mirror of
https://sourceware.org/git/glibc.git
synced 2024-11-10 07:10:06 +00:00
math: Use an improved algorithm for hypotl (ldbl-96)
This implementation is 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. The main advantage of the new algorithm is its precision. With a random 1e8 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc current implementation shows around 0.02% results with an error of 1 ulp (23158 results) while the new implementation only shows 0.0001% of total (111). [1] https://arxiv.org/pdf/1904.09481.pdf
This commit is contained in:
Adhemerval Zanella
parent
ccfa865a82
commit
aa9c28cde3
@ -1,142 +1,107 @@
|
||||
/* e_hypotl.c -- long double version of e_hypot.c.
|
||||
*/
|
||||
/* Euclidean distance function. Long Double/Binary96 version.
|
||||
Copyright (C) 2021 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
|
||||
/*
|
||||
* ====================================================
|
||||
* 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.
|
||||
* ====================================================
|
||||
*/
|
||||
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.
|
||||
|
||||
/* __ieee754_hypotl(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 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/>. */
|
||||
|
||||
/* This implementation is 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 <libm-alias-finite.h>
|
||||
|
||||
long double __ieee754_hypotl(long double x, long double y)
|
||||
{
|
||||
long double a,b,t1,t2,y1,y2,w;
|
||||
uint32_t j,k,ea,eb;
|
||||
#define SCALE 0x8p-8257L
|
||||
#define LARGE_VAL 0xb.504f333f9de6484p+8188L
|
||||
#define TINY_VAL 0x8p-8194L
|
||||
#define EPS 0x8p-68L
|
||||
|
||||
GET_LDOUBLE_EXP(ea,x);
|
||||
ea &= 0x7fff;
|
||||
GET_LDOUBLE_EXP(eb,y);
|
||||
eb &= 0x7fff;
|
||||
if(eb > ea) {a=y;b=x;j=ea; ea=eb;eb=j;} else {a=x;b=y;}
|
||||
SET_LDOUBLE_EXP(a,ea); /* a <- |a| */
|
||||
SET_LDOUBLE_EXP(b,eb); /* b <- |b| */
|
||||
if((ea-eb)>0x46) {return a+b;} /* x/y > 2**70 */
|
||||
k=0;
|
||||
if(__builtin_expect(ea > 0x5f3f,0)) { /* a>2**8000 */
|
||||
if(ea == 0x7fff) { /* Inf or NaN */
|
||||
uint32_t exp __attribute__ ((unused));
|
||||
uint32_t high,low;
|
||||
w = a+b; /* for sNaN */
|
||||
if (issignaling (a) || issignaling (b))
|
||||
return w;
|
||||
GET_LDOUBLE_WORDS(exp,high,low,a);
|
||||
if(((high&0x7fffffff)|low)==0) w = a;
|
||||
GET_LDOUBLE_WORDS(exp,high,low,b);
|
||||
if(((eb^0x7fff)|(high&0x7fffffff)|low)==0) w = b;
|
||||
return w;
|
||||
}
|
||||
/* scale a and b by 2**-9600 */
|
||||
ea -= 0x2580; eb -= 0x2580; k += 9600;
|
||||
SET_LDOUBLE_EXP(a,ea);
|
||||
SET_LDOUBLE_EXP(b,eb);
|
||||
}
|
||||
if(__builtin_expect(eb < 0x20bf, 0)) { /* b < 2**-8000 */
|
||||
if(eb == 0) { /* subnormal b or 0 */
|
||||
uint32_t exp __attribute__ ((unused));
|
||||
uint32_t high,low;
|
||||
GET_LDOUBLE_WORDS(exp,high,low,b);
|
||||
if((high|low)==0) return a;
|
||||
