From: Wilco Dijkstra via Libc-alpha <libc-alpha@sourceware.org>
To: 'GNU C Library' <libc-alpha@sourceware.org>
Subject: [PATCH] Improve hypot performance
Date: Tue, 30 Nov 2021 13:04:27 +0000 [thread overview]
Message-ID: <VE1PR08MB5599DFC6FA620B3B3AFA212B83679@VE1PR08MB5599.eurprd08.prod.outlook.com> (raw)
Improve hypot performance significantly by using fma when available. The fma
version has twice the throughput of the previous version and 70% of the latency.
The non-fma version has 30% higher throughput and 10% higher latency.
Max ULP error is 0.949 with fma and 0.792 without fma.
Passes GLIBC testsuite.
---
diff --git a/sysdeps/ieee754/dbl-64/e_hypot.c b/sysdeps/ieee754/dbl-64/e_hypot.c
index 9ec4c1ced096160d7360afe2e002b8da81e0cca2..820da4ca6f87f7718510731318a8060705b61ea3 100644
--- a/sysdeps/ieee754/dbl-64/e_hypot.c
+++ b/sysdeps/ieee754/dbl-64/e_hypot.c
@@ -1,164 +1,147 @@
-/* @(#)e_hypot.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-/* __ieee754_hypot(x,y)
- *
- * Method :
- * If (assume round-to-nearest) z=x*x+y*y
- * has error less than sqrt(2)/2 ulp, than
- * sqrt(z) has error less than 1 ulp (exercise).
- *
- * So, compute sqrt(x*x+y*y) with some care as
- * follows to get the error below 1 ulp:
- *
- * Assume x>y>0;
- * (if possible, set rounding to round-to-nearest)
- * 1. if x > 2y use
- * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
- * where x1 = x with lower 32 bits cleared, x2 = x-x1; else
- * 2. if x <= 2y use
- * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
- * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
- * y1= y with lower 32 bits chopped, y2 = y-y1.
- *
- * NOTE: scaling may be necessary if some argument is too
- * large or too tiny
- *
- * Special cases:
- * hypot(x,y) is INF if x or y is +INF or -INF; else
- * hypot(x,y) is NAN if x or y is NAN.
- *
- * Accuracy:
- * hypot(x,y) returns sqrt(x^2+y^2) with error less
- * than 1 ulps (units in the last place)
- */
+/* Euclidean distance function. Double/Binary64 version.
+ Copyright (C) 2021 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ The GNU C Library 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.
+
+ The GNU C Library 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 the GNU C Library; if not, see
+ <https://www.gnu.org/licenses/>. */
+
+/* Fast ~0.949 ULP hypot implementation using fma and fmin/fmax.
+
+ The error of sqrt (x^2 + y^2) is below 1 ULP if x^2 + y^2 is computed
+ with less than 0.707 ULP error. If |x| >= |2y|, fma (x, x, y^2) has
+ ~0.625 ULP. If |x| < |2y|, fma (|2x|, |y|, (x - y)^2) has ~0.625 ULP.
+
+ If fma is not available, the implementation uses a correction based on
+ 'An Improved Algorithm for hypot(a,b)' by Carlos F. Borges [1] using
+ the MyHypot3 with the following changes:
+
+ - Handle qNaN and sNaN.
+ - Tune the 'widely varying operands' to avoid spurious underflow
+ due the multiplication and fix the return value for upwards
+ rounding mode.
+ - Handle required underflow exception for subnormal results.
+
+ [1] https://arxiv.org/pdf/1904.09481.pdf */
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
+#include <math-narrow-eval.h>
#include <libm-alias-finite.h>
+#include <libm-alias-double.h>
+#include <math_config.h>
+#include <math-svid-compat.h>
+#include <errno.h>
-double
-__ieee754_hypot (double x, double y)
+#define SCALE 0x1p-600
+#define LARGE_VAL 0x1p+511
+#define TINY_VAL 0x1p-459
+#define EPS 0x1p-54
+
+#ifndef FAST_FMINMAX
+# define FAST_FMINMAX 0
+#endif
+
+static inline double
+handle_errno (double r)
{
- double a, b, t1, t2, y1, y2, w;
- int32_t j, k, ha, hb;
-
- GET_HIGH_WORD (ha, x);
- ha &= 0x7fffffff;
- GET_HIGH_WORD (hb, y);
- hb &= 0x7fffffff;
- if (hb > ha)
- {
- a = y; b = x; j = ha; ha = hb; hb = j;
- }
+ r = math_narrow_eval (r);
+ if (isinf (r))
+ __set_errno (ERANGE);
+ return r;
+}
+
+/* Hypot kernel. The inputs must be adjusted so that ax >= ay >= 0
+ and squaring ax, ay and (ax - ay) does not overflow or underflow. */
+static inline double
+kernel (double ax, double ay)
+{
+ double t1, t2;
+#ifdef __FP_FAST_FMA
+ t1 = ay + ay;
+ t2 = ax - ay;
+
+ if (t1 >= ax)
