + (cc1 + u23.d * c1))
- {
- retval = y;
- goto ret;
- }
+ /* tan */
+ y = a + t2;
+ retval = y;
+ /* Max ULP is 0.507. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
}
/* (XI) The case 1e8 < abs(x) < 2**1024, 0.0608 < abs(y) <= 0.787 */
- /* First stage */
- i = ((int) (mfftnhf.d + TWO8 * ya));
+ i = ((int) (mfftnhf.d + 256 * ya));
z = (z0 = (ya - xfg[i][0].d)) + yya;
z2 = z * z;
pz = z + z * z2 * (e0.d + z2 * e1.d);
@@ -739,97 +351,25 @@ __tan (double x)
{
/* -cot */
t2 = pz * (fi + gi) / (fi + pz);
- if ((y = gi - (t2 - gi * u26.d)) == gi - (t2 + gi * u26.d))
- {
- retval = (-sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = gi * ua26.d + t3 * ub26.d;
- if ((y = gi - (t2 - t4)) == gi - (t2 + t4))
- {
- retval = (-sy * y);
- goto ret;
- }
+ y = gi - t2;
+ retval = (-sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
else
{
/* tan */
t2 = pz * (gi + fi) / (gi - pz);
- if ((y = fi + (t2 - fi * u25.d)) == fi + (t2 + fi * u25.d))
- {
- retval = (sy * y);
- goto ret;
- }
- t3 = (t2 < 0.0) ? -t2 : t2;
- t4 = fi * ua25.d + t3 * ub25.d;
- if ((y = fi + (t2 - t4)) == fi + (t2 + t4))
- {
- retval = (sy * y);
- goto ret;
- }
- }
-
- /* Second stage */
- ffi = xfg[i][3].d;
- EADD (z0, yya, z, zz);
- MUL2 (z, zz, z, zz, z2, zz2, t1, t2);
- c1 = z2 * (a7.d + z2 * (a9.d + z2 * a11.d));
- ADD2 (a5.d, aa5.d, c1, 0.0, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- ADD2 (a3.d, aa3.d, c1, cc1, c2, cc2, t1, t2);
- MUL2 (z2, zz2, c2, cc2, c1, cc1, t1, t2);
- MUL2 (z, zz, c1, cc1, c2, cc2, t1, t2);
- ADD2 (z, zz, c2, cc2, c1, cc1, t1, t2);
-
- ADD2 (fi, ffi, c1, cc1, c2, cc2, t1, t2);
- MUL2 (fi, ffi, c1, cc1, c3, cc3, t1, t2);
- SUB2 (1.0, 0.0, c3, cc3, c1, cc1, t1, t2);
-
- if (n)
- {
- /* -cot */
- DIV2 (c1, cc1, c2, cc2, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u28.d * c3)) == c3 + (cc3 + u28.d * c3))
- {
- retval = (-sy * y);
- goto ret;
- }
- }
- else
- {
- /* tan */
- DIV2 (c2, cc2, c1, cc1, c3, cc3, t1, t2, t3, t4);
- if ((y = c3 + (cc3 - u27.d * c3)) == c3 + (cc3 + u27.d * c3))
- {
- retval = (sy * y);
- goto ret;
- }
+ y = fi + t2;
+ retval = (sy * y);
+ /* Max ULP is 0.62. */
+ goto ret;
}
- retval = tanMp (x);
- goto ret;
ret:
return retval;
}
-/* multiple precision stage */
-/* Convert x to multi precision number,compute tan(x) by mptan() routine */
-/* and converts result back to double */
-static double
-SECTION
-tanMp (double x)
-{
- int p;
- double y;
- mp_no mpy;
- p = 32;
- __mptan (x, &mpy, p);
- __mp_dbl (&mpy, &y, p);
- LIBC_PROBE (slowtan, 2, &x, &y);
- return y;
-}
-
#ifndef __tan
libm_alias_double (__tan, tan)
#endif
diff --git a/sysdeps/ieee754/dbl-64/utan.h b/sysdeps/ieee754/dbl-64/utan.h
index 85f4861b1f5d578c309822279120f64f21ac5c05..37e4be87b314a21aacdf262d14fc3bf70e4af6c5 100644
--- a/sysdeps/ieee754/dbl-64/utan.h
+++ b/sysdeps/ieee754/dbl-64/utan.h
@@ -28,7 +28,7 @@
#define UTAN_H
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x3FD55555, 0x55555555} }, /* 0.333... */
/**/ d5 = {{0x3FC11111, 0x111107C6} }, /* 0.133... */
@@ -36,46 +36,6 @@
/**/ d9 = {{0x3F9664ED, 0x49CFC666} }, /* . */
/**/ d11 = {{0x3F82385A, 0x3CF2E4EA} }, /* . */
/* polynomial II */
-/**/ a3 = {{0x3fd55555, 0x55555555} }, /* 1/3 */
-/**/ aa3 = {{0x3c755555, 0x55555555} }, /* 1/3-a3 */
-/**/ a5 = {{0x3fc11111, 0x11111111} }, /* 2/15 */
-/**/ aa5 = {{0x3c411111, 0x11111111} }, /* 2/15-a5 */
-/**/ a7 = {{0x3faba1ba, 0x1ba1ba1c} }, /* 17/315 */
-/**/ aa7 = {{0xbc479179, 0x17917918} }, /* ()-a7 */
-/**/ a9 = {{0x3f9664f4, 0x882c10fa} }, /* 62/2835 */
-/**/ aa9 = {{0xbc09a528, 0x8b6c44fd} }, /* ()-a9 */
-/**/ a11 = {{0x3f8226e3, 0x55e6c23d} }, /* . */
-/**/ aa11 = {{0xbc2c292b, 0x8f1a2c13} }, /* . */
-/**/ a13 = {{0x3f6d6d3d, 0x0e157de0} }, /* . */
-/**/ aa13 = {{0xbc0280cf, 0xc968d971} }, /* . */
-/**/ a15 = {{0x3f57da36, 0x452b75e3} }, /* . */
-#if 0
-/**/ aa15 = {{0xbbf25789, 0xb285d2ed} }, /* . */
-#endif
-/**/ a17 = {{0x3f435582, 0x48036744} }, /* . */
-#if 0
-/**/ aa17 = {{0x3be488d9, 0x563f1f23} }, /* . */
-#endif
-/**/ a19 = {{0x3f2f57d7, 0x734d1664} }, /* . */
-#if 0
-/**/ aa19 = {{0x3bb0d55a, 0x913ccb50} }, /* . */
-#endif
-/**/ a21 = {{0x3f1967e1, 0x8afcafad} }, /* . */
-#if 0
-/**/ aa21 = {{0xbbbd7614, 0xa42d44e6} }, /* . */
-#endif
-/**/ a23 = {{0x3f0497d8, 0xeea25259} }, /* . */
-#if 0
-/**/ aa23 = {{0x3b99f2d0, 0x2e4d2863} }, /* . */
-#endif
-/**/ a25 = {{0x3ef0b132, 0xd39a6050} }, /* . */
-#if 0
-/**/ aa25 = {{0x3b93b274, 0xc2c19614} }, /* . */
-#endif
-/**/ a27 = {{0x3edb0f72, 0xd3ee24e9} }, /* . */
-#if 0
-/**/ aa27 = {{0x3b61688d, 0xdd595609} }, /* . */
-#endif
/* polynomial III */
/**/ e0 = {{0x3FD55555, 0x55554DBD} }, /* . */
/**/ e1 = {{0x3FC11112, 0xE0A6B45F} }, /* . */
@@ -88,52 +48,8 @@
/**/ g3 = {{0x3fe92f1a, 0x00000000} }, /* 0.787 */
/**/ g4 = {{0x40390000, 0x00000000} }, /* 25.0 */
/**/ g5 = {{0x4197d784, 0x00000000} }, /* 1e8 */
-/**/ gy1 = {{0x3e7ad7f2, 0x9abcaf48} }, /* 1e-7 */
/**/ gy2 = {{0x3faf212d, 0x00000000} }, /* 0.0608 */
-/**/ u1 = {{0x3cc8c33a, 0x00000000} }, /* 6.873e-16 */
-/**/ u2 = {{0x3983dc4d, 0x00000000} }, /* 1.224e-31 */
-/**/ u3 = {{0x3c78e14b, 0x00000000} }, /* 2.158e-17 */
-/**/ ua3 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub3 = {{0x3cc81898, 0x00000000} }, /* 6.688e-16 */
-/**/ u4 = {{0x399856c2, 0x00000000} }, /* 3e-31 */
-/**/ u5 = {{0x3c39d80a, 0x00000000} }, /* 1.401e-18 */
-/**/ u6 = {{0x3c374c5a, 0x00000000} }, /* 1.263e-18 */
-/**/ u7 = {{0x39903beb, 0x00000000} }, /* 2.001e-31 */
-/**/ u8 = {{0x399c56ae, 0x00000000} }, /* 3.493e-31 */
-/**/ u9 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua9 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub9 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u10 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua10 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub10 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u11 = {{0x39e509b6, 0x00000000} }, /* 8.298e-30 */
-/**/ u12 = {{0x39e509b6, 0x00000000} }, /* 8.298e-30 */
-/**/ u13 = {{0x3c39d80a, 0x00000000} }, /* 1.401e-18 */
-/**/ u14 = {{0x3c374c5a, 0x00000000} }, /* 1.263e-18 */
-/**/ u15 = {{0x3ab5767a, 0x00000000} }, /* 6.935e-26 */
-/**/ u16 = {{0x3ab57744, 0x00000000} }, /* 6.936e-26 */
-/**/ u17 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua17 = {{0x3bfdb11f, 0x00000000} }, /* 1.006e-19 */
-/**/ ub17 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u18 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua18 = {{0x3bfdb11f, 0x00000000} }, /* 1.006e-19 */
-/**/ ub18 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u19 = {{0x39a13b61, 0x00000000} }, /* 4.248e-31 */
-/**/ u20 = {{0x39a13b61, 0x00000000} }, /* 4.248e-31 */
-/**/ u21 = {{0x3c3bb9b8, 0x00000000} }, /* 1.503e-18 */
-/**/ u22 = {{0x3c392e08, 0x00000000} }, /* 1.365e-18 */
-/**/ u23 = {{0x3a0ce706, 0x00000000} }, /* 4.560e-29 */
-/**/ u24 = {{0x3a0cff5d, 0x00000000} }, /* 4.575e-29 */
-/**/ u25 = {{0x3c7d0ac7, 0x00000000} }, /* 2.519e-17 */
-/**/ ua25 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub25 = {{0x3ccc2375, 0x00000000} }, /* 7.810e-16 */
-/**/ u26 = {{0x3c7e40af, 0x00000000} }, /* 2.624e-17 */
-/**/ ua26 = {{0x3bfd8b58, 0x00000000} }, /* 1.001e-19 */
-/**/ ub26 = {{0x3ccc6405, 0x00000000} }, /* 7.880e-16 */
-/**/ u27 = {{0x3ad421cb, 0x00000000} }, /* 2.602e-25 */
-/**/ u28 = {{0x3ad421cb, 0x00000000} }, /* 2.602e-25 */
-
/**/ mp1 = {{0x3FF921FB, 0x58000000} },
/**/ mp2 = {{0xBE4DDE97, 0x3C000000} },
/**/ mp3 = {{0xBC8CB3B3, 0x99D747F2} },
@@ -145,7 +61,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0x3FD55555} }, /* 0.333... */
/**/ d5 = {{0x111107C6, 0x3FC11111} }, /* 0.133... */
@@ -153,46 +69,6 @@
/**/ d9 = {{0x49CFC666, 0x3F9664ED} }, /* . */
/**/ d11 = {{0x3CF2E4EA, 0x3F82385A} }, /* . */
/* polynomial II */
-/**/ a3 = {{0x55555555, 0x3fd55555} }, /* 1/3 */
-/**/ aa3 = {{0x55555555, 0x3c755555} }, /* 1/3-a3 */
-/**/ a5 = {{0x11111111, 0x3fc11111} }, /* 2/15 */
-/**/ aa5 = {{0x11111111, 0x3c411111} }, /* 2/15-a5 */
-/**/ a7 = {{0x1ba1ba1c, 0x3faba1ba} }, /* 17/315 */
-/**/ aa7 = {{0x17917918, 0xbc479179} }, /* ()-a7 */
-/**/ a9 = {{0x882c10fa, 0x3f9664f4} }, /* 62/2835 */
-/**/ aa9 = {{0x8b6c44fd, 0xbc09a528} }, /* ()-a9 */
-/**/ a11 = {{0x55e6c23d, 0x3f8226e3} }, /* . */
-/**/ aa11 = {{0x8f1a2c13, 0xbc2c292b} }, /* . */
-/**/ a13 = {{0x0e157de0, 0x3f6d6d3d} }, /* . */
-/**/ aa13 = {{0xc968d971, 0xbc0280cf} }, /* . */
-/**/ a15 = {{0x452b75e3, 0x3f57da36} }, /* . */
-#if 0
-/**/ aa15 = {{0xb285d2ed, 0xbbf25789} }, /* . */
-#endif
-/**/ a17 = {{0x48036744, 0x3f435582} }, /* . */
-#if 0
-/**/ aa17 = {{0x563f1f23, 0x3be488d9} }, /* . */
-#endif
-/**/ a19 = {{0x734d1664, 0x3f2f57d7} }, /* . */
-#if 0
-/**/ aa19 = {{0x913ccb50, 0x3bb0d55a} }, /* . */
-#endif
-/**/ a21 = {{0x8afcafad, 0x3f1967e1} }, /* . */
-#if 0
-/**/ aa21 = {{0xa42d44e6, 0xbbbd7614} }, /* . */
-#endif
-/**/ a23 = {{0xeea25259, 0x3f0497d8} }, /* . */
-#if 0
-/**/ aa23 = {{0x2e4d2863, 0x3b99f2d0} }, /* . */
-#endif
-/**/ a25 = {{0xd39a6050, 0x3ef0b132} }, /* . */
-#if 0
-/**/ aa25 = {{0xc2c19614, 0x3b93b274} }, /* . */
-#endif
-/**/ a27 = {{0xd3ee24e9, 0x3edb0f72} }, /* . */
-#if 0
-/**/ aa27 = {{0xdd595609, 0x3b61688d} }, /* . */
-#endif
/* polynomial III */
/**/ e0 = {{0x55554DBD, 0x3FD55555} }, /* . */
/**/ e1 = {{0xE0A6B45F, 0x3FC11112} }, /* . */
@@ -205,52 +81,8 @@
/**/ g3 = {{0x00000000, 0x3fe92f1a} }, /* 0.787 */
/**/ g4 = {{0x00000000, 0x40390000} }, /* 25.0 */
/**/ g5 = {{0x00000000, 0x4197d784} }, /* 1e8 */
-/**/ gy1 = {{0x9abcaf48, 0x3e7ad7f2} }, /* 1e-7 */
/**/ gy2 = {{0x00000000, 0x3faf212d} }, /* 0.0608 */
-/**/ u1 = {{0x00000000, 0x3cc8c33a} }, /* 6.873e-16 */
-/**/ u2 = {{0x00000000, 0x3983dc4d} }, /* 1.224e-31 */
-/**/ u3 = {{0x00000000, 0x3c78e14b} }, /* 2.158e-17 */
-/**/ ua3 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub3 = {{0x00000000, 0x3cc81898} }, /* 6.688e-16 */
-/**/ u4 = {{0x00000000, 0x399856c2} }, /* 3e-31 */
-/**/ u5 = {{0x00000000, 0x3c39d80a} }, /* 1.401e-18 */
-/**/ u6 = {{0x00000000, 0x3c374c5a} }, /* 1.263e-18 */
-/**/ u7 = {{0x00000000, 0x39903beb} }, /* 2.001e-31 */
-/**/ u8 = {{0x00000000, 0x399c56ae} }, /* 3.493e-31 */
-/**/ u9 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua9 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub9 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u10 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua10 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub10 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u11 = {{0x00000000, 0x39e509b6} }, /* 8.298e-30 */
-/**/ u12 = {{0x00000000, 0x39e509b6} }, /* 8.298e-30 */
-/**/ u13 = {{0x00000000, 0x3c39d80a} }, /* 1.401e-18 */
-/**/ u14 = {{0x00000000, 0x3c374c5a} }, /* 1.263e-18 */
-/**/ u15 = {{0x00000000, 0x3ab5767a} }, /* 6.935e-26 */
-/**/ u16 = {{0x00000000, 0x3ab57744} }, /* 6.936e-26 */
-/**/ u17 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua17 = {{0x00000000, 0x3bfdb11f} }, /* 1.006e-19 */
-/**/ ub17 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u18 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua18 = {{0x00000000, 0x3bfdb11f} }, /* 1.006e-19 */
-/**/ ub18 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u19 = {{0x00000000, 0x39a13b61} }, /* 4.248e-31 */
-/**/ u20 = {{0x00000000, 0x39a13b61} }, /* 4.248e-31 */
-/**/ u21 = {{0x00000000, 0x3c3bb9b8} }, /* 1.503e-18 */
-/**/ u22 = {{0x00000000, 0x3c392e08} }, /* 1.365e-18 */
-/**/ u23 = {{0x00000000, 0x3a0ce706} }, /* 4.560e-29 */
-/**/ u24 = {{0x00000000, 0x3a0cff5d} }, /* 4.575e-29 */
-/**/ u25 = {{0x00000000, 0x3c7d0ac7} }, /* 2.519e-17 */
-/**/ ua25 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub25 = {{0x00000000, 0x3ccc2375} }, /* 7.810e-16 */
-/**/ u26 = {{0x00000000, 0x3c7e40af} }, /* 2.624e-17 */
-/**/ ua26 = {{0x00000000, 0x3bfd8b58} }, /* 1.001e-19 */
-/**/ ub26 = {{0x00000000, 0x3ccc6405} }, /* 7.880e-16 */
-/**/ u27 = {{0x00000000, 0x3ad421cb} }, /* 2.602e-25 */
-/**/ u28 = {{0x00000000, 0x3ad421cb} }, /* 2.602e-25 */
-
/**/ mp1 = {{0x58000000, 0x3FF921FB} },
/**/ mp2 = {{0x3C000000, 0xBE4DDE97} },
/**/ mp3 = {{0x99D747F2, 0xBC8CB3B3} },
diff --git a/sysdeps/ieee754/dbl-64/utan.tbl b/sysdeps/ieee754/dbl-64/utan.tbl
index 325abdeab300027e99c9160c8d083c6840c73b0b..f05e67f48847b740f2aac2ac3c36ff1265fe9d2f 100644
--- a/sysdeps/ieee754/dbl-64/utan.tbl
+++ b/sysdeps/ieee754/dbl-64/utan.tbl
@@ -23,7 +23,7 @@
#ifdef BIG_ENDI
-static const number
+static const mynumber
xfg[186][4] = { /* xi,Fi,Gi,FFi, i=16..201 */
/**/ {{{0x3fb00000, 0x1e519d60} },
/**/ {{0x3fb00557, 0x96c4e240} },
@@ -773,7 +773,7 @@ static const number
#else
#ifdef LITTLE_ENDI
-static const number
+static const mynumber
xfg[186][4] = { /* xi,Fi,Gi,FFi, i=16..201 */
/**/ {{{0x1e519d60, 0x3fb00000} },
/**/ {{0x96c4e240, 0x3fb00557} },
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH v2 3/5] Remove slow paths from atan
2021-01-04 12:19 ` [PATCH 3/5] Remove slow paths from atan Wilco Dijkstra via Libc-alpha
2021-01-04 16:43 ` Paul Zimmermann
@ 2021-01-07 19:26 ` Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 19:26 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths from atan. Add ULP annotations.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/atnat.h b/sysdeps/ieee754/dbl-64/atnat.h
index af4374019f1fce9b56b387b93a69d1330b9c8666..2b12481f0491ade9379c71ae98935ae8e67cb755 100644
--- a/sysdeps/ieee754/dbl-64/atnat.h
+++ b/sysdeps/ieee754/dbl-64/atnat.h
@@ -29,7 +29,7 @@
#define M 4
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@@ -79,7 +79,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */
diff --git a/sysdeps/ieee754/dbl-64/s_atan.c b/sysdeps/ieee754/dbl-64/s_atan.c
index 291b988318a67444b473044b030619d2f7851585..977183eb928f62d110e2c9e9ca6dd1f9286b1b97 100644
--- a/sysdeps/ieee754/dbl-64/s_atan.c
+++ b/sysdeps/ieee754/dbl-64/s_atan.c
@@ -20,25 +20,15 @@
/* MODULE_NAME: atnat.c */
/* */
/* FUNCTIONS: uatan */
-/* atanMp */
/* signArctan */
/* */
-/* */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat.h */
-/* mpatan.c mpatan2.c mpsqrt.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h atnat.h */
/* uatan.tbl */
/* */
-/* An ultimate atan() routine. Given an IEEE double machine number x */
-/* it computes the correctly rounded (to nearest) value of atan(x). */
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/************************************************************************/
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
+#include "mydefs.h"
#include "uatan.tbl"
#include "atnat.h"
#include <fenv.h>
@@ -47,10 +37,8 @@
#include <math.h>
#include <fenv_private.h>
#include <math-underflow.h>
-#include <stap-probe.h>
-void __mpatan (mp_no *, mp_no *, int); /* see definition in mpatan.c */
-static double atanMp (double, const int[]);
+#define TWO52 0x1.0p52
/* Fix the sign of y and return */
static double
@@ -59,17 +47,14 @@ __signArctan (double x, double y)
return copysign (y, x);
}
-
-/* An ultimate atan() routine. Given an IEEE double machine number x, */
-/* routine computes the correctly rounded (to nearest) value of atan(x). */
