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math: Use an improved algorithm for hypot (dbl-64)
This implementation is based on the '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 denormal results.
The main advantage of the new algorithm is its precision: with a
random 1e9 input pairs in the range of [DBL_MIN, DBL_MAX], glibc
current implementation shows around 0.34% results with an error of
1 ulp (3424869 results) while the new implementation only shows
0.002% of total (18851).
The performance result are also only slight worse than current
implementation. On x86_64 (Ryzen 5900X) with gcc 12:
Before:
"hypot": {
"workload-random": {
"duration": 3.73319e+09,
"iterations": 1.12e+08,
"reciprocal-throughput": 22.8737,
"latency": 43.7904,
"max-throughput": 4.37184e+07,
"min-throughput": 2.28361e+07
}
}
After:
"hypot": {
"workload-random": {
"duration": 3.7597e+09,
"iterations": 9.8e+07,
"reciprocal-throughput": 23.7547,
"latency": 52.9739,
"max-throughput": 4.2097e+07,
"min-throughput": 1.88772e+07
}
}
Co-Authored-By: Adhemerval Zanella <adhemerval.zanella@linaro.org>
Checked on x86_64-linux-gnu and aarch64-linux-gnu.
[1] https://arxiv.org/pdf/1904.09481.pdf
This commit is contained in:
Wilco Dijkstra
Adhemerval Zanella
parent
7fe0ace3e2
commit
6c848d7038
@ -1,164 +1,113 @@
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/* @(#)e_hypot.c 5.1 93/09/24 */
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/* Euclidean distance function. Double/Binary64 version.
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/*
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Copyright (C) 2021 Free Software Foundation, Inc.
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* ====================================================
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This file is part of the GNU C Library.
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* __ieee754_hypot(x,y)
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The GNU C Library is free software; you can redistribute it and/or
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*
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modify it under the terms of the GNU Lesser General Public
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* Method :
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License as published by the Free Software Foundation; either
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* If (assume round-to-nearest) z=x*x+y*y
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version 2.1 of the License, or (at your option) any later version.
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* has error less than sqrt(2)/2 ulp, than
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* sqrt(z) has error less than 1 ulp (exercise).
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The GNU C Library is distributed in the hope that it will be useful,
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*
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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* So, compute sqrt(x*x+y*y) with some care as
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* follows to get the error below 1 ulp:
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Lesser General Public License for more details.
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*
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* Assume x>y>0;
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You should have received a copy of the GNU Lesser General Public
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* (if possible, set rounding to round-to-nearest)
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License along with the GNU C Library; if not, see
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* 1. if x > 2y use
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<https://www.gnu.org/licenses/>. */
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* x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
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* where x1 = x with lower 32 bits cleared, x2 = x-x1; else
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/* The implementation uses a correction based on 'An Improved Algorithm for
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* 2. if x <= 2y use
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hypot(a,b)' by Carlos F. Borges [1] usingthe MyHypot3 with the following
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* t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
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changes:
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* where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
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* y1= y with lower 32 bits chopped, y2 = y-y1.
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- Handle qNaN and sNaN.
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*
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- Tune the 'widely varying operands' to avoid spurious underflow
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* NOTE: scaling may be necessary if some argument is too
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due the multiplication and fix the return value for upwards
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* large or too tiny
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rounding mode.
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*
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- Handle required underflow exception for subnormal results.
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* Special cases:
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* hypot(x,y) is INF if x or y is +INF or -INF; else
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The expected ULP is ~0.792.
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* hypot(x,y) is NAN if x or y is NAN.