+ return sqrt (fma (t1, ax, t2 * t2));
else
+ return sqrt (fma (ax, ax, ay * ay));
+
+#else
+ double h = sqrt (ax * ax + ay * ay);
+
+ if (h <= 2.0 * ay)
{
- a = x; b = y;
- }
- SET_HIGH_WORD (a, ha); /* a <- |a| */
- SET_HIGH_WORD (b, hb); /* b <- |b| */
- if ((ha - hb) > 0x3c00000)
- {
- return a + b;
- } /* x/y > 2**60 */
- k = 0;
- if (__glibc_unlikely (ha > 0x5f300000)) /* a>2**500 */
- {
- if (ha >= 0x7ff00000) /* Inf or NaN */
- {
- uint32_t low;
- w = a + b; /* for sNaN */
- if (issignaling (a) || issignaling (b))
- return w;
- GET_LOW_WORD (low, a);
- if (((ha & 0xfffff) | low) == 0)
- w = a;
- GET_LOW_WORD (low, b);
- if (((hb ^ 0x7ff00000) | low) == 0)
- w = b;
- return w;
- }
- /* scale a and b by 2**-600 */
- ha -= 0x25800000; hb -= 0x25800000; k += 600;
- SET_HIGH_WORD (a, ha);
- SET_HIGH_WORD (b, hb);
+ double delta = h - ay;
+ t1 = ax * (2.0 * delta - ax);
+ t2 = (delta - 2.0 * (ax - ay)) * delta;
}
- if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */
+ else
{
- if (hb <= 0x000fffff) /* subnormal b or 0 */
- {
- uint32_t low;
- GET_LOW_WORD (low, b);
- if ((hb | low) == 0)
- return a;
- t1 = 0;
- SET_HIGH_WORD (t1, 0x7fd00000); /* t1=2^1022 */
- b *= t1;
- a *= t1;
- k -= 1022;
- GET_HIGH_WORD (ha, a);
- GET_HIGH_WORD (hb, b);
- if (hb > ha)
- {
- t1 = a;
- a = b;
- b = t1;
- j = ha;
- ha = hb;
- hb = j;
- }
- }
- else /* scale a and b by 2^600 */
- {
- ha += 0x25800000; /* a *= 2^600 */
- hb += 0x25800000; /* b *= 2^600 */
- k -= 600;
- SET_HIGH_WORD (a, ha);
- SET_HIGH_WORD (b, hb);
- }
+ double delta = h - ax;
+ t1 = 2.0 * delta * (ax - 2.0 * ay);
+ t2 = (4.0 * delta - ay) * ay + delta * delta;
}
- /* medium size a and b */
- w = a - b;
- if (w > b)
+
+ h -= (t1 + t2) / (2.0 * h);
+ return h;
+#endif
+}
+
+
+double
+__ieee754_hypot (double x, double y)
+{
+ if (!isfinite (x) || !isfinite (y))
{
- t1 = 0;
- SET_HIGH_WORD (t1, ha);
- t2 = a - t1;
- w = sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1)));
+ if ((isinf (x) || isinf (y))
+ && !issignaling_inline (x) && !issignaling_inline (y))
+ return INFINITY;
+ return x + y;
}
- else
+
+ x = fabs (x);
+ y = fabs (y);
+
+ /* Ensure ax >= ay >= 0 without branches if possible. */
+ double ax = FAST_FMINMAX ? fmax (x, y) : (x < y ? y : x);
+ double ay = FAST_FMINMAX ? fmin (x, y) : (x < y ? x : y);
+
+ /* If ax is huge, scale both inputs down. */
+ if (__glibc_unlikely (ax > LARGE_VAL))
{
- a = a + a;
- y1 = 0;
- SET_HIGH_WORD (y1, hb);
- y2 = b - y1;
- t1 = 0;
- SET_HIGH_WORD (t1, ha + 0x00100000);
- t2 = a - t1;
- w = sqrt (t1 * y1 - (w * (-w) - (t1 * y2 + t2 * b)));
+ if (__glibc_unlikely (ay <= ax * EPS))
+ return handle_errno (ax + ay);
+
+ return handle_errno (kernel (ax * SCALE, ay * SCALE) / SCALE);
}
- if (k != 0)
+
+ /* If ay is tiny, scale both inputs up. */
+ if (__glibc_unlikely (ay < TINY_VAL))
{
- uint32_t high;
- t1 = 1.0;
- GET_HIGH_WORD (high, t1);
- SET_HIGH_WORD (t1, high + (k << 20));
- w *= t1;
- math_check_force_underflow_nonneg (w);
- return w;
+ if (__glibc_unlikely (ax >= ay / EPS))
+ return math_narrow_eval (ax + ay);
+
+ ax = math_narrow_eval (kernel (ax / SCALE, ay / SCALE) * SCALE);
+ math_check_force_underflow_nonneg (ax);
+ return ax;
}
- else
- return w;
+
+ /* Common case: ax is not huge and ay is not tiny. */
+ if (__glibc_unlikely (ay <= ax * EPS))
+ return math_narrow_eval (ax + ay);
+
+ return math_narrow_eval (kernel (ax, ay));
}
#ifndef __ieee754_hypot
libm_alias_finite (__ieee754_hypot, __hypot)
next reply other threads:[~2021-11-30 13:05 UTC|newest]
Thread overview: 9+ messages / expand[flat|nested] mbox.gz Atom feed top
2021-11-30 13:04 Wilco Dijkstra via Libc-alpha [this message]
2021-11-30 16:27 ` [PATCH] Improve hypot performance Adhemerval Zanella via Libc-alpha
2021-11-30 18:19 ` Adhemerval Zanella via Libc-alpha
2021-12-01 11:12 ` Paul Zimmermann
2021-12-01 12:06 ` Adhemerval Zanella via Libc-alpha
2021-12-01 13:20 ` Wilco Dijkstra via Libc-alpha
2021-12-01 13:26 ` Paul Zimmermann
2021-12-01 14:08 ` Wilco Dijkstra via Libc-alpha
2021-12-01 17:14 ` Wilco Dijkstra via Libc-alpha
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