+/* atan with max ULP of ~0.523 based on random sampling. */
double
__atan (double x)
{
- double cor, s1, ss1, s2, ss2, t1, t2, t3, t4, u, u2, u3,
- v, vv, w, ww, y, yy, z, zz;
+ double cor, t1, t2, t3, u,
+ v, w, ww, y, yy, z;
int i, ux, dx;
- static const int pr[M] = { 6, 8, 10, 32 };
- number num;
+ mynumber num;
num.d = x;
ux = num.i[HIGH_HALF];
@@ -102,36 +87,14 @@ __atan (double x)
yy = d3.d + v * yy;
yy *= x * v;
- if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
- return y;
-
- EMULV (x, x, v, vv); /* v+vv=x^2 */
-
- s1 = f17.d + v * f19.d;
- s1 = f15.d + v * s1;
- s1 = f13.d + v * s1;
- s1 = f11.d + v * s1;
- s1 *= v;
-
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2);
- ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
- if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
- return y;
-
- return atanMp (x, pr);
+ y = x + yy;
+ /* Max ULP is 0.511. */
+ return y;
}
}
else
{ /* B <= u < C */
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
z = u - cij[i][0].d;
yy = cij[i][5].d + z * cij[i][6].d;
@@ -141,44 +104,9 @@ __atan (double x)
yy *= z;
t1 = cij[i][1].d;
- if (i < 112)
- {
- if (i < 48)
- u2 = U21; /* u < 1/4 */
- else
- u2 = U22;
- } /* 1/4 <= u < 1/2 */
- else
- {
- if (i < 176)
- u2 = U23; /* 1/2 <= u < 3/4 */
- else
- u2 = U24;
- } /* 3/4 <= u <= 1 */
- if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
- return __signArctan (x, y);
-
- z = u - hij[i][0].d;
-
- s1 = hij[i][14].d + z * hij[i][15].d;
- s1 = hij[i][13].d + z * s1;
- s1 = hij[i][12].d + z * s1;
- s1 = hij[i][11].d + z * s1;
- s1 *= z;
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t1 + yy;
+ /* Max ULP is 0.56. */
+ return __signArctan (x, y);
}
}
else
@@ -188,7 +116,7 @@ __atan (double x)
w = 1 / u;
EMULV (w, u, t1, t2);
ww = w * ((1 - t1) - t2);
- i = (TWO52 + TWO8 * w) - TWO52;
+ i = (TWO52 + 256 * w) - TWO52;
i -= 16;
z = (w - cij[i][0].d) + ww;
@@ -199,37 +127,9 @@ __atan (double x)
yy = HPI1 - z * yy;
t1 = HPI - cij[i][1].d;
- if (i < 112)
- u3 = U31; /* w < 1/2 */
- else
- u3 = U32; /* w >= 1/2 */
- if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
- return __signArctan (x, y);
-
- DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
- t1 = w - hij[i][0].d;
- EADD (t1, ww, z, zz);
-
- s1 = hij[i][14].d + z * hij[i][15].d;
- s1 = hij[i][13].d + z * s1;
- s1 = hij[i][12].d + z * s1;
- s1 = hij[i][11].d + z * s1;
- s1 *= z;
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
- if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t1 + yy;
+ /* Max ULP is 0.503. */
+ return __signArctan (x, y);
}
else
{
@@ -249,34 +149,9 @@ __atan (double x)
ww = w * ((1 - t1) - t2);
ESUB (HPI, w, t3, cor);
yy = ((HPI1 + cor) - ww) - yy;
- if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
- return __signArctan (x, y);
-
- DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4);
- MUL2 (w, ww, w, ww, v, vv, t1, t2);
-
- s1 = f17.d + v * f19.d;
- s1 = f15.d + v * s1;
- s1 = f13.d + v * s1;
- s1 = f11.d + v * s1;
- s1 *= v;
-
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2);
- ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
- SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
-
- if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
- return __signArctan (x, y);
-
- return atanMp (x, pr);
+ y = t3 + yy;
+ /* Max ULP is 0.5003. */
+ return __signArctan (x, y);
}
else
{
@@ -290,35 +165,6 @@ __atan (double x)
}
}
- /* Final stages. Compute atan(x) by multiple precision arithmetic */
-static double
-atanMp (double x, const int pr[])
-{
- mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
- double y1, y2;
- int i, p;
-
- for (i = 0; i < M; i++)
- {
- p = pr[i];
- __dbl_mp (x, &mpx, p);
- __mpatan (&mpx, &mpy, p);
- __dbl_mp (u9[i].d, &mpt1, p);
- __mul (&mpy, &mpt1, &mperr, p);
- __add (&mpy, &mperr, &mpy1, p);
- __sub (&mpy, &mperr, &mpy2, p);
- __mp_dbl (&mpy1, &y1, p);
- __mp_dbl (&mpy2, &y2, p);
- if (y1 == y2)
- {
- LIBC_PROBE (slowatan, 3, &p, &x, &y1);
- return y1;
- }
- }
- LIBC_PROBE (slowatan_inexact, 3, &p, &x, &y1);
- return y1; /*if impossible to do exact computing */
-}
-
#ifndef __atan
libm_alias_double (__atan, atan)
#endif
diff --git a/sysdeps/ieee754/dbl-64/uatan.tbl b/sysdeps/ieee754/dbl-64/uatan.tbl
index 8a477caa385d3f447abacf8490a8b45278d86610..fdc3e53304112eb31863a3144d6acb2f65a77a2e 100644
--- a/sysdeps/ieee754/dbl-64/uatan.tbl
+++ b/sysdeps/ieee754/dbl-64/uatan.tbl
@@ -25,7 +25,7 @@
#ifdef BIG_ENDI
- static const number
+ static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X3FB04006, 0X65E0244E} },
/**/ {{0X3FB03A73, 0X7B53DD20} },
@@ -1716,7 +1716,7 @@
/**/ {{0XBF99B9A7, 0X18A3BA58} } },
};
- static const number
+ static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x3fb04000, 0x00000000} },
/**/ {{0x3fb03a6d, 0x1c06693d} },
@@ -5579,7 +5579,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
cij[241][7] = { /* x0,cij for (1/16,1) */
/**/ {{{0X65E0244E, 0X3FB04006} },
/**/ {{0X7B53DD20, 0X3FB03A73} },
@@ -7270,7 +7270,7 @@
/**/ {{0X18A3BA58, 0XBF99B9A7} } },
};
- static const number
+ static const mynumber
hij[241][16] = { /* x0,hij for (1/16,1) */
/**/ {{{0x00000000, 0x3fb04000} },
/**/ {{0x1c06693d, 0x3fb03a6d} },
^ permalink raw reply related [flat|nested] 19+ messages in thread
* [PATCH v2 4/5] Remove slow paths from atan2
2021-01-04 12:21 ` [PATCH 4/5] Remove slow paths from atan2 Wilco Dijkstra via Libc-alpha
2021-01-04 16:49 ` Paul Zimmermann
@ 2021-01-07 19:28 ` Wilco Dijkstra via Libc-alpha
1 sibling, 0 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 19:28 UTC (permalink / raw
To: 'GNU C Library'
Remove slow paths from atan2. Add ULP annotations.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/atnat2.h b/sysdeps/ieee754/dbl-64/atnat2.h
index f1c0d23f9ecd3084a79a9dc18f6550153eba06db..f9f8debd55d85e77163a1bd3ebbf339ce0ac3bc7 100644
--- a/sysdeps/ieee754/dbl-64/atnat2.h
+++ b/sysdeps/ieee754/dbl-64/atnat2.h
@@ -34,7 +34,7 @@
#define MM 5
#ifdef BIG_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0xbfd55555, 0x55555555} }, /* -0.333... */
/**/ d5 = {{0x3fc99999, 0x999997fd} }, /* 0.199... */
@@ -96,7 +96,7 @@
#else
#ifdef LITTLE_ENDI
- static const number
+ static const mynumber
/* polynomial I */
/**/ d3 = {{0x55555555, 0xbfd55555} }, /* -0.333... */
/**/ d5 = {{0x999997fd, 0x3fc99999} }, /* 0.199... */
diff --git a/sysdeps/ieee754/dbl-64/e_atan2.c b/sysdeps/ieee754/dbl-64/e_atan2.c
index e6b98142fbbc7c6bd4a51df0257b361fa46181ec..b77ad2dc6b9561e6396a76db11f6db686f170bb1 100644
--- a/sysdeps/ieee754/dbl-64/e_atan2.c
+++ b/sysdeps/ieee754/dbl-64/e_atan2.c
@@ -20,25 +20,14 @@
/* MODULE_NAME: atnat2.c */
/* */
/* FUNCTIONS: uatan2 */
-/* atan2Mp */
/* signArctan2 */
-/* normalized */
/* */
-/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h */
-/* mpatan.c mpatan2.c mpsqrt.c */
+/* FILES NEEDED: dla.h endian.h mydefs.h atnat2.h */
/* uatan.tbl */
/* */
-/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
-/* x it computes the correctly rounded (to nearest) value of atan2(y,x).*/
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/* */
/************************************************************************/
#include <dla.h>
-#include "mpa.h"
-#include "MathLib.h"
#include "mydefs.h"
#include "uatan.tbl"
#include "atnat2.h"
@@ -48,20 +37,15 @@
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
-#include <stap-probe.h>
#include <libm-alias-finite.h>
#ifndef SECTION
# define SECTION
#endif
-/************************************************************************/
-/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
-/* it computes the correctly rounded (to nearest) value of atan2(y,x). */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round to nearest mode of IEEE 754 standard. */
-/************************************************************************/
-static double atan2Mp (double, double, const int[]);
+#define TWO52 0x1.0p52
+#define TWOM1022 0x1.0p-1022
+
/* Fix the sign and return after stage 1 or stage 2 */
static double
signArctan2 (double y, double z)
@@ -69,18 +53,15 @@ signArctan2 (double y, double z)
return copysign (z, y);
}
-static double normalized (double, double, double, double);
-void __mpatan2 (mp_no *, mp_no *, mp_no *, int);
-
+/* atan2 with max ULP of ~0.524 based on random sampling. */
double
SECTION
__ieee754_atan2 (double y, double x)
{
int i, de, ux, dx, uy, dy;
- static const int pr[MM] = { 6, 8, 10, 20, 32 };
- double ax, ay, u, du, u9, ua, v, vv, dv, t1, t2, t3,
- z, zz, cor, s1, ss1, s2, ss2;
- number num;
+ double ax, ay, u, du, v, vv, dv, t1, t2, t3,
+ z, zz, cor;
+ mynumber num;
static const int ep = 59768832, /* 57*16**5 */
em = -59768832; /* -57*16**5 */
@@ -208,10 +189,8 @@ __ieee754_atan2 (double y, double x)
if (x > 0)
{
double ret;
- if ((z = ay / ax) < TWOM1022)
- ret = normalized (ax, ay, y, z);
- else
- ret = signArctan2 (y, z);
+ z = ay / ax;
+ ret = signArctan2 (y, z);
if (fabs (ret) < DBL_MIN)
{
double vret = ret ? ret : DBL_MIN;
@@ -270,30 +249,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d
+ v * d13.d)))));
- if ((z = u + (zz - u1.d * u)) == u + (zz + u1.d * u))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d + v * (f13.d
- + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - u5.d * s1)) == s1 + (ss1 + u5.d * s1))
- return signArctan2 (y, z);
-
- return atan2Mp (x, y, pr);
+ z = u + zz;
+ /* Max ULP is 0.504. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
t3 = u - cij[i][0].d;
EADD (t3, du, v, dv);
@@ -304,43 +265,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
- if (i < 112)
- {
- if (i < 48)
- u9 = u91.d; /* u < 1/4 */
- else
- u9 = u92.d;
- } /* 1/4 <= u < 1/2 */
- else
- {
- if (i < 176)
- u9 = u93.d; /* 1/2 <= u < 3/4 */
- else
- u9 = u94.d;
- } /* 3/4 <= u <= 1 */
- if ((z = t1 + (zz - u9 * t1)) == t1 + (zz + u9 * t1))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - ub.d * s2)) == s2 + (ss2 + ub.d * s2))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.56. */
+ return signArctan2 (y, z);
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
@@ -355,31 +282,12 @@ __ieee754_atan2 (double y, double x)
+ v * d13.d)))));
ESUB (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) - du) - zz;
- if ((z = t2 + (t3 - u2.d)) == t2 + (t3 + u2.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d
- + v * (f13.d
- + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- SUB2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u6.d)) == s2 + (ss2 + u6.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
@@ -389,36 +297,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d - cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
-
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.503. */
+ return signArctan2 (y, z);
}
/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
@@ -434,30 +315,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d11.d + v * d13.d)))));
EADD (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) + du) + zz;
- if ((z = t2 + (t3 - u3.d)) == t2 + (t3 + u3.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d
- + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u7.d)) == s2 + (ss2 + u7.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d + v * (cij[i][2].d
@@ -466,34 +329,9 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d + cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
- EADD (t1, du, v, vv);
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d
- + v * hij[i][15].d))));
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- ADD2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.503. */
+ return signArctan2 (y, z);
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
@@ -506,29 +344,12 @@ __ieee754_atan2 (double y, double x)
+ v * (d9.d + v * (d11.d + v * d13.d)))));
ESUB (opi.d, u, t2, cor);
t3 = ((opi1.d + cor) - du) - zz;
- if ((z = t2 + (t3 - u4.d)) == t2 + (t3 + u4.d))
- return signArctan2 (y, z);
-
- MUL2 (u, du, u, du, v, vv, t1, t2);
- s1 = v * (f11.d + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
- ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- MUL2 (u, du, s1, ss1, s2, ss2, t1, t2);
- ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
- SUB2 (opi.d, opi1.d, s1, ss1, s2, ss2, t1, t2);
-
- if ((z = s2 + (ss2 - u8.d)) == s2 + (ss2 + u8.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t2 + t3;
+ /* Max ULP is 0.501. */
+ return signArctan2 (y, z);
}
- i = (TWO52 + TWO8 * u) - TWO52;
+ i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = opi1.d - v * (cij[i][2].d
@@ -536,86 +357,11 @@ __ieee754_atan2 (double y, double x)
+ v * (cij[i][4].d
+ v * (cij[i][5].d + v * cij[i][6].d))));
t1 = opi.d - cij[i][1].d;
- if (i < 112)
- ua = ua1.d; /* w < 1/2 */
- else
- ua = ua2.d; /* w >= 1/2 */
- if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
- return signArctan2 (y, z);
-
- t1 = u - hij[i][0].d;
-
- EADD (t1, du, v, vv);
-
- s1 = v * (hij[i][11].d
- + v * (hij[i][12].d
- + v * (hij[i][13].d
- + v * (hij[i][14].d + v * hij[i][15].d))));
-
- ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
- MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2);
- ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
- SUB2 (opi.d, opi1.d, s2, ss2, s1, ss1, t1, t2);
-
- if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
- return signArctan2 (y, z);
- return atan2Mp (x, y, pr);
+ z = t1 + zz;
+ /* Max ULP is 0.502. */
+ return signArctan2 (y, z);
}
#ifndef __ieee754_atan2
libm_alias_finite (__ieee754_atan2, __atan2)
#endif
-
-/* Treat the Denormalized case */
-static double
-SECTION
-normalized (double ax, double ay, double y, double z)
-{
- int p;
- mp_no mpx, mpy, mpz, mperr, mpz2, mpt1;
- p = 6;
- __dbl_mp (ax, &mpx, p);
- __dbl_mp (ay, &mpy, p);
- __dvd (&mpy, &mpx, &mpz, p);
- __dbl_mp (ue.d, &mpt1, p);
- __mul (&mpz, &mpt1, &mperr, p);
- __sub (&mpz, &mperr, &mpz2, p);
- __mp_dbl (&mpz2, &z, p);
- return signArctan2 (y, z);
-}
-
-/* Stage 3: Perform a multi-Precision computation */
-static double
-SECTION
-atan2Mp (double x, double y, const int pr[])
-{
- double z1, z2;
- int i, p;
- mp_no mpx, mpy, mpz, mpz1, mpz2, mperr, mpt1;
- for (i = 0; i < MM; i++)
- {
- p = pr[i];
- __dbl_mp (x, &mpx, p);
- __dbl_mp (y, &mpy, p);
- __mpatan2 (&mpy, &mpx, &mpz, p);
- __dbl_mp (ud[i].d, &mpt1, p);
- __mul (&mpz, &mpt1, &mperr, p);
- __add (&mpz, &mperr, &mpz1, p);
- __sub (&mpz, &mperr, &mpz2, p);
- __mp_dbl (&mpz1, &z1, p);
- __mp_dbl (&mpz2, &z2, p);
- if (z1 == z2)
- {
- LIBC_PROBE (slowatan2, 4, &p, &x, &y, &z1);
- return z1;
- }
- }
- LIBC_PROBE (slowatan2_inexact, 4, &p, &x, &y, &z1);
- return z1; /*if impossible to do exact computing */
-}
^ permalink raw reply related [flat|nested] 19+ messages in thread
* Re: [PATCH 1/5] Remove slow paths from asin
2021-01-04 17:41 ` Paul Zimmermann
@ 2021-01-07 19:36 ` Wilco Dijkstra via Libc-alpha
2021-01-08 10:29 ` Paul Zimmermann
0 siblings, 1 reply; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 19:36 UTC (permalink / raw
To: Paul Zimmermann, Szabolcs Nagy; +Cc: libc-alpha@sourceware.org
Hi Paul,
Thanks for the reviews, I have updated the patches accordingly.