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*
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[1] https://arxiv.org/pdf/1904.09481.pdf */
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* Accuracy:
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* hypot(x,y) returns sqrt(x^2+y^2) with error less
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* than 1 ulps (units in the last place)
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*/
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#include <math.h>
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#include <math.h>
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#include <math_private.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <math-underflow.h>
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#include <math-narrow-eval.h>
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#include <libm-alias-finite.h>
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#include <libm-alias-finite.h>
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#include "math_config.h"
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#define SCALE 0x1p-600
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#define LARGE_VAL 0x1p+511
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#define TINY_VAL 0x1p-459
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#define EPS 0x1p-54
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/* Hypot kernel. The inputs must be adjusted so that ax >= ay >= 0
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and squaring ax, ay and (ax - ay) does not overflow or underflow. */
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static inline double
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kernel (double ax, double ay)
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{
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double t1, t2;
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double h = sqrt (ax * ax + ay * ay);
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if (h <= 2.0 * ay)
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{
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double delta = h - ay;
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t1 = ax * (2.0 * delta - ax);
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t2 = (delta - 2.0 * (ax - ay)) * delta;
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}
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else
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{
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double delta = h - ax;
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t1 = 2.0 * delta * (ax - 2.0 * ay);
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t2 = (4.0 * delta - ay) * ay + delta * delta;
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}
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h -= (t1 + t2) / (2.0 * h);
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return h;
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}
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double
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double
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__ieee754_hypot (double x, double y)
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__ieee754_hypot (double x, double y)
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{
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{
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double a, b, t1, t2, y1, y2, w;
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if (!isfinite(x) || !isfinite(y))
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int32_t j, k, ha, hb;
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{
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if ((isinf (x) || isinf (y))
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&& !issignaling_inline (x) && !issignaling_inline (y))
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return INFINITY;
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return x + y;
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}
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GET_HIGH_WORD (ha, x);
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x = fabs (x);
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ha &= 0x7fffffff;
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y = fabs (y);
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GET_HIGH_WORD (hb, y);
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hb &= 0x7fffffff;
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double ax = x < y ? y : x;
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if (hb > ha)
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double ay = x < y ? x : y;
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/* If ax is huge, scale both inputs down. */
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if (__glibc_unlikely (ax > LARGE_VAL))
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{
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{
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a = y; b = x; j = ha; ha = hb; hb = j;
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if (__glibc_unlikely (ay <= ax * EPS))
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return math_narrow_eval (ax + ay);
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return math_narrow_eval (kernel (ax * SCALE, ay * SCALE) / SCALE);
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}
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}
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else
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/* If ay is tiny, scale both inputs up. */
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if (__glibc_unlikely (ay < TINY_VAL))
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{
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{
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a = x; b = y;
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if (__glibc_unlikely (ax >= ay / EPS))
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return math_narrow_eval (ax + ay);
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ax = math_narrow_eval (kernel (ax / SCALE, ay / SCALE) * SCALE);
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math_check_force_underflow_nonneg (ax);
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return ax;
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}
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}
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SET_HIGH_WORD (a, ha); /* a <- |a| */
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SET_HIGH_WORD (b, hb); /* b <- |b| */
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/* Common case: ax is not huge and ay is not tiny. */
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if ((ha - hb) > 0x3c00000)
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if (__glibc_unlikely (ay <= ax * EPS))
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{
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return ax + ay;
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return a + b;
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} /* x/y > 2**60 */
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return kernel (ax, ay);
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k = 0;
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if (__glibc_unlikely (ha > 0x5f300000)) /* a>2**500 */
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{
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if (ha >= 0x7ff00000) /* Inf or NaN */
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{
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uint32_t low;
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w = a + b; /* for sNaN */
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if (issignaling (a) || issignaling (b))
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return w;
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GET_LOW_WORD (low, a);
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if (((ha & 0xfffff) | low) == 0)
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w = a;
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GET_LOW_WORD (low, b);
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if (((hb ^ 0x7ff00000) | low) == 0)
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w = b;
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return w;
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}
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/* scale a and b by 2**-600 */
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ha -= 0x25800000; hb -= 0x25800000; k += 600;
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SET_HIGH_WORD (a, ha);
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SET_HIGH_WORD (b, hb);
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}
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if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */
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{
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if (hb <= 0x000fffff) /* subnormal b or 0 */
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{
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uint32_t low;
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GET_LOW_WORD (low, b);
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if ((hb | low) == 0)
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return a;
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t1 = 0;
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SET_HIGH_WORD (t1, 0x7fd00000); /* t1=2^1022 */
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b *= t1;
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a *= t1;
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k -= 1022;
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GET_HIGH_WORD (ha, a);
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GET_HIGH_WORD (hb, b);
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if (hb > ha)
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{
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t1 = a;
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a = b;
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b = t1;
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j = ha;
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ha = hb;
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hb = j;
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}
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}
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else /* scale a and b by 2^600 */
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{
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ha += 0x25800000; /* a *= 2^600 */
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hb += 0x25800000; /* b *= 2^600 */
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k -= 600;
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SET_HIGH_WORD (a, ha);
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SET_HIGH_WORD (b, hb);
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}
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}
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/* medium size a and b */
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w = a - b;
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if (w > b)
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{
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t1 = 0;
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SET_HIGH_WORD (t1, ha);
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t2 = a - t1;
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w = sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1)));
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}
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else
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{
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a = a + a;
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y1 = 0;
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SET_HIGH_WORD (y1, hb);
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y2 = b - y1;
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t1 = 0;
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SET_HIGH_WORD (t1, ha + 0x00100000);
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t2 = a - t1;
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w = sqrt (t1 * y1 - (w * (-w) - (t1 * y2 + t2 * b)));
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}
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if (k != 0)
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{
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uint32_t high;
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t1 = 1.0;
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GET_HIGH_WORD (high, t1);
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SET_HIGH_WORD (t1, high + (k << 20));
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w *= t1;
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math_check_force_underflow_nonneg (w);
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return w;
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}
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else
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return w;
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}
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}
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#ifndef __ieee754_hypot
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#ifndef __ieee754_hypot
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libm_alias_finite (__ieee754_hypot, __hypot)
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libm_alias_finite (__ieee754_hypot, __hypot)
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