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math: Use an improved algorithm for hypot (dbl-64)

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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 2021-03-08 17:07:39 -03:00 committed by Adhemerval Zanella
parent 7fe0ace3e2
commit 6c848d7038
1 changed files with 95 additions and 146 deletions
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241
sysdeps/ieee754/dbl-64/e_hypot.c
View file

@ -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
<https://www.gnu.org/licenses/>. */
/* 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 <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <math-narrow-eval.h>
#include <libm-alias-finite.h>
#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)