> the patch you mention (commit f67f9c9) was already entitled
> "ieee754: Remove slow paths from asin and acos". This was confusing.
> I believe it did only remove the "very slow" paths.
I've updated the title in v2.
> here are the claimed bounds and the maximal errors I found so far after
> applying Wilco's patches (rounded up to 3 digits):
>
> claimed found so far
> acos 0.52 0.523
> asin 0.523 0.516
> atan 0.56 0.523
> tan 0.62 0.612
> atan2 0.56 0.524
Thanks for that, I have added these bounds at the start of each function.
> I have only one further remark:
>
> -typedef union { int4 i[2]; double x; } mynumber;
> +typedef union { int4 i[2]; double x; double d; } mynumber;
>
> Why did you add another 'double' field d? Only the 'd' field is used in
> e_atan2.c, why not replace it by 'x'? Maybe to avoid a too large diff?
Yes, the purpose is just to remove the slow paths and minimise the
changes, so this was the quickest and simplest way to remove the uses
of 'number'.
Cheers,
Wilco
^ permalink raw reply [flat|nested] 19+ messages in thread
* [PATCH v2 5/5] Remove mpa files
2021-01-04 12:25 ` [PATCH 5/5] Remove mpa files Wilco Dijkstra via Libc-alpha
@ 2021-01-07 20:12 ` Wilco Dijkstra via Libc-alpha
0 siblings, 0 replies; 19+ messages in thread
From: Wilco Dijkstra via Libc-alpha @ 2021-01-07 20:12 UTC (permalink / raw
To: 'GNU C Library'
Finally remove all mpa related files, headers, declarations, probes, unused tables and
update makefiles.
Passes GLIBC testsuite and buildmanyglibc.
---
diff --git a/manual/probes.texi b/manual/probes.texi
index 61254bc43018d0618c17776603c95c2f8d489678..4aae76b81921f3474e381b2d0c982e9bec1c6380 100644
--- a/manual/probes.texi
+++ b/manual/probes.texi
@@ -16,7 +16,6 @@ arguments.
@menu
* Memory Allocation Probes:: Probes in the memory allocation subsystem
-* Mathematical Function Probes:: Probes in mathematical functions
* Non-local Goto Probes:: Probes in setjmp and longjmp
@end menu
@@ -255,90 +254,6 @@ is the memory location as passed to @code{free}, Argument @var{$arg2}
is the tcache bin it resides in.
@end deftp
-@node Mathematical Function Probes
-@section Mathematical Function Probes
-
-Some mathematical functions fall back to multiple precision arithmetic for
-some inputs to get last bit precision for their return values. This multiple
-precision fallback is much slower than the default algorithms and may have a
-significant impact on application performance. The systemtap probe markers
-described in this section may help you determine if your application calls
-mathematical functions with inputs that may result in multiple-precision
-arithmetic.
-
-Unless explicitly mentioned otherwise, a precision of 1 implies 24 bits of
-precision in the mantissa of the multiple precision number. Hence, a precision
-level of 32 implies 768 bits of precision in the mantissa.
-
-@deftp Probe slowatan2 (int @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
-This probe is triggered when the @code{atan2} function is called with
-an input that results in multiple precision computation. Argument
-@var{$arg1} is the precision with which computation succeeded.
-Arguments @var{$arg2} and @var{$arg3} are inputs to the @code{atan2}
-function and @var{$arg4} is the computed result.
-@end deftp
-
-@deftp Probe slowatan2_inexact (int @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation and none of
-the multiple precision computations result in an accurate result.
-Argument @var{$arg1} is the maximum precision with which computations
-were performed. Arguments @var{$arg2} and @var{$arg3} are inputs to
-the @code{atan2} function and @var{$arg4} is the computed result.
-@end deftp
-
-@deftp Probe slowatan (int @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation. Argument
-@var{$arg1} is the precision with which computation succeeded.
-Argument @var{$arg2} is the input to the @code{atan} function and
-@var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowatan_inexact (int @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{atan} function is called with
-an input that results in multiple precision computation and none of
-the multiple precision computations result in an accurate result.
-Argument @var{$arg1} is the maximum precision with which computations
-were performed. Argument @var{$arg2} is the input to the @code{atan}
-function and @var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowtan (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{tan} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowsin (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{sin} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowcos (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{cos} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function and @var{$arg2}
-is the computed result.
-@end deftp
-
-@deftp Probe slowsin_dx (double @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{sin} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function, @var{$arg2} is
-the error bound of @var{$arg1} and @var{$arg3} is the computed result.
-@end deftp
-
-@deftp Probe slowcos_dx (double @var{$arg1}, double @var{$arg2}, double @var{$arg3})
-This probe is triggered when the @code{cos} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input to the function, @var{$arg2} is
-the error bound of @var{$arg1} and @var{$arg3} is the computed result.
-@end deftp
-
@node Non-local Goto Probes
@section Non-local Goto Probes
diff --git a/math/Makefile b/math/Makefile
index 687aa5d510bb9e3611367ff0059c58551c30ce01..d6aad43937e49216df656c36c76ec26da364a537 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -126,8 +126,7 @@ type-ldouble-yes := ldouble
# double support
type-double-suffix :=
-type-double-routines := branred doasin dosincos mpa mpatan2 \
- k_rem_pio2 mpatan mpsqrt mptan sincos32 \
+type-double-routines := branred k_rem_pio2 \
sincostab math_err e_exp_data e_log_data \
e_log2_data e_pow_log_data
diff --git a/sysdeps/generic/math_private.h b/sysdeps/generic/math_private.h
index 10a6fde28dbcbfe55148c744e13515b7c9d0707c..98b54875f51304f966db6d7b304b12073f9b2b2f 100644
--- a/sysdeps/generic/math_private.h
+++ b/sysdeps/generic/math_private.h
@@ -199,11 +199,5 @@ do { \
extern double __sin (double __x);
extern double __cos (double __x);
extern int __branred (double __x, double *__a, double *__aa);
-extern void __doasin (double __x, double __dx, double __v[]);
-extern void __dubsin (double __x, double __dx, double __v[]);
-extern void __dubcos (double __x, double __dx, double __v[]);
-extern double __mpsin (double __x, double __dx, bool __range_reduce);
-extern double __mpcos (double __x, double __dx, bool __range_reduce);
-extern void __docos (double __x, double __dx, double __v[]);
#endif /* _MATH_PRIVATE_H_ */
diff --git a/sysdeps/ieee754/dbl-64/MathLib.h b/sysdeps/ieee754/dbl-64/MathLib.h
deleted file mode 100644
index 4bc012163bc63f53227d21c6753adfb2dc238ed3..0000000000000000000000000000000000000000
--- a/sysdeps/ieee754/dbl-64/MathLib.h
+++ /dev/null
@@ -1,100 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * Written by International Business Machines Corp.
- * Copyright (C) 2001-2021 Free Software Foundation, Inc.
- *
- * This program is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Lesser General Public License as published by
- * the Free Software Foundation; either version 2.1 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public License
- * along with this program; if not, see <https://www.gnu.org/licenses/>.
- */
-
-/********************************************************************/
-/* Ultimate math functions. Each function computes the exact */
-/* theoretical value of its argument rounded to nearest or even. */
-/* */
-/* Assumption: Machine arithmetic operations are performed in */
-/* round nearest mode of IEEE 754 standard. */
-/********************************************************************/
-
-#ifndef UMATH_LIB
-#define UMATH_LIB
-/********************************************************************/
-/* Function changes the precision mode to IEEE 754 double precision */
-/* and the rounding mode to nearest or even. */
-/* It returns the original status of these modes. */
-/* See further explanations of usage in DPChange.h */
-/********************************************************************/
-unsigned short Init_Lib (void);
-
-/********************************************************************/
-/* Function that changes the precision and rounding modes to the */
-/* specified by the argument received. See further explanations in */
-/* DPChange.h */
-/********************************************************************/
-void Exit_Lib (unsigned short);
-
-
-/* The asin() function calculates the arc sine of its argument. */
-/* The function returns the arc sine in radians */
-/* (between -PI/2 and PI/2). */
-/* If the argument is greater than 1 or less than -1 it returns */
-/* a NaN. */
-double uasin (double);
-
-
-/* The acos() function calculates the arc cosine of its argument. */
-/* The function returns the arc cosine in radians */
-/* (between -PI/2 and PI/2). */
-/* If the argument is greater than 1 or less than -1 it returns */
-/* a NaN. */
-double uacos (double);
-
-/* The atan() function calculates the arctanget of its argument. */
-/* The function returns the arc tangent in radians */
-/* (between -PI/2 and PI/2). */
-double uatan (double);
-
-
-/* The uatan2() function calculates the arc tangent of the two arguments x */
-/* and y (x is the right argument and y is the left one).The signs of both */
-/* arguments are used to determine the quadrant of the result. */
-/* The function returns the result in radians, which is between -PI and PI */
-double uatan2 (double, double);
-
-/* Compute log(x). The base of log is e (natural logarithm) */
-double ulog (double);
-
-/* Compute e raised to the power of argument x. */
-double uexp (double);
-
-/* Compute sin(x). The argument x is assumed to be given in radians.*/
-double usin (double);
-
-/* Compute cos(x). The argument x is assumed to be given in radians.*/
-double ucos (double);
-
-/* Compute tan(x). The argument x is assumed to be given in radians.*/
-double utan (double);
-
-/* Compute the square root of non-negative argument x. */
-/* If x is negative the returned value is NaN. */
-double usqrt (double);
-
-/* Compute x raised to the power of y, where x is the left argument */
-/* and y is the right argument. The function returns a NaN if x<0. */
-/* If x equals zero it returns -inf */
-double upow (double, double);
-
-/* Computing x mod y, where x is the left argument and y is the */
-/* right one. */
-double uremainder (double, double);
-#endif
diff --git a/sysdeps/ieee754/dbl-64/e_remainder.c b/sysdeps/ieee754/dbl-64/e_remainder.c
index d7028a60b3867cb3a93f472df02a6461af2ba143..d076a37168e9a0a788986490a0f170ddc49b3e5c 100644
--- a/sysdeps/ieee754/dbl-64/e_remainder.c
+++ b/sysdeps/ieee754/dbl-64/e_remainder.c
@@ -32,7 +32,6 @@
#include "endian.h"
#include "mydefs.h"
#include "urem.h"
-#include "MathLib.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
diff --git a/sysdeps/ieee754/dbl-64/e_sqrt.c b/sysdeps/ieee754/dbl-64/e_sqrt.c
index 2ff1bf257204e8cff0f11fe6e24617f57eb4cf08..b3e63861c6ccae582039360e9a5766e86fa92d71 100644
--- a/sysdeps/ieee754/dbl-64/e_sqrt.c
+++ b/sysdeps/ieee754/dbl-64/e_sqrt.c
@@ -35,7 +35,6 @@
#include "endian.h"
#include "mydefs.h"
#include <dla.h>
-#include "MathLib.h"
#include "root.tbl"
#include <math-barriers.h>
#include <math_private.h>
diff --git a/sysdeps/ieee754/dbl-64/s_sin.c b/sysdeps/ieee754/dbl-64/s_sin.c
index 13c08227201e304807c6b1515f27375d2a2c9af2..7d89e3dfc271bfc21d5ac2a2a38b133c532579f4 100644
--- a/sysdeps/ieee754/dbl-64/s_sin.c
+++ b/sysdeps/ieee754/dbl-64/s_sin.c
@@ -38,7 +38,6 @@
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
-#include "MathLib.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
diff --git a/sysdeps/ieee754/dbl-64/uatan.tbl b/sysdeps/ieee754/dbl-64/uatan.tbl
index fdc3e53304112eb31863a3144d6acb2f65a77a2e..8639292405d413da6ef32b622bf36978bd435fa8 100644
--- a/sysdeps/ieee754/dbl-64/uatan.tbl
+++ b/sysdeps/ieee754/dbl-64/uatan.tbl
@@ -1716,3866 +1716,6 @@
/**/ {{0XBF99B9A7, 0X18A3BA58} } },
};
- static const mynumber
- hij[241][16] = { /* x0,hij for (1/16,1) */
-/**/ {{{0x3fb04000, 0x00000000} },
-/**/ {{0x3fb03a6d, 0x1c06693d} },
-/**/ {{0xbc428a02, 0xd4e7f128} },
-/**/ {{0x3fefdf1f, 0xe92592ae} },
