From 6c848d70383e1dbe932ef41723ac0abfdeec7ca8 Mon Sep 17 00:00:00 2001 From: Wilco Dijkstra Date: Mon, 8 Mar 2021 17:07:39 -0300 Subject: [PATCH] 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 Checked on x86_64-linux-gnu and aarch64-linux-gnu. [1] https://arxiv.org/pdf/1904.09481.pdf --- sysdeps/ieee754/dbl-64/e_hypot.c | 241 ++++++++++++------------------- 1 file changed, 95 insertions(+), 146 deletions(-) diff --git a/sysdeps/ieee754/dbl-64/e_hypot.c b/sysdeps/ieee754/dbl-64/e_hypot.c index 9ec4c1ced0..75bce2df4e 100644 --- a/sysdeps/ieee754/dbl-64/e_hypot.c +++ b/sysdeps/ieee754/dbl-64/e_hypot.c @@ -1,164 +1,113 @@ -/* @(#)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. - * ==================================================== - */ +/* Euclidean distance function. Double/Binary64 version. + Copyright (C) 2021 Free Software Foundation, Inc. + This file is part of the GNU C Library. -/* __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) - */ + 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 + . */ + +/* The implementation uses a correction based on 'An Improved Algorithm for + hypot(a,b)' by Carlos F. Borges [1] usingthe 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. + + The expected ULP is ~0.792. + + [1] https://arxiv.org/pdf/1904.09481.pdf */ #include #include #include +#include #include +#include "math_config.h" + +#define SCALE 0x1p-600 +#define LARGE_VAL 0x1p+511 +#define TINY_VAL 0x1p-459 +#define EPS 0x1p-54 + +/* 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; + double h = sqrt (ax * ax + ay * ay); + if (h <= 2.0 * ay) + { + double delta = h - ay; + t1 = ax * (2.0 * delta - ax); + t2 = (delta - 2.0 * (ax - ay)) * delta; + } + else + { + double delta = h - ax; + t1 = 2.0 * delta * (ax - 2.0 * ay); + t2 = (4.0 * delta - ay) * ay + delta * delta; + } + + h -= (t1 + t2) / (2.0 * h); + return h; +} double __ieee754_hypot (double x, double y) { - double a, b, t1, t2, y1, y2, w; - int32_t j, k, ha, hb; + if (!isfinite(x) || !isfinite(y)) + { + if ((isinf (x) || isinf (y)) + && !issignaling_inline (x) && !issignaling_inline (y)) + return INFINITY; + return x + y; + } - GET_HIGH_WORD (ha, x); - ha &= 0x7fffffff; - GET_HIGH_WORD (hb, y); - hb &= 0x7fffffff; - if (hb > ha) + x = fabs (x); + y = fabs (y); + + double ax = x < y ? y : x; + double ay = x < y ? x : y; + + /* If ax is huge, scale both inputs down. */ + if (__glibc_unlikely (ax > LARGE_VAL)) { - a = y; b = x; j = ha; ha = hb; hb = j; + if (__glibc_unlikely (ay <= ax * EPS)) + return math_narrow_eval (ax + ay); + + return math_narrow_eval (kernel (ax * SCALE, ay * SCALE) / SCALE); } - else + + /* If ay is tiny, scale both inputs up. */ + if (__glibc_unlikely (ay < TINY_VAL)) { - a = x; b = y; + 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; } - 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); - } - if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */ - { - 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); - } - } - /* medium size a and b */ - w = a - b; - if (w > b) - { - t1 = 0; - SET_HIGH_WORD (t1, ha); - t2 = a - t1; - w = sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1))); - } - else - { - 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 (k != 0) - { - 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; - } - else - return w; + + /* Common case: ax is not huge and ay is not tiny. */ + if (__glibc_unlikely (ay <= ax * EPS)) + return ax + ay; + + return kernel (ax, ay); } #ifndef __ieee754_hypot libm_alias_finite (__ieee754_hypot, __hypot)