-/**/ {{0x3c88bfc0, 0xb5490162} },
-/**/ {{0xbfb01ead, 0x8f7e4151} },
-/**/ {{0xbc5395e8, 0x0b64d205} },
-/**/ {{0xbfd4d29f, 0x433dd49b} },
-/**/ {{0xbc75b19d, 0x4aa42633} },
-/**/ {{0x3fafda41, 0xce35961d} },
-/**/ {{0x3c4e6a5f, 0x425d7696} },
-/**/ {{0x3fc814dd, 0x6c1bb5e2} },
-/**/ {{0xbfaf4cb7, 0x2b33739f} },
-/**/ {{0xbfc048b2, 0xc267d8ec} },
-/**/ {{0x3fae9649, 0xe8ababc6} },
-/**/ {{0x3fb78293, 0xfe802692} } },
-/**/ {{{0x3fb10000, 0x00000000} },
-/**/ {{0x3fb0f99e, 0xa71d52a7} },
-/**/ {{0xbc22069f, 0xeec3624f} },
-/**/ {{0x3fefdc08, 0x9a49d2a9} },
-/**/ {{0x3c7780f7, 0x68b2ce25} },
-/**/ {{0xbfb0d9de, 0x9da73e1d} },
-/**/ {{0x3c4ebf46, 0xa1a487bf} },
-/**/ {{0xbfd4c669, 0xd13ea108} },
-/**/ {{0x3c7354bc, 0xebb4528c} },
-/**/ {{0x3fb0a137, 0x789374c1} },
-/**/ {{0xbc56c223, 0xc3f2c5c2} },
-/**/ {{0x3fc7f0e7, 0x79c60cda} },
-/**/ {{0xbfb05062, 0xcdcc7b81} },
-/**/ {{0xbfc019e4, 0xc5266783} },
-/**/ {{0x3fafd0b2, 0xf2540289} },
-/**/ {{0x3fb71107, 0xf6d3cd8a} } },
-/**/ {{{0x3fb20000, 0x00000000} },
-/**/ {{0x3fb1f86d, 0xbf082d59} },
-/**/ {{0xbc4095dc, 0x7732ef81} },
-/**/ {{0x3fefd7b3, 0x01722b81} },
-/**/ {{0xbc5e618c, 0x8a212e02} },
-/**/ {{0xbfb1d2c5, 0xee4e9cfa} },
-/**/ {{0x3c426273, 0x29abece0} },
-/**/ {{0xbfd4b551, 0x37eb7f46} },
-/**/ {{0x3c73b360, 0x01d8bf12} },
-/**/ {{0x3fb18fa7, 0x6adb6a7c} },
-/**/ {{0xbc5c00d8, 0x398999ad} },
-/**/ {{0x3fc7bea5, 0xf4a7cff3} },
-/**/ {{0xbfb13008, 0x61f84829} },
-/**/ {{0xbfbfb14f, 0xa8e135a1} },
-/**/ {{0x3fb0b532, 0x4324f177} },
-/**/ {{0x3fb6734a, 0x3498dd9d} } },
-/**/ {{{0x3fb30000, 0x00000000} },
-/**/ {{0x3fb2f719, 0x318a4a9a} },
-/**/ {{0x3c03fd17, 0x79b9801f} },
-/**/ {{0x3fefd31f, 0x48e238fe} },
-/**/ {{0xbc876a7a, 0xd8c45327} },
-/**/ {{0xbfb2cada, 0x852096e2} },
-/**/ {{0x3c460860, 0x11efd787} },
-/**/ {{0xbfd4a34b, 0x2e476a39} },
-/**/ {{0x3c7254f2, 0xeb11ee51} },
-/**/ {{0x3fb27c13, 0xc54ae225} },
-/**/ {{0x3c513096, 0x4ae66f0c} },
-/**/ {{0x3fc789ca, 0xef0d59d0} },
-/**/ {{0xbfb20c06, 0x6d9aaa8c} },
-/**/ {{0xbfbf2885, 0x846ba912} },
-/**/ {{0x3fb17c5f, 0xc697ef5e} },
-/**/ {{0x3fb5ce93, 0xcad31e6e} } },
-/**/ {{{0x3fb40000, 0x00000000} },
-/**/ {{0x3fb3f59f, 0x0e7c559d} },
-/**/ {{0x3c5ac4ce, 0x285df847} },
-/**/ {{0x3fefce4d, 0xa6ab93e9} },
-/**/ {{0xbc6be46b, 0x18a97736} },
-/**/ {{0xbfb3c211, 0x4d22b635} },
-/**/ {{0x3c42033c, 0x6950679f} },
-/**/ {{0xbfd49059, 0xc4d74033} },
-/**/ {{0x3c57dd7c, 0xd7e376aa} },
-/**/ {{0x3fb36662, 0xc0896a7c} },
-/**/ {{0xbc36cf6a, 0xd79232cf} },
-/**/ {{0x3fc75261, 0xa13a97a2} },
-/**/ {{0xbfb2e431, 0x5fdd1509} },
-/**/ {{0xbfbe9999, 0x6e52db32} },
-/**/ {{0x3fb23da4, 0xb0a71e9f} },
-/**/ {{0x3fb52335, 0xe3bc8178} } },
-/**/ {{{0x3fb50000, 0x00000000} },
-/**/ {{0x3fb4f3fd, 0x677292fb} },
-/**/ {{0x3c4008d3, 0x6264979e} },
-/**/ {{0x3fefc93e, 0x53a1ee0d} },
-/**/ {{0xbc64421a, 0x20fd2bdf} },
-/**/ {{0xbfb4b85f, 0x4aba88e3} },
-/**/ {{0x3c54f184, 0x3c9d1e89} },
-/**/ {{0xbfd47c7f, 0x25ae4668} },
-/**/ {{0xbc7d7581, 0x816630d1} },
-/**/ {{0x3fb44e7b, 0x07f85056} },
-/**/ {{0x3c56d63c, 0x910bdf4f} },
-/**/ {{0x3fc71875, 0xc439029c} },
-/**/ {{0xbfb3b85e, 0xf2bcfa10} },
-/**/ {{0xbfbe04bb, 0x9707b205} },
-/**/ {{0x3fb2f8c6, 0x95e3e0cc} },
-/**/ {{0x3fb47184, 0x8093431b} } },
-/**/ {{{0x3fb60000, 0x00000000} },
-/**/ {{0x3fb5f232, 0x4fd2d7b2} },
-/**/ {{0x3c58a8da, 0x4401318e} },
-/**/ {{0x3fefc3f1, 0x8b549418} },
-/**/ {{0x3c34d896, 0x836f8130} },
-/**/ {{0xbfb5adb9, 0x9cdd92e7} },
-/**/ {{0x3c4d4161, 0xeb397cc3} },
-/**/ {{0xbfd467bd, 0x93f8f1dc} },
-/**/ {{0xbc609d7b, 0xffc760ad} },
-/**/ {{0x3fb53443, 0xbea6b2fe} },
-/**/ {{0x3c5eb03c, 0x4b24f5db} },
-/**/ {{0x3fc6dc13, 0x8de3d005} },
-/**/ {{0xbfb48866, 0x37d2d99d} },
-/**/ {{0xbfbd6a1d, 0xf6663fcb} },
-/**/ {{0x3fb3ad8e, 0x0adff464} },
-/**/ {{0x3fb3b9d6, 0x4159c223} } },
-/**/ {{{0x3fb70000, 0x00000000} },
-/**/ {{0x3fb6f03b, 0xdcea4b0d} },
-/**/ {{0xbc33f00e, 0x512fa17d} },
-/**/ {{0x3fefbe67, 0x8c07a436} },
-/**/ {{0xbc84baaa, 0x46250d6f} },
-/**/ {{0xbfb6a215, 0x7e3ba4c7} },
-/**/ {{0xbc3504e7, 0x54503f8d} },
-/**/ {{0xbfd45217, 0x6b82d03a} },
-/**/ {{0x3c7d1f0d, 0xbebdd1db} },
-/**/ {{0x3fb617a4, 0x841d5604} },
-/**/ {{0xbc47168b, 0x6681c436} },
-/**/ {{0x3fc69d47, 0xaccec6ce} },
-/**/ {{0xbfb5541f, 0xa4715800} },
-/**/ {{0xbfbcc9f4, 0x335a1c1b} },
-/**/ {{0x3fb45bc6, 0xbac0061f} },
-/**/ {{0x3fb2fc84, 0x2b3853b6} } },
-/**/ {{{0x3fb80000, 0x00000000} },
-/**/ {{0x3fb7ee18, 0x2602f10f} },
-/**/ {{0xbc5cfb65, 0x4c0c3d98} },
-/**/ {{0x3fefb8a0, 0x96acfacc} },
-/**/ {{0xbc82962e, 0x18495af3} },
-/**/ {{0xbfb79568, 0x46635c89} },
-/**/ {{0x3c5ac468, 0xa6bfd498} },
-/**/ {{0xbfd43b8f, 0x2037b997} },
-/**/ {{0xbc72ad53, 0xe2f12373} },
-/**/ {{0x3fb6f885, 0x7900c4ee} },
-/**/ {{0x3c53145d, 0x0aef1f9d} },
-/**/ {{0x3fc65c1f, 0x4409ba0e} },
-/**/ {{0xbfb61b65, 0x1d176e0c} },
-/**/ {{0xbfbc2473, 0x8ad65152} },
-/**/ {{0x3fb5033f, 0x7bc246c1} },
-/**/ {{0x3fb239e9, 0x6db30b46} } },
-/**/ {{{0x3fb90000, 0x00000000} },
-/**/ {{0x3fb8ebc5, 0x4478fb28} },
-/**/ {{0x3c473288, 0x0cad24cc} },
-/**/ {{0x3fefb29c, 0xeedcd6d7} },
-/**/ {{0x3c8efa9e, 0x23ea50f0} },
-/**/ {{0xbfb887a7, 0x6ae09982} },
-/**/ {{0x3c5b2275, 0x53801511} },
-/**/ {{0xbfd42427, 0x3da0757c} },
-/**/ {{0xbc7199e5, 0x311c7ac8} },
-/**/ {{0x3fb7d6cf, 0x4388717b} },
-/**/ {{0xbc5c4eb2, 0x3dd070b4} },
-/**/ {{0x3fc618a7, 0xe6c2b5f3} },
-/**/ {{0xbfb6de12, 0x00313569} },
-/**/ {{0xbfbb79d2, 0xb6316619} },
-/**/ {{0x3fb5a3ca, 0x61af5c21} },
-/**/ {{0x3fb17263, 0x26e60289} } },
-/**/ {{{0x3fba0000, 0x00000000} },
-/**/ {{0x3fb9e941, 0x53cfdcf1} },
-/**/ {{0x3c5a332e, 0x1d69c47e} },
-/**/ {{0x3fefac5c, 0xdace3776} },
-/**/ {{0xbc8c9a78, 0x1ad91ab5} },
-/**/ {{0xbfb978c8, 0x8054ad75} },
-/**/ {{0xbc5e35b8, 0x8ed66c17} },
-/**/ {{0xbfd40be2, 0x665afed1} },
-/**/ {{0x3c62eeef, 0x08ef10fb} },
-/**/ {{0x3fb8b26b, 0x13c989d2} },
-/**/ {{0x3c329f11, 0xbfeab3ba} },
-/**/ {{0x3fc5d2ef, 0x93c8f97c} },
-/**/ {{0xbfb79c03, 0x30234881} },
-/**/ {{0xbfbaca49, 0xd0f650c8} },
-/**/ {{0x3fb63d3c, 0xce2dcccc} },
-/**/ {{0x3fb0a650, 0x26fb0af2} } },
-/**/ {{{0x3fbb0000, 0x00000000} },
-/**/ {{0x3fbae68a, 0x71c722b8} },
-/**/ {{0x3c4c014e, 0x6910b9db} },
-/**/ {{0x3fefa5e0, 0xa34ef42b} },
-/**/ {{0xbc836583, 0xeb56d5b9} },
-/**/ {{0xbfba68c1, 0x3b881779} },
-/**/ {{0xbc473a0d, 0x13a09314} },
-/**/ {{0xbfd3f2c3, 0x538e939c} },
-/**/ {{0xbc68ed49, 0xee53e648} },
-/**/ {{0x3fb98b42, 0xa7d45973} },
-/**/ {{0xbc523943, 0x461ca7c4} },
-/**/ {{0x3fc58b04, 0xb0f2e2bb} },
-/**/ {{0xbfb85517, 0x1c9d23dc} },
-/**/ {{0xbfba1612, 0x3e3b5a66} },
-/**/ {{0x3fb6cf6f, 0x7ef1d0b9} },
-/**/ {{0x3fafac21, 0x6617b315} } },
-/**/ {{{0x3fbc0000, 0x00000000} },
-/**/ {{0x3fbbe39e, 0xbe6f07c3} },
-/**/ {{0x3c5f7b8f, 0x29a05987} },
-/**/ {{0x3fef9f28, 0x93bb9192} },
-/**/ {{0x3c78260b, 0x7cd1bdab} },
-/**/ {{0xbfbb5787, 0x72759741} },
-/**/ {{0x3c52f93f, 0xa6767247} },
-/**/ {{0xbfd3d8cc, 0xd45bbe91} },
-/**/ {{0x3c664839, 0x2edc0762} },
-/**/ {{0x3fba6140, 0x4fa31d26} },
-/**/ {{0x3c400647, 0x97891510} },
-/**/ {{0x3fc540f6, 0x0668fd66} },
-/**/ {{0xbfb9092d, 0xcb2f6e8f} },
-/**/ {{0xbfb95d66, 0x8d902073} },
-/**/ {{0x3fb75a3e, 0x99c53d16} },
-/**/ {{0x3fae040c, 0x8f475e61} } },
-/**/ {{{0x3fbd0000, 0x00000000} },
-/**/ {{0x3fbce07c, 0x5c3cca32} },
-/**/ {{0x3c4138e6, 0x425918a7} },
-/**/ {{0x3fef9834, 0xf9f6d421} },
-/**/ {{0x3c6f3089, 0x8c22a239} },
-/**/ {{0xbfbc4511, 0x1d4e69a5} },
-/**/ {{0x3c254c0f, 0xd2083ce8} },
-/**/ {{0xbfd3be01, 0xcd488978} },
-/**/ {{0x3c5612db, 0x6362ec0f} },
-/**/ {{0x3fbb344e, 0xf0d94873} },
-/**/ {{0xbc182beb, 0xfdf7db72} },
-/**/ {{0x3fc4f4d2, 0xb9d86c04} },
-/**/ {{0xbfb9b828, 0xdf238807} },
-/**/ {{0xbfb8a082, 0x5f93ffd6} },
-/**/ {{0x3fb7dd89, 0xb6650b0c} },
-/**/ {{0x3fac5526, 0xb62676ef} } },
-/**/ {{{0x3fbe0000, 0x00000000} },
-/**/ {{0x3fbddd21, 0x701eba6e} },
-/**/ {{0x3c594eff, 0xcd76fe58} },
-/**/ {{0x3fef9106, 0x266112ba} },
-/**/ {{0x3c74c302, 0x6b7e18b1} },
-/**/ {{0xbfbd3154, 0x5777816c} },
-/**/ {{0x3c5dc7e4, 0x1f9dbddd} },
-/**/ {{0xbfd3a265, 0x37a90881} },
-/**/ {{0xbc75bd61, 0xeb7ba840} },
-/**/ {{0x3fbc045a, 0x0a52514b} },
-/**/ {{0xbc35ca88, 0xcff49a99} },
-/**/ {{0x3fc4a6aa, 0x498eeb56} },
-/**/ {{0xbfba61eb, 0xa09232cf} },
-/**/ {{0xbfb7dfa2, 0x4a464027} },
-/**/ {{0x3fb85933, 0xe633c053} },
-/**/ {{0x3faaa036, 0x3f920107} } },
-/**/ {{{0x3fbf0000, 0x00000000} },
-/**/ {{0x3fbed98c, 0x2190043b} },
-/**/ {{0xbc23a598, 0x592c7b13} },
-/**/ {{0x3fef899c, 0x6bcf4ad8} },
-/**/ {{0x3c55fd73, 0x912c09b0} },
-/**/ {{0xbfbe1c47, 0x607f91a0} },
-/**/ {{0x3c576677, 0x5b5db022} },
-/**/ {{0xbfd385fa, 0x21046f5f} },
-/**/ {{0x3c7f01c3, 0x4487f4b8} },
-/**/ {{0x3fbcd14d, 0xb77f2d51} },
-/**/ {{0x3c57a86d, 0x30a2ccfe} },
-/**/ {{0x3fc4568c, 0x8782b530} },
-/**/ {{0xbfbb065b, 0x02b7ad2d} },
-/**/ {{0xbfb71b03, 0xbd215555} },
-/**/ {{0x3fb8cd23, 0xb9c1c1de} },
-/**/ {{0x3fa8e602, 0x8dbfa69b} } },
-/**/ {{{0x3fc00000, 0x00000000} },
-/**/ {{0x3fbfd5ba, 0x9aac2f6e} },
-/**/ {{0xbc4cd376, 0x86760c17} },
-/**/ {{0x3fef81f8, 0x1f81f820} },
-/**/ {{0xbc8f81f8, 0x1f81f820} },
-/**/ {{0xbfbf05e0, 0x9d0dc11b} },
-/**/ {{0xbc35a199, 0x1d821725} },
-/**/ {{0xbfd368c3, 0xaa76e1d7} },
-/**/ {{0xbc672d4c, 0xc796f8cd} },
-/**/ {{0x3fbd9b16, 0xb391c2e3} },
-/**/ {{0x3c58051b, 0x8086c51d} },
-/**/ {{0x3fc40489, 0x94488c86} },
-/**/ {{0xbfbba55d, 0xa98401c8} },
-/**/ {{0xbfb652e4, 0xe5127e64} },
-/**/ {{0x3fb93943, 0x442e53ae} },
-/**/ {{0x3fa72753, 0x86286f75} } },
-/**/ {{{0x3fc08000, 0x00000000} },
-/**/ {{0x3fc068d5, 0x84212b3e} },
-/**/ {{0xbc69e2d2, 0x83019bfd} },
-/**/ {{0x3fef7a19, 0x991bb133} },
-/**/ {{0x3c7a956a, 0x66627723} },
-/**/ {{0xbfbfee16, 0x97c8e137} },
-/**/ {{0x3c4d9399, 0x66dbe7af} },
-/**/ {{0xbfd34ac5, 0x0810323a} },
-/**/ {{0x3c6a1a57, 0x6bc6c512} },
-/**/ {{0x3fbe61a2, 0x5c75a6f9} },
-/**/ {{0xbc492b99, 0xd75c8f85} },
-/**/ {{0x3fc3b0b1, 0xd9fa3f20} },
-/**/ {{0xbfbc3edb, 0xee66d309} },
-/**/ {{0xbfb58784, 0x905eeb33} },
-/**/ {{0x3fb99d80, 0x1c65bb14} },
-/**/ {{0x3fa564f1, 0x18a09884} } },
-/**/ {{{0x3fc10000, 0x00000000} },
-/**/ {{0x3fc0e6ad, 0xccf40882} },
-/**/ {{0xbc6d71a3, 0x1bb98d0d} },
-/**/ {{0x3fef7201, 0x32978bad} },
-/**/ {{0x3c816476, 0x599381e9} },
-/**/ {{0xbfc06a70, 0x011b81fd} },
-/**/ {{0xbc422f5d, 0x9ba697ca} },
-/**/ {{0xbfd32c01, 0x802fc0a5} },
-/**/ {{0x3c7d8e47, 0x08a20868} },
-/**/ {{0x3fbf24de, 0xb59597fe} },
-/**/ {{0xbc43288f, 0x410d31eb} },
-/**/ {{0x3fc35b16, 0x070feb24} },
-/**/ {{0xbfbcd2bf, 0xe4565b78} },
-/**/ {{0xbfb4b922, 0x128768c6} },
-/**/ {{0x3fb9f9cb, 0x5c42a097} },
-/**/ {{0x3fa39fa2, 0xc7f97f2e} } },
-/**/ {{{0x3fc18000, 0x00000000} },
-/**/ {{0x3fc16465, 0x41060850} },
-/**/ {{0x3c66bcee, 0x8ae7ea92} },
-/**/ {{0x3fef69af, 0x483f492b} },
-/**/ {{0xbc6e3280, 0x57db963e} },
-/**/ {{0xbfc0dd19, 0xdacaa844} },
-/**/ {{0xbc6133c7, 0xad7fc21e} },
-/**/ {{0xbfd30c7c, 0x6addaea8} },
-/**/ {{0xbc71443d, 0x89161c76} },
-/**/ {{0x3fbfe4ba, 0x6a6d3cd2} },
-/**/ {{0x3c50d4b8, 0x423ee67a} },
-/**/ {{0x3fc303c7, 0x092e569a} },
-/**/ {{0xbfbd60f5, 0x5b11d3b6} },
-/**/ {{0xbfb3e7fd, 0x283b5c55} },
-/**/ {{0x3fba4e19, 0x9d9a6ab7} },
-/**/ {{0x3fa1d82f, 0x3487cc29} } },
-/**/ {{{0x3fc20000, 0x00000000} },
-/**/ {{0x3fc1e1fa, 0xfb043727} },
-/**/ {{0xbc4b4859, 0x14dacf8c} },
-/**/ {{0x3fef6124, 0x38a14f5e} },
-/**/ {{0x3c798e9e, 0x001f6124} },
-/**/ {{0xbfc14f04, 0x59d3fb7c} },
-/**/ {{0x3c531efa, 0x4cc99cb2} },
-/**/ {{0xbfd2ec39, 0x31219b34} },
-/**/ {{0xbc618697, 0x6e004611} },
-/**/ {{0x3fc05092, 0x68736312} },
-/**/ {{0x3c67aad4, 0x8a06e4b5} },
-/**/ {{0x3fc2aad6, 0x07eca5ec} },
-/**/ {{0xbfbde969, 0xe19fe31c} },
-/**/ {{0xbfb31455, 0xdb6b9127} },
-/**/ {{0x3fba9a62, 0xf53dd9ee} },
-/**/ {{0x3fa00f5b, 0xa8e4ede0} } },
-/**/ {{{0x3fc28000, 0x00000000} },
-/**/ {{0x3fc25f6e, 0x171a535c} },
-/**/ {{0x3c67c6d7, 0xbde1a310} },
-/**/ {{0x3fef5860, 0x64866d22} },
-/**/ {{0x3c88c6ff, 0xd1f6326c} },
-/**/ {{0xbfc1c02b, 0x13c11396} },
-/**/ {{0xbc51b469, 0xffeb1a0f} },
-/**/ {{0xbfd2cb3b, 0x4c571b0f} },
-/**/ {{0x3c6e4f76, 0x2fb0b163} },
-/**/ {{0x3fc0ad06, 0xf5c213ab} },
-/**/ {{0x3c625bf2, 0xabea9e66} },
-/**/ {{0x3fc25054, 0x5f93bbb2} },
-/**/ {{0xbfbe6c0c, 0xc80a32c8} },
-/**/ {{0xbfb23e6c, 0x678d0d1e} },
-/**/ {{0x3fbadea2, 0xebf8ae4b} },
-/**/ {{0x3f9c8bd7, 0x527f133b} } },
-/**/ {{{0x3fc30000, 0x00000000} },
-/**/ {{0x3fc2dcbd, 0xb2fba1ff} },
-/**/ {{0x3c58f287, 0x05561534} },
-/**/ {{0x3fef4f64, 0x2ee76e94} },
-/**/ {{0x3c80ec89, 0xc6da5865} },
-/**/ {{0xbfc23089, 0xb322f867} },
-/**/ {{0x3c4c2b54, 0x5fcd0d6f} },
-/**/ {{0xbfd2a986, 0x45802261} },
-/**/ {{0xbc79a132, 0x5ae78b8a} },
-/**/ {{0x3fc107b3, 0x35a9d974} },
-/**/ {{0x3c5ef22d, 0xb725e335} },
-/**/ {{0x3fc1f453, 0x9bd98832} },
-/**/ {{0xbfbee8cf, 0x2057aad4} },
-/**/ {{0xbfb16681, 0x1e1bc3a1} },
-/**/ {{0x3fbb1ad8, 0x759c8f58} },
-/**/ {{0x3f98f941, 0x0b15b4aa} } },
-/**/ {{{0x3fc38000, 0x00000000} },
-/**/ {{0x3fc359e8, 0xedeb99a4} },
-/**/ {{0xbc6a5fd7, 0x4e4604c6} },
-/**/ {{0x3fef462f, 0xfce28238} },
-/**/ {{0x3c83dc01, 0xd90595d1} },
-/**/ {{0xbfc2a01b, 0xf7edfa6d} },
-/**/ {{0xbc6b11fb, 0x4a3b5c9a} },
-/**/ {{0xbfd2871d, 0xb4959402} },
-/**/ {{0xbc4a3702, 0x2fcf7ea3} },
-/**/ {{0x3fc1608f, 0xd8d7fe8c} },
-/**/ {{0x3c61ac60, 0xf8f1d41c} },
-/**/ {{0x3fc196e5, 0x729a89ca} },
-/**/ {{0xbfbf5fa3, 0xbec74f31} },
-/**/ {{0xbfb08cd4, 0x4b6c9767} },
-/**/ {{0x3fbb4f05, 0xe624ce15} },
-/**/ {{0x3f956871, 0xddb2020c} } },
-/**/ {{{0x3fc40000, 0x00000000} },
-/**/ {{0x3fc3d6ee, 0xe8c6626c} },
-/**/ {{0x3c661a3b, 0x0ce9281b} },
-/**/ {{0x3fef3cc4, 0x35b0713c} },
-/**/ {{0x3c81d0a7, 0xe69ea094} },
-/**/ {{0xbfc30edd, 0xb7d169f0} },
-/**/ {{0x3c6b3394, 0xae999b97} },
-/**/ {{0xbfd26405, 0x3fd62b3c} },
-/**/ {{0x3c73e339, 0xc0736df9} },
-/**/ {{0x3fc1b795, 0xe8e57ee3} },
-/**/ {{0xbc6130dc, 0x0a42c7f6} },
-/**/ {{0x3fc1381b, 0xbe93b8e5} },
-/**/ {{0xbfbfd07f, 0x394e1bf7} },
-/**/ {{0xbfaf634c, 0x37bb5315} },
-/**/ {{0x3fbb7b30, 0xe501e57b} },
-/**/ {{0x3f91dae1, 0x20503792} } },
-/**/ {{{0x3fc48000, 0x00000000} },
-/**/ {{0x3fc453ce, 0xc6092a9e} },
-/**/ {{0x3c61f653, 0xb3a5a78b} },
-/**/ {{0x3fef3321, 0x4299ace8} },
-/**/ {{0xbc87414c, 0x3a742b30} },
-/**/ {{0xbfc37cca, 0xde8b2323} },
-/**/ {{0x3c649378, 0x7b50aedf} },
-/**/ {{0xbfd24040, 0x9b13f4d0} },
-/**/ {{0x3c7e271f, 0xb7dc85c0} },
-/**/ {{0x3fc20cbe, 0xc9024068} },
-/**/ {{0x3c50921f, 0x88ef3da7} },
-/**/ {{0x3fc0d808, 0x7a1f1270} },
-/**/ {{0xbfc01dab, 0xf32d5436} },
-/**/ {{0xbfadaa6d, 0x02e6f09c} },
-/**/ {{0x3fbb9f62, 0x5e9cd766} },
-/**/ {{0x3f8ca3fe, 0xab964c04} } },
-/**/ {{{0x3fc50000, 0x00000000} },
-/**/ {{0x3fc4d087, 0xa9da4f17} },
-/**/ {{0x3c61f323, 0xf1adf158} },
-/**/ {{0x3fef2947, 0x8eeb3352} },
-/**/ {{0x3c871eb0, 0x8799a164} },
-/**/ {{0xbfc3e9df, 0x6e36e75c} },
-/**/ {{0x3c541555, 0x4e37666f} },
-/**/ {{0xbfd21bd3, 0x87008bd0} },
-/**/ {{0xbc609e14, 0xc24ff75f} },
-/**/ {{0x3fc26004, 0x36860504} },
-/**/ {{0xbc58f8ca, 0x1ebc8c40} },
-/**/ {{0x3fc076bd, 0xb9f4ead3} },
-/**/ {{0xbfc05012, 0xed70ddd5} },
-/**/ {{0xbfabef8a, 0x33e194b1} },
-/**/ {{0x3fbbbba6, 0x7423a91f} },
-/**/ {{0x3f859e6a, 0xdd99da12} } },
-/**/ {{{0x3fc58000, 0x00000000} },
-/**/ {{0x3fc54d18, 0xba11570a} },
-/**/ {{0x3c618282, 0xf2884073} },
-/**/ {{0x3fef1f37, 0x87eb4d7d} },
-/**/ {{0x3c8476f0, 0xedda13e6} },
-/**/ {{0xbfc45617, 0x7f997c7c} },
-/**/ {{0xbc46bf5b, 0x6423ceda} },
-/**/ {{0xbfd1f6c1, 0xd0784ec7} },
-/**/ {{0xbc74ec12, 0xd106a8e0} },
-/**/ {{0x3fc2b160, 0x4967338d} },
-/**/ {{0x3c5309c0, 0x61339c25} },
-/**/ {{0x3fc0144d, 0xa7f42962} },
-/**/ {{0xbfc07f71, 0x73dbaeec} },
-/**/ {{0xbfaa3322, 0x2aeda9a4} },
-/**/ {{0x3fbbd00c, 0x69b152b3} },
-/**/ {{0x3f7d4f90, 0x4c782821} } },
-/**/ {{{0x3fc60000, 0x00000000} },
-/**/ {{0x3fc5c981, 0x1e3ec26a} },
-/**/ {{0xbc5054ab, 0x2c010f3d} },
-/**/ {{0x3fef14f1, 0x9cce28eb} },
-/**/ {{0xbc8b7c25, 0x2708cd6e} },
-/**/ {{0xbfc4c16f, 0x42678d07} },
-/**/ {{0x3c5f55ba, 0xc1560017} },
-/**/ {{0xbfd1d10f, 0x4fccc153} },
-/**/ {{0x3c529588, 0x1bcc361d} },
-/**/ {{0x3fc300cd, 0x74979f8c} },
-/**/ {{0xbc6b1da5, 0x0bc1e891} },
-/**/ {{0x3fbf6194, 0xfbe70208} },
-/**/ {{0xbfc0abc5, 0x4b1c266f} },
-/**/ {{0xbfa875b2, 0x3b74e858} },
-/**/ {{0x3fbbdca6, 0x92e46f11} },
-/**/ {{0x3f6f0b17, 0x9de94aef} } },
-/**/ {{{0x3fc68000, 0x00000000} },
-/**/ {{0x3fc645bf, 0xffb3aa74} },
-/**/ {{0xbc3f536b, 0x677c2cb4} },
-/**/ {{0x3fef0a76, 0x3eaa4ed6} },
-/**/ {{0x3c888c52, 0x0b06c761} },
-/**/ {{0xbfc52be2, 0xfd884489} },
-/**/ {{0x3c67ec59, 0xbe5c728a} },
-/**/ {{0xbfd1aabf, 0xe80e4e0a} },
-/**/ {{0xbc71320e, 0xe90c909e} },
-/**/ {{0x3fc34e46, 0x864781ca} },
-/**/ {{0x3c42fcb3, 0x126138ee} },
-/**/ {{0x3fbe988d, 0x013b5d4f} },
-/**/ {{0xbfc0d50d, 0x122409a2} },
-/**/ {{0xbfa6b7b6, 0x7bb562c1} },
-/**/ {{0x3fbbe18a, 0x3df8dee8} },
-/**/ {{0x3f3e4009, 0x8809e1ef} } },
-/**/ {{{0x3fc70000, 0x00000000} },
-/**/ {{0x3fc6c1d4, 0x898933d9} },
-/**/ {{0xbc52954a, 0x7603c427} },
-/**/ {{0x3feeffc5, 0xe06cfb34} },
-/**/ {{0xbc85c037, 0x379877c2} },
-/**/ {{0xbfc5956f, 0x0f53a52c} },
-/**/ {{0x3c4d46a2, 0xe566376c} },
-/**/ {{0xbfd183d7, 0x86559c11} },
-/**/ {{0x3c7d2520, 0x64734c7f} },
-/**/ {{0x3fc399c6, 0xa80eddd5} },
-/**/ {{0x3c616c26, 0x40fbef6f} },
-/**/ {{0x3fbdcda7, 0xf4b571a7} },
-/**/ {{0xbfc0fb48, 0x3fd42996} },
-/**/ {{0xbfa4f9a9, 0x95c85118} },
-/**/ {{0x3fbbdecf, 0x9d795df4} },
-/**/ {{0xbf672003, 0xb85bf719} } },
-/**/ {{{0x3fc78000, 0x00000000} },
-/**/ {{0x3fc73dbd, 0xe8a7d202} },
-/**/ {{0xbc55ad0f, 0x6d4a665d} },
-/**/ {{0x3feef4e0, 0xf6ce5590} },
-/**/ {{0xbc833df6, 0x556900ef} },
-/**/ {{0xbfc5fe0f, 0xedcc9488} },
-/**/ {{0x3c5078de, 0xd2b9e35c} },
-/**/ {{0xbfd15c5a, 0x210cab36} },
-/**/ {{0x3c67fa93, 0xf55e532a} },
-/**/ {{0x3fc3e349, 0x5efd9a41} },
-/**/ {{0xbc6cf709, 0xc8573a12} },
-/**/ {{0x3fbd010a, 0x6c903aef} },
-/**/ {{0xbfc11e77, 0x20571328} },
-/**/ {{0xbfa33c04, 0x9a1875dd} },
-/**/ {{0x3fbbd491, 0xb09ec0ce} },
-/**/ {{0xbf78d197, 0x35537a65} } },
-/**/ {{{0x3fc80000, 0x00000000} },
-/**/ {{0x3fc7b97b, 0x4bce5b02} },
-/**/ {{0x3c5347b0, 0xb4f881ca} },
-/**/ {{0x3feee9c7, 0xf8458e02} },
-/**/ {{0xbc616380, 0x7ba71fe1} },
-/**/ {{0xbfc665c2, 0x26d69eeb} },
-/**/ {{0xbc572a33, 0xfdb5eea8} },
-/**/ {{0xbfd1344b, 0xb737e8f3} },
-/**/ {{0xbc757b70, 0x62badf41} },
-/**/ {{0x3fc42aca, 0x8b929b0b} },
-/**/ {{0x3c43cdb5, 0x7a8b7d91} },
-/**/ {{0x3fbc32d8, 0xf683981c} },
-/**/ {{0xbfc13e9a, 0xd22d5ecc} },
-/**/ {{0xbfa17f3e, 0xd35c8c33} },
-/**/ {{0x3fbbc2ee, 0x2a73307e} },
-/**/ {{0xbf82ee04, 0x2bddc834} } },
-/**/ {{{0x3fc88000, 0x00000000} },
-/**/ {{0x3fc8350b, 0xe398ebc8} },
-/**/ {{0xbc55a913, 0x32b9c90d} },
-/**/ {{0x3feede7b, 0x5cfce04c} },
-/**/ {{0x3c8507c2, 0x3b51a72f} },
-/**/ {{0xbfc6cc82, 0x6067718b} },
-/**/ {{0x3c6d00ca, 0xdbfc430f} },
-/**/ {{0xbfd10bb0, 0x4fbf6fe8} },
-/**/ {{0x3c321748, 0x53749c72} },
-/**/ {{0x3fc47046, 0x699a36ad} },
-/**/ {{0xbc63924c, 0x3994d40c} },
-/**/ {{0x3fbb6338, 0x0dfb7483} },
-/**/ {{0xbfc15bb5, 0x42ee5820} },
-/**/ {{0xbf9f879b, 0x385194fc} },
-/**/ {{0x3fbbaa05, 0x57d040e9} },
-/**/ {{0xbf895566, 0xada71ca0} } },
-/**/ {{{0x3fc90000, 0x00000000} },
-/**/ {{0x3fc8b06e, 0xe2879c29} },
-/**/ {{0xbc6118cd, 0x30308c4f} },
-/**/ {{0x3feed2fb, 0x9ec57f51} },
-/**/ {{0xbc83fdc5, 0xc0d106ba} },
-/**/ {{0xbfc7324d, 0x58b40d27} },
-/**/ {{0x3c68e240, 0xfc062163} },
-/**/ {{0xbfd0e28b, 0xf8b8a2bf} },
-/**/ {{0xbc7b8d8a, 0x64c55b39} },
-/**/ {{0x3fc4b3b9, 0x8ff46730} },
-/**/ {{0xbc5af146, 0x988563da} },
-/**/ {{0x3fba924c, 0x1277a10d} },
-/**/ {{0xbfc175c9, 0x2bbfd54d} },
-/**/ {{0xbf9c1448, 0x6c522340} },
-/**/ {{0x3fbb89fa, 0x044f2f6b} },
-/**/ {{0xbf8f9cc7, 0xaaecc742} } },
-/**/ {{{0x3fc98000, 0x00000000} },
-/**/ {{0x3fc92ba3, 0x7d050272} },
-/**/ {{0xbc60d3de, 0xd0ff4764} },
-/**/ {{0x3feec749, 0x390b6afe} },
-/**/ {{0xbc5c3d17, 0x4e3659ca} },
-/**/ {{0xbfc7971f, 0xe659b3de} },
-/**/ {{0x3c4cab11, 0x373f554d} },
-/**/ {{0xbfd0b8e2, 0xc6b052a4} },
-/**/ {{0x3c7da014, 0x6f3b74bc} },
-/**/ {{0x3fc4f520, 0xf0432146} },
-/**/ {{0xbc6769ad, 0xa8027290} },
-/**/ {{0x3fb9c039, 0x3e17b570} },
-/**/ {{0xbfc18cda, 0x0d8833a4} },
-/**/ {{0xbf98a567, 0x4627d340} },
-/**/ {{0x3fbb62f1, 0x5e42eff7} },
-/**/ {{0xbf92e10a, 0x7ee3bed3} } },
-/**/ {{{0x3fca0000, 0x00000000} },
-/**/ {{0x3fc9a6a8, 0xe96c8626} },
-/**/ {{0x3c4cf601, 0xe7b4348e} },
-/**/ {{0x3feebb64, 0xa8c932d7} },
-/**/ {{0x3c20538d, 0x79aae302} },
-/**/ {{0xbfc7faf6, 0xf88295fe} },
-/**/ {{0xbc687a81, 0x932909e9} },
-/**/ {{0xbfd08eb8, 0xd3f5a07b} },
-/**/ {{0xbc620a05, 0xfb7d6aaa} },
-/**/ {{0x3fc53479, 0xd6814372} },
-/**/ {{0xbc53c682, 0x0a0c6620} },
-/**/ {{0x3fb8ed23, 0x9c562d77} },
-/**/ {{0xbfc1a0ec, 0x2cdd89fd} },
-/**/ {{0xbf953bd4, 0xfec9df82} },
-/**/ {{0x3fbb3512, 0xd9d3f0f6} },
-/**/ {{0xbf95e1ab, 0x4534ccf5} } },
-/**/ {{{0x3fca8000, 0x00000000} },
-/**/ {{0x3fca217e, 0x601081a6} },
-/**/ {{0xbc60def8, 0xa60af374} },
-/**/ {{0x3feeaf4e, 0x6c7ba732} },
-/**/ {{0x3c89fa72, 0xe91fffe1} },
-/**/ {{0xbfc85dcf, 0x970642c3} },
-/**/ {{0xbc5732c2, 0x5b7f0ad0} },
-/**/ {{0xbfd06412, 0x3fe5c74d} },
-/**/ {{0xbc7d0053, 0x4a82f9b1} },
-/**/ {{0x3fc571c1, 0xe882973d} },
-/**/ {{0x3c59d9a3, 0x9090f12c} },
-/**/ {{0x3fb8192f, 0x00f5d0e0} },
-/**/ {{0xbfc1b204, 0x8db53983} },
-/**/ {{0xbf91d869, 0xbdd7b47e} },
-/**/ {{0x3fbb0088, 0x1355a903} },
-/**/ {{0xbf98cf57, 0x724a2ad9} } },
-/**/ {{{0x3fcb0000, 0x00000000} },
-/**/ {{0x3fca9c23, 0x1b403279} },
-/**/ {{0x3c60e8bb, 0xe89cca85} },
-/**/ {{0x3feea307, 0x04157b4f} },
-/**/ {{0x3c8ad743, 0xfd8bf1f0} },
-/**/ {{0xbfc8bfa6, 0xe285e2fd} },
-/**/ {{0xbc6ce765, 0x9c834c8f} },
-/**/ {{0xbfd038f3, 0x2e38fd26} },
-/**/ {{0x3c6a42ec, 0xef212a80} },
-/**/ {{0x3fc5acf7, 0x255d65d5} },
-/**/ {{0xbc619fba, 0xbe486771} },
-/**/ {{0x3fb7447e, 0xff244e15} },
-/**/ {{0xbfc1c028, 0xeed71b69} },
-/**/ {{0xbf8cf7f0, 0xaceecf68} },
-/**/ {{0x3fbac57c, 0xb0ee161b} },
-/**/ {{0xbf9ba92d, 0xefc8f53e} } },
-/**/ {{{0x3fcb8000, 0x00000000} },
-/**/ {{0x3fcb1696, 0x574d780c} },
-/**/ {{0xbc585ab8, 0xfc15a673} },
-/**/ {{0x3fee968e, 0xf0f2da5a} },
-/**/ {{0xbc6fffe1, 0x69710f0d} },
-/**/ {{0xbfc9207a, 0x148444b5} },
-/**/ {{0xbc66661a, 0x1802fa91} },
-/**/ {{0xbfd00d5f, 0xc65096ca} },
-/**/ {{0x3c7f2a2e, 0x8920e744} },
-/**/ {{0x3fc5e617, 0xe4be288d} },
-/**/ {{0x3c67fa48, 0x99be934f} },
-/**/ {{0x3fb66f36, 0xe0d4c87a} },
-/**/ {{0xbfc1cb5f, 0xc5179ce8} },
-/**/ {{0xbf864e9c, 0x1011bb6c} },
-/**/ {{0x3fba841e, 0x43a75476} },
-/**/ {{0xbf9e6e5b, 0x845fc859} } },
-/**/ {{{0x3fcc0000, 0x00000000} },
-/**/ {{0x3fcb90d7, 0x529260a2} },
-/**/ {{0x3c217b10, 0xd2e0e5ab} },
-/**/ {{0x3fee89e6, 0xb5ccf172} },
-/**/ {{0x3c820357, 0x153be26a} },
-/**/ {{0xbfc98046, 0x7f79bfd6} },
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-/**/ {{{0x3fd94000, 0x00000000} },
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-/**/ {{0x3f835d1d, 0x48e09ecd} },
-/**/ {{0xbfacd0b5, 0x6b6f9aca} } },
-/**/ {{{0x3fd98000, 0x00000000} },
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-/**/ {{0x3f7eb3f1, 0xe5326389} },
-/**/ {{0xbfac3ebd, 0x234575e8} } },
-/**/ {{{0x3fd9c000, 0x00000000} },
-/**/ {{0x3fd87b4b, 0x0c1ebedc} },
-/**/ {{0xbc76dcfa, 0xa2fa470f} },
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-/**/ {{0xbfaba82f, 0x006e38b2} } },
-/**/ {{{0x3fda0000, 0x00000000} },
-/**/ {{0x3fd8b24d, 0x394a1b25} },
-/**/ {{0x3c7b6d0b, 0xa3748fa8} },
-/**/ {{0x3feb7786, 0x1d9cdc98} },
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-/**/ {{0x3f6e4b7a, 0x46aa98b6} },
-/**/ {{0xbfab0d64, 0x3bad3a76} } },
-/**/ {{{0x3fda4000, 0x00000000} },
-/**/ {{0x3fd8e929, 0x16f5cde8} },
-/**/ {{0x3c74c0a7, 0xe12bfafb} },
-/**/ {{0x3feb6454, 0x32024b37} },
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-/**/ {{0x3f5e80cd, 0x04721907} },
-/**/ {{0xbfaa6eb5, 0x0dc212a5} } },
-/**/ {{{0x3fda8000, 0x00000000} },
-/**/ {{0x3fd91fde, 0x7cd0c662} },
-/**/ {{0x3c710741, 0x88054b53} },
-/**/ {{0x3feb510e, 0x6454751c} },
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-/**/ {{0xbfb2c82d, 0xe771fa17} },
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-/**/ {{0x3f11f322, 0xa23eb6eb} },
-/**/ {{0xbfa9cc78, 0x8ae579b1} } },
-/**/ {{{0x3fdac000, 0x00000000} },
-/**/ {{0x3fd9566d, 0x43a34907} },
-/**/ {{0x3c69b015, 0x37e0af2b} },
-/**/ {{0x3feb3db5, 0x40ddf8d3} },
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-/**/ {{0xbfb2404f, 0xc8407216} },
-/**/ {{0x3fb367bf, 0x255139f9} },
-/**/ {{0xbf5b8a0d, 0x65acd6da} },
-/**/ {{0xbfa92704, 0x8052f51d} } },
-/**/ {{{0x3fdb0000, 0x00000000} },
-/**/ {{0x3fd98cd5, 0x454d6b18} },
-/**/ {{0x3c79e6c9, 0x88fd0a77} },
-/**/ {{0x3feb2a49, 0x5323eb6a} },
-/**/ {{0xbc572202, 0x70cc9678} },
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-/**/ {{0x3c6978a3, 0xda42aa4e} },
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-/**/ {{0xbc6aa72f, 0xa4ac9df8} },
-/**/ {{0xbfac0db0, 0xd0665a46} },
-/**/ {{0xbfb1b889, 0xb428e30d} },
-/**/ {{0x3fb3628d, 0x134448b0} },
-/**/ {{0xbf6bbbc1, 0x67619c9c} },
-/**/ {{0xbfa87ead, 0x53e1f653} } },
-/**/ {{{0x3fdb4000, 0x00000000} },
-/**/ {{0x3fd9c316, 0x5cc58107} },
-/**/ {{0x3c4b6696, 0x02250cfb} },
-/**/ {{0x3feb16cb, 0x25df55f4} },
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-/**/ {{0xbfd3871f, 0x00742189} },
-/**/ {{0xbc725ae2, 0xc05df451} },
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-/**/ {{0xbf74a987, 0x53bec2ff} },
-/**/ {{0xbfa7d3c5, 0xe5af58b6} } },
-/**/ {{{0x3fdb8000, 0x00000000} },
-/**/ {{0x3fd9f930, 0x66168002} },
-/**/ {{0xbc7c8270, 0x47c9439a} },
-/**/ {{0x3feb033b, 0x42f6e2c9} },
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-/**/ {{0xbf7b44ca, 0x1def6f17} },
-/**/ {{0xbfa7269f, 0x73d683b8} } },
-/**/ {{{0x3fdbc000, 0x00000000} },
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-/**/ {{0x3c5814d5, 0xf797086b} },
-/**/ {{0x3feaef9a, 0x3378ba79} },
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-/**/ {{0x3c571f2f, 0xb2996038} },
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-/**/ {{0xbfa67789, 0x8037e242} } },
-/**/ {{{0x3fdc0000, 0x00000000} },
-/**/ {{0x3fda64ee, 0xc3cc23fd} },
-/**/ {{0xbc724dec, 0x1b50b7ff} },
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-/**/ {{0xbf83f3f0, 0x25d7ff49} },
-/**/ {{0xbfa5c6d1, 0xb848e1d1} } },
-/**/ {{{0x3fdc4000, 0x00000000} },
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-/**/ {{0x3c42e42d, 0xff75d817} },
-/**/ {{0x3feac826, 0xae95dea9} },
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-/**/ {{0xbf86f764, 0x3ce97a26} },
-/**/ {{0xbfa514c3, 0xdee8421b} } },
-/**/ {{{0x3fdc8000, 0x00000000} },
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-/**/ {{0xbfa3adca, 0xf73e36f0} } },
-/**/ {{{0x3fdd0000, 0x00000000} },
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-/**/ {{{0x3fdd4000, 0x00000000} },
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-/**/ {{0xbf6fb0d0, 0x9bd39a5f} },
-/**/ {{0x3f7b7748, 0x8813998f} } },
-/**/ {{{0x3fed6000, 0x00000000} },
-/**/ {{0x3fe7c3d3, 0x11a6092b} },
-/**/ {{0x3c8bb3cb, 0x2d303288} },
-/**/ {{0x3fe15dbb, 0x1dc61b17} },
-/**/ {{0xbc8f0487, 0xbb77dc56} },
-/**/ {{0xbfd14d7e, 0xee0771ca} },
-/**/ {{0x3c72d38b, 0xdc2fcbd0} },
-/**/ {{0x3fb4d716, 0xd6080f0e} },
-/**/ {{0xbc5cb5bc, 0xa9fbc2c3} },
-/**/ {{0x3f89a7f9, 0xfc42e02f} },
-/**/ {{0xbc201eec, 0x857be8a4} },
-/**/ {{0xbfa2af2b, 0x44ceebb3} },
-/**/ {{0x3f9a62b5, 0x08511639} },
-/**/ {{0xbf8018ad, 0xc8de23de} },
-/**/ {{0xbf6dc8a2, 0xc964501a} },
-/**/ {{0x3f7ac6f9, 0xeb913697} } },
-/**/ {{{0x3fed8000, 0x00000000} },
-/**/ {{0x3fe7d528, 0x289fa093} },
-/**/ {{0x3c856082, 0x1e2f3aa9} },
-/**/ {{0x3fe14c75, 0x711551bb} },
-/**/ {{0xbc80c88e, 0x71970f2c} },
-/**/ {{0xbfd13dd8, 0xe4aa5095} },
-/**/ {{0x3c66dd31, 0xb4b7ae12} },
-/**/ {{0x3fb4e38d, 0xead4c211} },
-/**/ {{0x3c513fb0, 0xe392a31e} },
-/**/ {{0x3f88355f, 0xf6b74576} },
-/**/ {{0x3ba8cb44, 0xf3561ab7} },
-/**/ {{0xbfa26058, 0x0de0faaa} },
-/**/ {{0x3f9a29f8, 0x989371f0} },
-/**/ {{0xbf805261, 0x2b085d9a} },
-/**/ {{0xbf6beccb, 0x2511c555} },
-/**/ {{0x3f7a1863, 0x87b9d333} } },
-/**/ {{{0x3feda000, 0x00000000} },
-/**/ {{0x3fe7e66c, 0x01c114fe} },
-/**/ {{0xbc8c82b8, 0x8b760b8d} },
-/**/ {{0x3fe13b3f, 0x6f037c44} },
-/**/ {{0xbc635393, 0x8562c8c0} },
-/**/ {{0xbfd12e29, 0xc7182435} },
-/**/ {{0xbc73da80, 0x0d0fda95} },
-/**/ {{0x3fb4ef4d, 0x3ba21a8b} },
-/**/ {{0xbc17c450, 0x9aa41146} },
-/**/ {{0x3f86c8e7, 0xc39dff46} },
-/**/ {{0x3c1ddd70, 0x800ba9ae} },
-/**/ {{0xbfa21230, 0x34b94b56} },
-/**/ {{0x3f99f078, 0xa827f95a} },
-/**/ {{0xbf808869, 0x19caa997} },
-/**/ {{0xbf6a1d29, 0xf8c46d26} },
-/**/ {{0x3f796b9a, 0xae59da17} } },
-/**/ {{{0x3fedc000, 0x00000000} },
-/**/ {{0x3fe7f79e, 0xacb97898} },
-/**/ {{0x3c8fd5ca, 0x80ead221} },
-/**/ {{0x3fe12a19, 0x20604825} },
-/**/ {{0xbc5cc7d6, 0xa18970f8} },
-/**/ {{0xbfd11e72, 0x1dfe6ba4} },
-/**/ {{0x3c706717, 0x9d653d1c} },
-/**/ {{0x3fb4fa57, 0xd5fcbb3b} },
-/**/ {{0x3c1922c8, 0x5f50bc06} },
-/**/ {{0x3f856283, 0xe93a179f} },
-/**/ {{0xbc01c2ec, 0x5ea7135a} },
-/**/ {{0xbfa1c4b5, 0xf0c06b4f} },
-/**/ {{0x3f99b641, 0xe48a3b04} },
-/**/ {{0xbf80badd, 0xe1280a21} },
-/**/ {{0xbf68599e, 0x1be3c5dd} },
-/**/ {{0x3f78c0b3, 0x3a72c8e6} } },
-/**/ {{{0x3fede000, 0x00000000} },
-/**/ {{0x3fe808c0, 0x3940694b} },
-/**/ {{0xbc800f32, 0x7715f6a5} },
-/**/ {{0x3fe11902, 0x8d73d98e} },
-/**/ {{0x3c71d158, 0x30f8e290} },
-/**/ {{0xbfd10eb2, 0x6fc305eb} },
-/**/ {{0xbc7fd2e3, 0x3858c4b7} },
-/**/ {{0x3fb504b0, 0xc0a99255} },
-/**/ {{0x3c55c054, 0x142e134f} },
-/**/ {{0x3f840226, 0xc2f371cf} },
-/**/ {{0xbbfc85b0, 0xfc7d6225} },
-/**/ {{0xbfa177eb, 0x53d58f53} },
-/**/ {{0x3f997b60, 0xa6a1627d} },
-/**/ {{0xbf80e9d7, 0x89757c78} },
-/**/ {{0xbf66a205, 0x0d433cd6} },
-/**/ {{0x3f7817bf, 0x9c5dbd9f} } },
-/**/ {{{0x3fee0000, 0x00000000} },
-/**/ {{0x3fe819d0, 0xb7158a4d} },
-/**/ {{0xbc7bf762, 0x29d3b917} },
-/**/ {{0x3fe107fb, 0xbe011080} },
-/**/ {{0xbc8107fb, 0xbe011080} },
-/**/ {{0xbfd0feeb, 0x40894fcd} },
-/**/ {{0x3c76fbb9, 0xc155af9a} },
-/**/ {{0x3fb50e5a, 0xfb9125f7} },
-/**/ {{0x3c357762, 0x2f3313b0} },
-/**/ {{0x3f82a7c2, 0x843ba55a} },
-/**/ {{0x3c1f4994, 0x3fc197b7} },
-/**/ {{0xbfa12bd2, 0x4b4ae875} },
-/**/ {{0x3f993fe0, 0xf3b1b1ee} },
-/**/ {{0xbf81156d, 0xd4c2083b} },
-/**/ {{0xbf64f63b, 0x0c35aa9c} },
-/**/ {{0x3f7770d0, 0xe5d0462f} } },
-/**/ {{{0x3fee2000, 0x00000000} },
-/**/ {{0x3fe82ad0, 0x36000005} },
-/**/ {{0x3c74592f, 0xce924d24} },
-/**/ {{0x3fe0f704, 0xb947c8b7} },
-/**/ {{0x3c436cd7, 0x48a651b3} },
-/**/ {{0xbfd0ef1d, 0x1237505b} },
-/**/ {{0x3c69239b, 0x1b86b9d1} },
-/**/ {{0x3fb51759, 0x7fac4e21} },
-/**/ {{0xbc42a8cc, 0xbfce0e36} },
-/**/ {{0x3f815349, 0x3b5f3edd} },
-/**/ {{0xbc25e1f1, 0x88c702d9} },
-/**/ {{0xbfa0e06c, 0xa0df17a9} },
-/**/ {{0x3f9903ce, 0x7e56b8b1} },
-/**/ {{0xbf813db8, 0x3c701e30} },
-/**/ {{0xbf63561b, 0x30c99e47} },
-/**/ {{0x3f76cbf6, 0xd5bffce0} } },
-/**/ {{{0x3fee4000, 0x00000000} },
-/**/ {{0x3fe83bbe, 0xc5cdee22} },
-/**/ {{0x3c631071, 0x04ffc6c3} },
-/**/ {{0x3fe0e61d, 0x86071468} },
-/**/ {{0xbc70ccc4, 0x59be09c9} },
-/**/ {{0xbfd0df48, 0x647af38b} },
-/**/ {{0x3c7dd47c, 0x427c295b} },
-/**/ {{0x3fb51faf, 0x3ef25277} },
-/**/ {{0x3bdf056a, 0xa81026a7} },
-/**/ {{0x3f8004ac, 0xd443a18b} },
-/**/ {{0x3c027610, 0x8178f329} },
-/**/ {{0xbfa095bb, 0xfbb3a658} },
-/**/ {{0x3f98c734, 0xa7859d46} },
-/**/ {{0xbf8162cd, 0xeefe9a81} },
-/**/ {{0xbf61c17f, 0x8330eac0} },
-/**/ {{0x3f76293f, 0xe421c20a} } },
-/**/ {{{0x3fee6000, 0x00000000} },
-/**/ {{0x3fe84c9c, 0x7653f7eb} },
-/**/ {{0xbc383611, 0xfe0a3e8f} },
-/**/ {{0x3fe0d546, 0x2a7f71b5} },
-/**/ {{0x3c757061, 0x596848c6} },
-/**/ {{0xbfd0cf6d, 0xb4cf51a6} },
-/**/ {{0x3c4c99ab, 0x5b18bb8c} },
-/**/ {{0x3fb5275f, 0x24486227} },
-/**/ {{0x3c5b4a59, 0xbb1f4f56} },
-/**/ {{0x3f7d77be, 0x36238bb2} },
-/**/ {{0x3c1ddbd1, 0xcaec6ba2} },
-/**/ {{0xbfa04bc1, 0xe1406cd0} },
-/**/ {{0x3f988a1e, 0x7f96d6ca} },
-/**/ {{0xbf8184c5, 0xcdffc380} },
-/**/ {{0xbf603841, 0x12561f8b} },
-/**/ {{0x3f7588b9, 0x4d81a668} } },
-/**/ {{{0x3fee8000, 0x00000000} },
-/**/ {{0x3fe85d69, 0x576cc2c5} },
-/**/ {{0x3c66b66e, 0x7fc8b8c3} },
-/**/ {{0x3fe0c47e, 0xac74fadc} },
-/**/ {{0xbc8035f8, 0x77bb1887} },
-/**/ {{0xbfd0bf8d, 0x7e8202a9} },
-/**/ {{0x3c798048, 0x1f4d2357} },
-/**/ {{0x3fb52e6c, 0x13725c73} },
-/**/ {{0xbc34c3af, 0xf5b19ded} },
-/**/ {{0x3f7af1a3, 0x7d9c2711} },
-/**/ {{0x3bea7ec7, 0x1af1098d} },
-/**/ {{0xbfa0027f, 0xb643d11f} },
-/**/ {{0x3f984c96, 0xc756b7d7} },
-/**/ {{0xbf81a3b6, 0x6c3ca3ae} },
-/**/ {{0xbf5d7470, 0x13459246} },
-/**/ {{0x3f74ea6f, 0x1e70d9a4} } },
-/**/ {{{0x3feea000, 0x00000000} },
-/**/ {{0x3fe86e25, 0x78f87ae5} },
-/**/ {{0x3c8022b1, 0x375cfe34} },
-/**/ {{0x3fe0b3c7, 0x11319104} },
-/**/ {{0x3c8ac394, 0x25152519} },
-/**/ {{0xbfd0afa8, 0x3ab87c8a} },
-/**/ {{0x3c724f26, 0x27b31384} },
-/**/ {{0x3fb534d8, 0xe904e078} },
-/**/ {{0xbc55bfde, 0xf8948323} },
-/**/ {{0x3f7876ec, 0xa7bb2dfb} },
-/**/ {{0xbc197116, 0x8a87be50} },
-/**/ {{0xbf9f73ed, 0x7f5f95b4} },
-/**/ {{0x3f980ea7, 0xf11c3266} },
-/**/ {{0xbf81bfb6, 0x0c032389} },
-/**/ {{0xbf5a8e77, 0x8bf305a1} },
-/**/ {{0x3f744e6c, 0x3ec72e6d} } },
-/**/ {{{0x3feec000, 0x00000000} },
-/**/ {{0x3fe87ed0, 0xeadc5a2a} },
-/**/ {{0x3c70af5a, 0xd957f4bc} },
-/**/ {{0x3fe0a31f, 0x5d8701b3} },
-/**/ {{0xbc869b25, 0x263ce937} },
-/**/ {{0xbfd09fbe, 0x60757b83} },
-/**/ {{0x3c767aff, 0xa96db9ef} },
-/**/ {{0x3fb53aa8, 0x7a589afb} },
-/**/ {{0xbc4b7e8e, 0x0844ff86} },
-/**/ {{0x3f76077c, 0xacf1a65c} },
-/**/ {{0xbc19a3b2, 0xb13331a9} },
-/**/ {{0xbf9ee450, 0x472733eb} },
-/**/ {{0x3f97d05c, 0x21e541d7} },
-/**/ {{0xbf81d8da, 0x9d9d4dfc} },
-/**/ {{0xbf57be45, 0xd3ce1b4a} },
-/**/ {{0x3f73b4ba, 0x7cb60047} } },
-/**/ {{{0x3feee000, 0x00000000} },
-/**/ {{0x3fe88f6b, 0xbd023119} },
-/**/ {{0xbc532d1d, 0x25aba660} },
-/**/ {{0x3fe09287, 0x95d126c6} },
-/**/ {{0x3c85aad3, 0xeccc37a6} },
-/**/ {{0xbfd08fd0, 0x649e7367} },
-/**/ {{0x3c71e96c, 0xed21a127} },
-/**/ {{0x3fb53fdd, 0x957ec910} },
-/**/ {{0xbc339c23, 0xaf97a601} },
-/**/ {{0x3f73a336, 0x5a18e5a2} },
-/**/ {{0xbc1f7225, 0x477571de} },
-/**/ {{0xbf9e5629, 0xd4044135} },
-/**/ {{0x3f9791bd, 0x32786dc4} },
-/**/ {{0xbf81ef39, 0xbdf030c4} },
-/**/ {{0xbf550386, 0xe21b8bcb} },
-/**/ {{0x3f731d62, 0x97aa7fb2} } },
-/**/ {{{0x3fef0000, 0x00000000} },
-/**/ {{0x3fe89ff5, 0xff57f1f8} },
-/**/ {{0xbc855b9a, 0x5e177a1b} },
-/**/ {{0x3fe081ff, 0xbdf80108} },
-/**/ {{0x3c6ffbdf, 0x80108200} },
-/**/ {{0xbfd07fde, 0xba010928} },
-/**/ {{0x3c38d37f, 0x7bae0295} },
-/**/ {{0x3fb5447b, 0x0136e69f} },
-/**/ {{0x3c50316a, 0x0dda278d} },
-/**/ {{0x3f7149fc, 0x55103947} },
-/**/ {{0x3c176e96, 0x849e505f} },
-/**/ {{0xbf9dc97b, 0xfbe9a2ee} },
-/**/ {{0x3f9752d4, 0xb08adda9} },
-/**/ {{0xbf8202e8, 0xb540d106} },
-/**/ {{0xbf525de5, 0x859de3e9} },
-/**/ {{0x3f72886c, 0x4afd9f21} } },
-/**/ {{{0x3fef2000, 0x00000000} },
-/**/ {{0x3fe8b06f, 0xc1cf3dff} },
-/**/ {{0xbc80fb31, 0x2656db6d} },
-/**/ {{0x3fe07187, 0xd971cd38} },
-/**/ {{0x3c89baa4, 0x202c20ac} },
-/**/ {{0xbfd06fe9, 0xd15893ab} },
-/**/ {{0xbc7a864b, 0xdc0cb586} },
-/**/ {{0x3fb54883, 0x7ce57fed} },
-/**/ {{0xbc49498e, 0x294f4b18} },
-/**/ {{0x3f6df762, 0x426ebecc} },
-/**/ {{0xbc022f08, 0xf28644c0} },
-/**/ {{0xbf9d3e48, 0x5c564b44} },
-/**/ {{0x3f9713ab, 0xdfea7acf} },
-/**/ {{0xbf8213fc, 0x761db35c} },
-/**/ {{0xbf4f9a17, 0x10d60f49} },
-/**/ {{0x3f71f5de, 0x58700e9b} } },
-/**/ {{{0x3fef4000, 0x00000000} },
-/**/ {{0x3fe8c0d9, 0x145cf49d} },
-/**/ {{0x3c8bea40, 0x76dc4333} },
-/**/ {{0x3fe0611f, 0xeb45139a} },
-/**/ {{0x3c7e4998, 0x65aadb1f} },
-/**/ {{0xbfd05ff2, 0x1953a316} },
-/**/ {{0x3c759922, 0xa1b67b0f} },
-/**/ {{0x3fb54bf9, 0xc08c1d66} },
-/**/ {{0x3c5b9353, 0xd220330c} },
-/**/ {{0x3f69706e, 0x478cb604} },
-/**/ {{0xbbfdb6d3, 0xa22fd45a} },
-/**/ {{0xbf9cb490, 0x5c0d1d38} },
-/**/ {{0x3f96d44b, 0xbbaba2f2} },
-/**/ {{0xbf822289, 0x9c6b7de1} },
-/**/ {{0xbf4aa143, 0xa49803b6} },
-/**/ {{0x3f7165be, 0x9270e49e} } },
-/**/ {{{0x3fef6000, 0x00000000} },
-/**/ {{0x3fe8d132, 0x06f8c4cb} },
-/**/ {{0xbc7b018c, 0xbaa89a8b} },
-/**/ {{0x3fe050c7, 0xf60ab1f4} },
-/**/ {{0x3c63f8e2, 0xc6cf5796} },
-/**/ {{0xbfd04ff7, 0xfe998dc0} },
-/**/ {{0x3c77873c, 0x7dc56419} },
-/**/ {{0x3fb54ee0, 0x7cc24121} },
-/**/ {{0x3c313117, 0x8e5c84c5} },
-/**/ {{0x3f64fee1, 0x50066301} },
-/**/ {{0x3c043698, 0x017261a1} },
-/**/ {{0xbf9c2c55, 0x2cc5b4f1} },
-/**/ {{0x3f9694bc, 0xf759f369} },
-/**/ {{0xbf822ea4, 0x6c93426a} },
-/**/ {{0xbf45d0a1, 0x135d6c51} },
-/**/ {{0x3f70d811, 0xe62dc18f} } },
-/**/ {{{0x3fef8000, 0x00000000} },
-/**/ {{0x3fe8e17a, 0xa99cc05e} },
-/**/ {{0xbc7ec182, 0xab042f61} },
-/**/ {{0x3fe0407f, 0xfbefe001} },
-/**/ {{0x3c401ffe, 0xfbf80041} },
-/**/ {{0xbfd03ffb, 0xebd00209} },
-/**/ {{0xbc53ff3c, 0xb9004112} },
-/**/ {{0x3fb5513a, 0x5aaf6d91} },
-/**/ {{0x3c54a20d, 0xc0516ddb} },
-/**/ {{0x3f60a27f, 0xc6ac4038} },
-/**/ {{0x3bf06bee, 0x2a340912} },
-/**/ {{0xbf9ba597, 0xccd6032a} },
-/**/ {{0x3f965508, 0x002bb974} },
-/**/ {{0xbf823860, 0xd2d1068b} },
-/**/ {{0xbf41277e, 0x666265bc} },
-/**/ {{0x3f704cdc, 0x656b66ea} } },
-/**/ {{{0x3fefa000, 0x00000000} },
-/**/ {{0x3fe8f1b3, 0x0c44f167} },
-/**/ {{0x3c6dd1ca, 0xb93933fd} },
-/**/ {{0x3fe03047, 0xfeb82e4e} },
-/**/ {{0x3c69ee56, 0x5272e5ac} },
-/**/ {{0xbfd02ffe, 0x49a09c45} },
-/**/ {{0xbc700a59, 0xb26267bb} },
-/**/ {{0x3fb55309, 0xfc062d2f} },
-/**/ {{0x3c5dba48, 0xb11938e0} },
-/**/ {{0x3f58b61b, 0xe4f365be} },
-/**/ {{0x3bf8b585, 0xa79ad31a} },
-/**/ {{0xbf9b2059, 0x08d4ad17} },
-/**/ {{0x3f961534, 0xfe379940} },
-/**/ {{0xbf823fd2, 0x62a1270e} },
-/**/ {{0xbf394a53, 0x3f3a0aec} },
-/**/ {{0x3f6f8842, 0xa04bcae2} } },
-/**/ {{{0x3fefc000, 0x00000000} },
-/**/ {{0x3fe901db, 0x3eeef187} },
-/**/ {{0x3c868665, 0xe5603c8f} },
-/**/ {{0x3fe0201f, 0xffbf7f80} },
-/**/ {{0x3c20201f, 0xffbf7f80} },
-/**/ {{0xbfd01fff, 0x7ebe8004} },
-/**/ {{0xbc4213ff, 0xcf979001} },
-/**/ {{0x3fb55451, 0xfb0012db} },
-/**/ {{0xbc395606, 0xf73aa59f} },
-/**/ {{0x3f50509f, 0xfc757100} },
-/**/ {{0x3bebc7da, 0xfee554d0} },
-/**/ {{0xbf9a9c99, 0x7d3424d0} },
-/**/ {{0x3f95d54b, 0xd5ac0217} },
-/**/ {{0xbf82450c, 0x564b3c49} },
-/**/ {{0xbf3091df, 0xe6d3e986} },
-/**/ {{0x3f6e7bc6, 0x3bef5a22} } },
-/**/ {{{0x3fefe000, 0x00000000} },
-/**/ {{0x3fe911f3, 0x5199833b} },
-/**/ {{0x3c63ae8a, 0x0edbf522} },
-/**/ {{0x3fe01007, 0xfffbfbfe} },
-/**/ {{0x3ba01007, 0xfffbfbfe} },
-/**/ {{0xbfd00fff, 0xefebf400} },
-/**/ {{0xbc401209, 0xfff9f97d} },
-/**/ {{0x3fb55514, 0xea5aaaf6} },
-/**/ {{0xbc529baa, 0xb5b7b240} },
-/**/ {{0x3f402827, 0xffc7abc4} },
-/**/ {{0x3b5ba3d6, 0xbfee6ab3} },
-/**/ {{0xbf9a1a59, 0x97d67093} },
-/**/ {{0x3f959554, 0x28080aaf} },
-/**/ {{0xbf824821, 0x8e892ce2} },
-/**/ {{0xbf204877, 0xfe70a2a6} },
-/**/ {{0x3f6d7447, 0x0e8ddd67} } },
-/**/ {{{0x3feff800, 0x00000000} },
-/**/ {{0x3fe91dfa, 0xd439826e} },
-/**/ {{0xbc786a19, 0x6df48d55} },
-/**/ {{0x3fe00400, 0x7ffffbff} },
-/**/ {{0xbbeffffe, 0xffbff800} },
-/**/ {{0xbfd003ff, 0xffbfebfd} },
-/**/ {{0xbb600480, 0x9ffff9fe} },
-/**/ {{0x3fb55551, 0x53aa5aab} },
-/**/ {{0xbc542a4a, 0x9baaab5b} },
-/**/ {{0x3f200a02, 0x7fffc7eb} },
-/**/ {{0xbb7dfffe, 0x4770e940} },
-/**/ {{0xbf99b9a5, 0x9997d8d0} },
-/**/ {{0x3f956555, 0x50a80a03} },
-/**/ {{0xbf824914, 0x86456493} },
-/**/ {{0xbf001207, 0x7ffe7329} },
-/**/ {{0x3f6cb1ef, 0x1c63fe2a} } },
- };
-
#else
#ifdef LITTLE_ENDI
@@ -7270,3865 +3410,5 @@
/**/ {{0X18A3BA58, 0XBF99B9A7} } },
};
- static const mynumber
- hij[241][16] = { /* x0,hij for (1/16,1) */
-/**/ {{{0x00000000, 0x3fb04000} },
-/**/ {{0x1c06693d, 0x3fb03a6d} },
-/**/ {{0xd4e7f128, 0xbc428a02} },
-/**/ {{0xe92592ae, 0x3fefdf1f} },
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-/**/ {{0x30f8e290, 0x3c71d158} },
-/**/ {{0x6fc305eb, 0xbfd10eb2} },
-/**/ {{0x3858c4b7, 0xbc7fd2e3} },
-/**/ {{0xc0a99255, 0x3fb504b0} },
-/**/ {{0x142e134f, 0x3c55c054} },
-/**/ {{0xc2f371cf, 0x3f840226} },
-/**/ {{0xfc7d6225, 0xbbfc85b0} },
-/**/ {{0x53d58f53, 0xbfa177eb} },
-/**/ {{0xa6a1627d, 0x3f997b60} },
-/**/ {{0x89757c78, 0xbf80e9d7} },
-/**/ {{0x0d433cd6, 0xbf66a205} },
-/**/ {{0x9c5dbd9f, 0x3f7817bf} } },
-/**/ {{{0x00000000, 0x3fee0000} },
-/**/ {{0xb7158a4d, 0x3fe819d0} },
-/**/ {{0x29d3b917, 0xbc7bf762} },
-/**/ {{0xbe011080, 0x3fe107fb} },
-/**/ {{0xbe011080, 0xbc8107fb} },
-/**/ {{0x40894fcd, 0xbfd0feeb} },
-/**/ {{0xc155af9a, 0x3c76fbb9} },
-/**/ {{0xfb9125f7, 0x3fb50e5a} },
-/**/ {{0x2f3313b0, 0x3c357762} },
-/**/ {{0x843ba55a, 0x3f82a7c2} },
-/**/ {{0x3fc197b7, 0x3c1f4994} },
-/**/ {{0x4b4ae875, 0xbfa12bd2} },
-/**/ {{0xf3b1b1ee, 0x3f993fe0} },
-/**/ {{0xd4c2083b, 0xbf81156d} },
-/**/ {{0x0c35aa9c, 0xbf64f63b} },
-/**/ {{0xe5d0462f, 0x3f7770d0} } },
-/**/ {{{0x00000000, 0x3fee2000} },
-/**/ {{0x36000005, 0x3fe82ad0} },
-/**/ {{0xce924d24, 0x3c74592f} },
-/**/ {{0xb947c8b7, 0x3fe0f704} },
-/**/ {{0x48a651b3, 0x3c436cd7} },
-/**/ {{0x1237505b, 0xbfd0ef1d} },
-/**/ {{0x1b86b9d1, 0x3c69239b} },
-/**/ {{0x7fac4e21, 0x3fb51759} },
-/**/ {{0xbfce0e36, 0xbc42a8cc} },
-/**/ {{0x3b5f3edd, 0x3f815349} },
-/**/ {{0x88c702d9, 0xbc25e1f1} },
-/**/ {{0xa0df17a9, 0xbfa0e06c} },
-/**/ {{0x7e56b8b1, 0x3f9903ce} },
-/**/ {{0x3c701e30, 0xbf813db8} },
-/**/ {{0x30c99e47, 0xbf63561b} },
-/**/ {{0xd5bffce0, 0x3f76cbf6} } },
-/**/ {{{0x00000000, 0x3fee4000} },
-/**/ {{0xc5cdee22, 0x3fe83bbe} },
-/**/ {{0x04ffc6c3, 0x3c631071} },
-/**/ {{0x86071468, 0x3fe0e61d} },
-/**/ {{0x59be09c9, 0xbc70ccc4} },
-/**/ {{0x647af38b, 0xbfd0df48} },
-/**/ {{0x427c295b, 0x3c7dd47c} },
-/**/ {{0x3ef25277, 0x3fb51faf} },
-/**/ {{0xa81026a7, 0x3bdf056a} },
-/**/ {{0xd443a18b, 0x3f8004ac} },
-/**/ {{0x8178f329, 0x3c027610} },
-/**/ {{0xfbb3a658, 0xbfa095bb} },
-/**/ {{0xa7859d46, 0x3f98c734} },
-/**/ {{0xeefe9a81, 0xbf8162cd} },
-/**/ {{0x8330eac0, 0xbf61c17f} },
-/**/ {{0xe421c20a, 0x3f76293f} } },
-/**/ {{{0x00000000, 0x3fee6000} },
-/**/ {{0x7653f7eb, 0x3fe84c9c} },
-/**/ {{0xfe0a3e8f, 0xbc383611} },
-/**/ {{0x2a7f71b5, 0x3fe0d546} },
-/**/ {{0x596848c6, 0x3c757061} },
-/**/ {{0xb4cf51a6, 0xbfd0cf6d} },
-/**/ {{0x5b18bb8c, 0x3c4c99ab} },
-/**/ {{0x24486227, 0x3fb5275f} },
-/**/ {{0xbb1f4f56, 0x3c5b4a59} },
-/**/ {{0x36238bb2, 0x3f7d77be} },
-/**/ {{0xcaec6ba2, 0x3c1ddbd1} },
-/**/ {{0xe1406cd0, 0xbfa04bc1} },
-/**/ {{0x7f96d6ca, 0x3f988a1e} },
-/**/ {{0xcdffc380, 0xbf8184c5} },
-/**/ {{0x12561f8b, 0xbf603841} },
-/**/ {{0x4d81a668, 0x3f7588b9} } },
-/**/ {{{0x00000000, 0x3fee8000} },
-/**/ {{0x576cc2c5, 0x3fe85d69} },
-/**/ {{0x7fc8b8c3, 0x3c66b66e} },
-/**/ {{0xac74fadc, 0x3fe0c47e} },
-/**/ {{0x77bb1887, 0xbc8035f8} },
-/**/ {{0x7e8202a9, 0xbfd0bf8d} },
-/**/ {{0x1f4d2357, 0x3c798048} },
-/**/ {{0x13725c73, 0x3fb52e6c} },
-/**/ {{0xf5b19ded, 0xbc34c3af} },
-/**/ {{0x7d9c2711, 0x3f7af1a3} },
-/**/ {{0x1af1098d, 0x3bea7ec7} },
-/**/ {{0xb643d11f, 0xbfa0027f} },
-/**/ {{0xc756b7d7, 0x3f984c96} },
-/**/ {{0x6c3ca3ae, 0xbf81a3b6} },
-/**/ {{0x13459246, 0xbf5d7470} },
-/**/ {{0x1e70d9a4, 0x3f74ea6f} } },
-/**/ {{{0x00000000, 0x3feea000} },
-/**/ {{0x78f87ae5, 0x3fe86e25} },
-/**/ {{0x375cfe34, 0x3c8022b1} },
-/**/ {{0x11319104, 0x3fe0b3c7} },
-/**/ {{0x25152519, 0x3c8ac394} },
-/**/ {{0x3ab87c8a, 0xbfd0afa8} },
-/**/ {{0x27b31384, 0x3c724f26} },
-/**/ {{0xe904e078, 0x3fb534d8} },
-/**/ {{0xf8948323, 0xbc55bfde} },
-/**/ {{0xa7bb2dfb, 0x3f7876ec} },
-/**/ {{0x8a87be50, 0xbc197116} },
-/**/ {{0x7f5f95b4, 0xbf9f73ed} },
-/**/ {{0xf11c3266, 0x3f980ea7} },
-/**/ {{0x0c032389, 0xbf81bfb6} },
-/**/ {{0x8bf305a1, 0xbf5a8e77} },
-/**/ {{0x3ec72e6d, 0x3f744e6c} } },
-/**/ {{{0x00000000, 0x3feec000} },
-/**/ {{0xeadc5a2a, 0x3fe87ed0} },
-/**/ {{0xd957f4bc, 0x3c70af5a} },
-/**/ {{0x5d8701b3, 0x3fe0a31f} },
-/**/ {{0x263ce937, 0xbc869b25} },
-/**/ {{0x60757b83, 0xbfd09fbe} },
-/**/ {{0xa96db9ef, 0x3c767aff} },
-/**/ {{0x7a589afb, 0x3fb53aa8} },
-/**/ {{0x0844ff86, 0xbc4b7e8e} },
-/**/ {{0xacf1a65c, 0x3f76077c} },
-/**/ {{0xb13331a9, 0xbc19a3b2} },
-/**/ {{0x472733eb, 0xbf9ee450} },
-/**/ {{0x21e541d7, 0x3f97d05c} },
-/**/ {{0x9d9d4dfc, 0xbf81d8da} },
-/**/ {{0xd3ce1b4a, 0xbf57be45} },
-/**/ {{0x7cb60047, 0x3f73b4ba} } },
-/**/ {{{0x00000000, 0x3feee000} },
-/**/ {{0xbd023119, 0x3fe88f6b} },
-/**/ {{0x25aba660, 0xbc532d1d} },
-/**/ {{0x95d126c6, 0x3fe09287} },
-/**/ {{0xeccc37a6, 0x3c85aad3} },
-/**/ {{0x649e7367, 0xbfd08fd0} },
-/**/ {{0xed21a127, 0x3c71e96c} },
-/**/ {{0x957ec910, 0x3fb53fdd} },
-/**/ {{0xaf97a601, 0xbc339c23} },
-/**/ {{0x5a18e5a2, 0x3f73a336} },
-/**/ {{0x477571de, 0xbc1f7225} },
-/**/ {{0xd4044135, 0xbf9e5629} },
-/**/ {{0x32786dc4, 0x3f9791bd} },
-/**/ {{0xbdf030c4, 0xbf81ef39} },
-/**/ {{0xe21b8bcb, 0xbf550386} },
-/**/ {{0x97aa7fb2, 0x3f731d62} } },
-/**/ {{{0x00000000, 0x3fef0000} },
-/**/ {{0xff57f1f8, 0x3fe89ff5} },
-/**/ {{0x5e177a1b, 0xbc855b9a} },
-/**/ {{0xbdf80108, 0x3fe081ff} },
-/**/ {{0x80108200, 0x3c6ffbdf} },
-/**/ {{0xba010928, 0xbfd07fde} },
-/**/ {{0x7bae0295, 0x3c38d37f} },
-/**/ {{0x0136e69f, 0x3fb5447b} },
-/**/ {{0x0dda278d, 0x3c50316a} },
-/**/ {{0x55103947, 0x3f7149fc} },
-/**/ {{0x849e505f, 0x3c176e96} },
-/**/ {{0xfbe9a2ee, 0xbf9dc97b} },
-/**/ {{0xb08adda9, 0x3f9752d4} },
-/**/ {{0xb540d106, 0xbf8202e8} },
-/**/ {{0x859de3e9, 0xbf525de5} },
-/**/ {{0x4afd9f21, 0x3f72886c} } },
-/**/ {{{0x00000000, 0x3fef2000} },
-/**/ {{0xc1cf3dff, 0x3fe8b06f} },
-/**/ {{0x2656db6d, 0xbc80fb31} },
-/**/ {{0xd971cd38, 0x3fe07187} },
-/**/ {{0x202c20ac, 0x3c89baa4} },
-/**/ {{0xd15893ab, 0xbfd06fe9} },
-/**/ {{0xdc0cb586, 0xbc7a864b} },
-/**/ {{0x7ce57fed, 0x3fb54883} },
-/**/ {{0x294f4b18, 0xbc49498e} },
-/**/ {{0x426ebecc, 0x3f6df762} },
-/**/ {{0xf28644c0, 0xbc022f08} },
-/**/ {{0x5c564b44, 0xbf9d3e48} },
-/**/ {{0xdfea7acf, 0x3f9713ab} },
-/**/ {{0x761db35c, 0xbf8213fc} },
-/**/ {{0x10d60f49, 0xbf4f9a17} },
-/**/ {{0x58700e9b, 0x3f71f5de} } },
-/**/ {{{0x00000000, 0x3fef4000} },
-/**/ {{0x145cf49d, 0x3fe8c0d9} },
-/**/ {{0x76dc4333, 0x3c8bea40} },
-/**/ {{0xeb45139a, 0x3fe0611f} },
-/**/ {{0x65aadb1f, 0x3c7e4998} },
-/**/ {{0x1953a316, 0xbfd05ff2} },
-/**/ {{0xa1b67b0f, 0x3c759922} },
-/**/ {{0xc08c1d66, 0x3fb54bf9} },
-/**/ {{0xd220330c, 0x3c5b9353} },
-/**/ {{0x478cb604, 0x3f69706e} },
-/**/ {{0xa22fd45a, 0xbbfdb6d3} },
-/**/ {{0x5c0d1d38, 0xbf9cb490} },
-/**/ {{0xbbaba2f2, 0x3f96d44b} },
-/**/ {{0x9c6b7de1, 0xbf822289} },
-/**/ {{0xa49803b6, 0xbf4aa143} },
-/**/ {{0x9270e49e, 0x3f7165be} } },
-/**/ {{{0x00000000, 0x3fef6000} },
-/**/ {{0x06f8c4cb, 0x3fe8d132} },
-/**/ {{0xbaa89a8b, 0xbc7b018c} },
-/**/ {{0xf60ab1f4, 0x3fe050c7} },
-/**/ {{0xc6cf5796, 0x3c63f8e2} },
-/**/ {{0xfe998dc0, 0xbfd04ff7} },
-/**/ {{0x7dc56419, 0x3c77873c} },
-/**/ {{0x7cc24121, 0x3fb54ee0} },
-/**/ {{0x8e5c84c5, 0x3c313117} },
-/**/ {{0x50066301, 0x3f64fee1} },
-/**/ {{0x017261a1, 0x3c043698} },
-/**/ {{0x2cc5b4f1, 0xbf9c2c55} },
-/**/ {{0xf759f369, 0x3f9694bc} },
-/**/ {{0x6c93426a, 0xbf822ea4} },
-/**/ {{0x135d6c51, 0xbf45d0a1} },
-/**/ {{0xe62dc18f, 0x3f70d811} } },
-/**/ {{{0x00000000, 0x3fef8000} },
-/**/ {{0xa99cc05e, 0x3fe8e17a} },
-/**/ {{0xab042f61, 0xbc7ec182} },
-/**/ {{0xfbefe001, 0x3fe0407f} },
-/**/ {{0xfbf80041, 0x3c401ffe} },
-/**/ {{0xebd00209, 0xbfd03ffb} },
-/**/ {{0xb9004112, 0xbc53ff3c} },
-/**/ {{0x5aaf6d91, 0x3fb5513a} },
-/**/ {{0xc0516ddb, 0x3c54a20d} },
-/**/ {{0xc6ac4038, 0x3f60a27f} },
-/**/ {{0x2a340912, 0x3bf06bee} },
-/**/ {{0xccd6032a, 0xbf9ba597} },
-/**/ {{0x002bb974, 0x3f965508} },
-/**/ {{0xd2d1068b, 0xbf823860} },
-/**/ {{0x666265bc, 0xbf41277e} },
-/**/ {{0x656b66ea, 0x3f704cdc} } },
-/**/ {{{0x00000000, 0x3fefa000} },
-/**/ {{0x0c44f167, 0x3fe8f1b3} },
-/**/ {{0xb93933fd, 0x3c6dd1ca} },
-/**/ {{0xfeb82e4e, 0x3fe03047} },
-/**/ {{0x5272e5ac, 0x3c69ee56} },
-/**/ {{0x49a09c45, 0xbfd02ffe} },
-/**/ {{0xb26267bb, 0xbc700a59} },
-/**/ {{0xfc062d2f, 0x3fb55309} },
-/**/ {{0xb11938e0, 0x3c5dba48} },
-/**/ {{0xe4f365be, 0x3f58b61b} },
-/**/ {{0xa79ad31a, 0x3bf8b585} },
-/**/ {{0x08d4ad17, 0xbf9b2059} },
-/**/ {{0xfe379940, 0x3f961534} },
-/**/ {{0x62a1270e, 0xbf823fd2} },
-/**/ {{0x3f3a0aec, 0xbf394a53} },
-/**/ {{0xa04bcae2, 0x3f6f8842} } },
-/**/ {{{0x00000000, 0x3fefc000} },
-/**/ {{0x3eeef187, 0x3fe901db} },
-/**/ {{0xe5603c8f, 0x3c868665} },
-/**/ {{0xffbf7f80, 0x3fe0201f} },
-/**/ {{0xffbf7f80, 0x3c20201f} },
-/**/ {{0x7ebe8004, 0xbfd01fff} },
-/**/ {{0xcf979001, 0xbc4213ff} },
-/**/ {{0xfb0012db, 0x3fb55451} },
-/**/ {{0xf73aa59f, 0xbc395606} },
-/**/ {{0xfc757100, 0x3f50509f} },
-/**/ {{0xfee554d0, 0x3bebc7da} },
-/**/ {{0x7d3424d0, 0xbf9a9c99} },
-/**/ {{0xd5ac0217, 0x3f95d54b} },
-/**/ {{0x564b3c49, 0xbf82450c} },
-/**/ {{0xe6d3e986, 0xbf3091df} },
-/**/ {{0x3bef5a22, 0x3f6e7bc6} } },
-/**/ {{{0x00000000, 0x3fefe000} },
-/**/ {{0x5199833b, 0x3fe911f3} },
-/**/ {{0x0edbf522, 0x3c63ae8a} },
-/**/ {{0xfffbfbfe, 0x3fe01007} },
-/**/ {{0xfffbfbfe, 0x3ba01007} },
-/**/ {{0xefebf400, 0xbfd00fff} },
-/**/ {{0xfff9f97d, 0xbc401209} },
-/**/ {{0xea5aaaf6, 0x3fb55514} },
-/**/ {{0xb5b7b240, 0xbc529baa} },
-/**/ {{0xffc7abc4, 0x3f402827} },
-/**/ {{0xbfee6ab3, 0x3b5ba3d6} },
-/**/ {{0x97d67093, 0xbf9a1a59} },
-/**/ {{0x28080aaf, 0x3f959554} },
-/**/ {{0x8e892ce2, 0xbf824821} },
-/**/ {{0xfe70a2a6, 0xbf204877} },
-/**/ {{0x0e8ddd67, 0x3f6d7447} } },
-/**/ {{{0x00000000, 0x3feff800} },
-/**/ {{0xd439826e, 0x3fe91dfa} },
-/**/ {{0x6df48d55, 0xbc786a19} },
-/**/ {{0x7ffffbff, 0x3fe00400} },
-/**/ {{0xffbff800, 0xbbeffffe} },
-/**/ {{0xffbfebfd, 0xbfd003ff} },
-/**/ {{0x9ffff9fe, 0xbb600480} },
-/**/ {{0x53aa5aab, 0x3fb55551} },
-/**/ {{0x9baaab5b, 0xbc542a4a} },
-/**/ {{0x7fffc7eb, 0x3f200a02} },
-/**/ {{0x4770e940, 0xbb7dfffe} },
-/**/ {{0x9997d8d0, 0xbf99b9a5} },
-/**/ {{0x50a80a03, 0x3f956555} },
-/**/ {{0x86456493, 0xbf824914} },
-/**/ {{0x7ffe7329, 0xbf001207} },
-/**/ {{0x1c63fe2a, 0x3f6cb1ef} } },
- };
-
#endif
#endif
diff --git a/sysdeps/x86_64/fpu/multiarch/Makefile b/sysdeps/x86_64/fpu/multiarch/Makefile
index 3836574f48400eb69e3b8996103cfb3b4f4d3ea8..57892c56bb7705737ac637bed851171dd125b6bc 100644
--- a/sysdeps/x86_64/fpu/multiarch/Makefile
+++ b/sysdeps/x86_64/fpu/multiarch/Makefile
@@ -9,25 +9,14 @@ libm-sysdep_routines += s_ceil-sse4_1 s_ceilf-sse4_1 s_floor-sse4_1 \
s_trunc-sse4_1 s_truncf-sse4_1
libm-sysdep_routines += e_exp-fma e_log-fma e_pow-fma s_atan-fma \
- e_asin-fma e_atan2-fma s_sin-fma s_tan-fma \
- mpa-fma \
- sincos32-fma doasin-fma dosincos-fma \
- mpatan2-fma mpatan-fma mpsqrt-fma mptan-fma
+ e_asin-fma e_atan2-fma s_sin-fma s_tan-fma
-CFLAGS-doasin-fma.c = -mfma -mavx2
-CFLAGS-dosincos-fma.c = -mfma -mavx2
CFLAGS-e_asin-fma.c = -mfma -mavx2
CFLAGS-e_atan2-fma.c = -mfma -mavx2
CFLAGS-e_exp-fma.c = -mfma -mavx2
CFLAGS-e_log-fma.c = -mfma -mavx2
CFLAGS-e_pow-fma.c = -mfma -mavx2
-CFLAGS-mpa-fma.c = -mfma -mavx2
-CFLAGS-mpatan-fma.c = -mfma -mavx2
-CFLAGS-mpatan2-fma.c = -mfma -mavx2
-CFLAGS-mpsqrt-fma.c = -mfma -mavx2
-CFLAGS-mptan-fma.c = -mfma -mavx2
CFLAGS-s_atan-fma.c = -mfma -mavx2
-CFLAGS-sincos32-fma.c = -mfma -mavx2
CFLAGS-s_sin-fma.c = -mfma -mavx2
CFLAGS-s_tan-fma.c = -mfma -mavx2
@@ -46,36 +35,23 @@ CFLAGS-s_cosf-fma.c = -mfma -mavx2
CFLAGS-s_sincosf-fma.c = -mfma -mavx2
libm-sysdep_routines += e_exp-fma4 e_log-fma4 e_pow-fma4 s_atan-fma4 \
- e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4 \
- mpa-fma4 \
- sincos32-fma4 doasin-fma4 dosincos-fma4 \
- mpatan2-fma4 mpatan-fma4 mpsqrt-fma4 mptan-fma4
+ e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4
-CFLAGS-doasin-fma4.c = -mfma4
-CFLAGS-dosincos-fma4.c = -mfma4
CFLAGS-e_asin-fma4.c = -mfma4
CFLAGS-e_atan2-fma4.c = -mfma4
CFLAGS-e_exp-fma4.c = -mfma4
CFLAGS-e_log-fma4.c = -mfma4
CFLAGS-e_pow-fma4.c = -mfma4
-CFLAGS-mpa-fma4.c = -mfma4
-CFLAGS-mpatan-fma4.c = -mfma4
-CFLAGS-mpatan2-fma4.c = -mfma4
-CFLAGS-mpsqrt-fma4.c = -mfma4
-CFLAGS-mptan-fma4.c = -mfma4
CFLAGS-s_atan-fma4.c = -mfma4
-CFLAGS-sincos32-fma4.c = -mfma4
CFLAGS-s_sin-fma4.c = -mfma4
CFLAGS-s_tan-fma4.c = -mfma4
libm-sysdep_routines += e_exp-avx e_log-avx s_atan-avx \
- e_atan2-avx s_sin-avx s_tan-avx \
- mpa-avx
+ e_atan2-avx s_sin-avx s_tan-avx
CFLAGS-e_atan2-avx.c = -msse2avx -DSSE2AVX
CFLAGS-e_exp-avx.c = -msse2avx -DSSE2AVX
CFLAGS-e_log-avx.c = -msse2avx -DSSE2AVX
-CFLAGS-mpa-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_atan-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_sin-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_tan-avx.c = -msse2avx -DSSE2AVX
diff --git a/sysdeps/x86_64/fpu/multiarch/doasin-fma.c b/sysdeps/x86_64/fpu/multiarch/doasin-fma.c
deleted file mode 100644
index 7a09865fcaa98bc50b5461e5aca181325de9d383..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/doasin-fma.c
+++ /dev/null
@@ -1,4 +0,0 @@
-#define __doasin __doasin_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/doasin.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c b/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c
deleted file mode 100644
index 53eb4194726e605af7acf7f6a6513203acbca5c8..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/doasin-fma4.c
+++ /dev/null
@@ -1,4 +0,0 @@
-#define __doasin __doasin_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/doasin.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c b/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c
deleted file mode 100644
index 5744586bdbda42646686d103074e6902087e85bf..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/dosincos-fma.c
+++ /dev/null
@@ -1,6 +0,0 @@
-#define __docos __docos_fma
-#define __dubcos __dubcos_fma
-#define __dubsin __dubsin_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/dosincos.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c b/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c
deleted file mode 100644
index 1578b2fce0be41bd396abf2da8153444f5578602..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/dosincos-fma4.c
+++ /dev/null
@@ -1,6 +0,0 @@
-#define __docos __docos_fma4
-#define __dubcos __dubcos_fma4
-#define __dubsin __dubsin_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/dosincos.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-avx.c b/sysdeps/x86_64/fpu/multiarch/mpa-avx.c
deleted file mode 100644
index 366b0b7134078e79d243502dcc23198f669cf73d..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-avx.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_avx
-#define __mul __mul_avx
-#define __sqr __sqr_avx
-#define __sub __sub_avx
-#define __dbl_mp __dbl_mp_avx
-#define __dvd __dvd_avx
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.avx")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-fma.c b/sysdeps/x86_64/fpu/multiarch/mpa-fma.c
deleted file mode 100644
index 177cc2517f66a593ecee4737804e4c771e77e848..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-fma.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_fma
-#define __mul __mul_fma
-#define __sqr __sqr_fma
-#define __sub __sub_fma
-#define __dbl_mp __dbl_mp_fma
-#define __dvd __dvd_fma
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c
deleted file mode 100644
index a4a759407e48c5cf1e375b32b1f9a54ce9f08c40..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpa-fma4.c
+++ /dev/null
@@ -1,14 +0,0 @@
-#define __add __add_fma4
-#define __mul __mul_fma4
-#define __sqr __sqr_fma4
-#define __sub __sub_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __dvd __dvd_fma4
-
-#define NO___CPY 1
-#define NO___MP_DBL 1
-#define NO___ACR 1
-#define NO__CONST 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpa.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c b/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c
deleted file mode 100644
index d216f9142de1cb646cc7c0e00816a97f88fc515f..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan-fma.c
+++ /dev/null
@@ -1,10 +0,0 @@
-#define __mpatan __mpatan_fma
-#define __add __add_fma
-#define __dvd __dvd_fma
-#define __mpsqrt __mpsqrt_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define AVOID_MPATAN_H 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c
deleted file mode 100644
index fbd3bd49a2b40a83a12f7589e7e8b233e9dee8c3..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan-fma4.c
+++ /dev/null
@@ -1,10 +0,0 @@
-#define __mpatan __mpatan_fma4
-#define __add __add_fma4
-#define __dvd __dvd_fma4
-#define __mpsqrt __mpsqrt_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define AVOID_MPATAN_H 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c b/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c
deleted file mode 100644
index 98df336f79fca93aef6d8dbcb5510c064d049ef3..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __mpatan2 __mpatan2_fma
-#define __add __add_fma
-#define __dvd __dvd_fma
-#define __mpatan __mpatan_fma
-#define __mpsqrt __mpsqrt_fma
-#define __mul __mul_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan2.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c
deleted file mode 100644
index e6e44d49b09f67d26507a13e4d1b55c4cb5a058b..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpatan2-fma4.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __mpatan2 __mpatan2_fma4
-#define __add __add_fma4
-#define __dvd __dvd_fma4
-#define __mpatan __mpatan_fma4
-#define __mpsqrt __mpsqrt_fma4
-#define __mul __mul_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpatan2.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c b/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c
deleted file mode 100644
index 44d7a23ae3332b9e4d0f3eabe115f3bf5fe099e9..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma.c
+++ /dev/null
@@ -1,8 +0,0 @@
-#define __mpsqrt __mpsqrt_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define AVOID_MPSQRT_H 1
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mpsqrt.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c b/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c
deleted file mode 100644
index f8a1ba2d92300ce3ace02bad3bc3311230037e02..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mpsqrt-fma4.c
+++ /dev/null
@@ -1,8 +0,0 @@
-#define __mpsqrt __mpsqrt_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define AVOID_MPSQRT_H 1
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mpsqrt.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mptan-fma.c b/sysdeps/x86_64/fpu/multiarch/mptan-fma.c
deleted file mode 100644
index d1a691413c58785b56fa54be0924d42103c2dc50..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mptan-fma.c
+++ /dev/null
@@ -1,7 +0,0 @@
-#define __mptan __mptan_fma
-#define __c32 __c32_fma
-#define __dvd __dvd_fma
-#define __mpranred __mpranred_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/mptan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c b/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c
deleted file mode 100644
index fb4a9d48cae2737af89d85611d64541e286138ed..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/mptan-fma4.c
+++ /dev/null
@@ -1,7 +0,0 @@
-#define __mptan __mptan_fma4
-#define __c32 __c32_fma4
-#define __dvd __dvd_fma4
-#define __mpranred __mpranred_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/mptan.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c b/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c
deleted file mode 100644
index 4152b8483535b253c6e9e75978cc0da26d13b7d7..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/sincos32-fma.c
+++ /dev/null
@@ -1,13 +0,0 @@
-#define __c32 __c32_fma
-#define __mpsin __mpsin_fma
-#define __mpsin1 __mpsin1_fma
-#define __mpcos __mpcos_fma
-#define __mpcos1 __mpcos1_fma
-#define __mpranred __mpranred_fma
-#define __add __add_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/sincos32.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c b/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c
deleted file mode 100644
index 643eedf1381a8ca8bcfd2d83448c410870bba8cf..0000000000000000000000000000000000000000
--- a/sysdeps/x86_64/fpu/multiarch/sincos32-fma4.c
+++ /dev/null
@@ -1,13 +0,0 @@
-#define __c32 __c32_fma4
-#define __mpsin __mpsin_fma4
-#define __mpsin1 __mpsin1_fma4
-#define __mpcos __mpcos_fma4
-#define __mpcos1 __mpcos1_fma4
-#define __mpranred __mpranred_fma4
-#define __add __add_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/sincos32.c>
^ permalink raw reply related [flat|nested] 19+ messages in thread
* Re: [PATCH 1/5] Remove slow paths from asin
2021-01-07 19:36 ` Wilco Dijkstra via Libc-alpha
@ 2021-01-08 10:29 ` Paul Zimmermann
0 siblings, 0 replies; 19+ messages in thread
From: Paul Zimmermann @ 2021-01-08 10:29 UTC (permalink / raw
To: Wilco Dijkstra; +Cc: libc-alpha
Dear Wilco,
the v2 of this series is ok for me, except in the meantime I have found a
larger error for tan:
tan 0 -1 0x1.c673a473b3503p+3 [0.619] 0.618365 0.6183652837454632
thus you could replace ~0.612 by ~0.619.
Let me summarize the largest errors we get with 2.32 and after this patch
(on random sampling):
2.32 after patch (with corresponding input)
asin 0.500 0.516 0x1.07fffeadab7ecp-3
acos 0.501 0.523 0x1.dffff776c7505p-1
tan 0.500 0.619 0x1.c673a473b3503p+3
atan 0.500 0.523 -0x1.f9004176e07bep-4
atan2 0.500 0.524 0x1.4aa3aada7c44fp-996,0x1.4f32ff56386e8p-993
Best regards,
Paul
^ permalink raw reply [flat|nested] 19+ messages in thread
end of thread, other threads:[~2021-01-08 10:29 UTC | newest]
Thread overview: 19+ messages (download: mbox.gz follow: Atom feed
-- links below jump to the message on this page --
2021-01-04 12:11 [PATCH 0/5] Remove remaining slow paths from math functions Wilco Dijkstra via Libc-alpha
2021-01-04 12:15 ` [PATCH 1/5] Remove slow paths from asin Wilco Dijkstra via Libc-alpha
2021-01-04 16:25 ` Paul Zimmermann
2021-01-04 17:06 ` Szabolcs Nagy via Libc-alpha
2021-01-04 17:41 ` Paul Zimmermann
2021-01-07 19:36 ` Wilco Dijkstra via Libc-alpha
2021-01-08 10:29 ` Paul Zimmermann
2021-01-07 19:23 ` [PATCH v2 1/5] Remove remaining slow paths from asin and acos Wilco Dijkstra via Libc-alpha
2021-01-04 12:17 ` [PATCH 2/5] Remove slow paths in tan Wilco Dijkstra via Libc-alpha
2021-01-04 16:36 ` Paul Zimmermann
2021-01-07 19:25 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:19 ` [PATCH 3/5] Remove slow paths from atan Wilco Dijkstra via Libc-alpha
2021-01-04 16:43 ` Paul Zimmermann
2021-01-07 19:26 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:21 ` [PATCH 4/5] Remove slow paths from atan2 Wilco Dijkstra via Libc-alpha
2021-01-04 16:49 ` Paul Zimmermann
2021-01-07 19:28 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
2021-01-04 12:25 ` [PATCH 5/5] Remove mpa files Wilco Dijkstra via Libc-alpha
2021-01-07 20:12 ` [PATCH v2 " Wilco Dijkstra via Libc-alpha
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