SET_LDOUBLE_WORDS(t1, 0x7ffd, 0x80000000, 0); /* t1=2^16382 */
|
||||
b *= t1;
|
||||
a *= t1;
|
||||
k -= 16382;
|
||||
GET_LDOUBLE_EXP (ea, a);
|
||||
GET_LDOUBLE_EXP (eb, b);
|
||||
if (eb > ea)
|
||||
{
|
||||
t1 = a;
|
||||
a = b;
|
||||
b = t1;
|
||||
j = ea;
|
||||
ea = eb;
|
||||
eb = j;
|
||||
}
|
||||
} else { /* scale a and b by 2^9600 */
|
||||
ea += 0x2580; /* a *= 2^9600 */
|
||||
eb += 0x2580; /* b *= 2^9600 */
|
||||
k -= 9600;
|
||||
SET_LDOUBLE_EXP(a,ea);
|
||||
SET_LDOUBLE_EXP(b,eb);
|
||||
}
|
||||
}
|
||||
/* medium size a and b */
|
||||
w = a-b;
|
||||
if (w>b) {
|
||||
uint32_t high;
|
||||
GET_LDOUBLE_MSW(high,a);
|
||||
SET_LDOUBLE_WORDS(t1,ea,high,0);
|
||||
t2 = a-t1;
|
||||
w = sqrtl(t1*t1-(b*(-b)-t2*(a+t1)));
|
||||
} else {
|
||||
uint32_t high;
|
||||
GET_LDOUBLE_MSW(high,b);
|
||||
a = a+a;
|
||||
SET_LDOUBLE_WORDS(y1,eb,high,0);
|
||||
y2 = b - y1;
|
||||
GET_LDOUBLE_MSW(high,a);
|
||||
SET_LDOUBLE_WORDS(t1,ea+1,high,0);
|
||||
t2 = a - t1;
|
||||
w = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b)));
|
||||
}
|
||||
if(k!=0) {
|
||||
uint32_t exp;
|
||||
t1 = 1.0;
|
||||
GET_LDOUBLE_EXP(exp,t1);
|
||||
SET_LDOUBLE_EXP(t1,exp+k);
|
||||
w *= t1;
|
||||
math_check_force_underflow_nonneg (w);
|
||||
return w;
|
||||
} else return w;
|
||||
/* 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 long double
|
||||
kernel (long double ax, long double ay)
|
||||
{
|
||||
long double t1, t2;
|
||||
long double h = sqrtl (ax * ax + ay * ay);
|
||||
if (h <= 2.0L * ay)
|
||||
{
|
||||
long double delta = h - ay;
|
||||
t1 = ax * (2.0L * delta - ax);
|
||||
t2 = (delta - 2.0L * (ax - ay)) * delta;
|
||||
}
|
||||
else
|
||||
{
|
||||
long double delta = h - ax;
|
||||
t1 = 2.0L * delta * (ax - 2.0L * ay);
|
||||
t2 = (4.0L * delta - ay) * ay + delta * delta;
|
||||
}
|
||||
|
||||
h -= (t1 + t2) / (2.0L * h);
|
||||
return h;
|
||||
}
|
||||
|
||||
long double
|
||||
__ieee754_hypotl (long double x, long double y)
|
||||
{
|
||||
if (!isfinite(x) || !isfinite(y))
|
||||
{
|
||||
if ((isinf (x) || isinf (y))
|
||||
&& !issignaling (x) && !issignaling (y))
|
||||
return INFINITY;
|
||||
return x + y;
|
||||
}
|
||||
|
||||
x = fabsl (x);
|
||||
y = fabsl (y);
|
||||
|
||||
long double ax = x < y ? y : x;
|
||||
long double ay = x < y ? x : y;
|
||||
|
||||
/* If ax is huge, scale both inputs down. */
|
||||
if (__glibc_unlikely (ax > LARGE_VAL))
|
||||
{
|
||||
if (__glibc_unlikely (ay <= ax * EPS))
|
||||
return ax + ay;
|
||||
|
||||
return kernel (ax * SCALE, ay * SCALE) / SCALE;
|
||||
}
|
||||
|
||||
/* If ay is tiny, scale both inputs up. */
|
||||
if (__glibc_unlikely (ay < TINY_VAL))
|
||||
{
|
||||
if (__glibc_unlikely (ax >= ay / EPS))
|
||||
return ax + ay;
|
||||
|
||||
ax = kernel (ax / SCALE, ay / SCALE) * SCALE;
|
||||
math_check_force_underflow_nonneg (ax);
|
||||
return ax;
|
||||
}
|
||||
|
||||
/* Common case: ax is not huge and ay is not tiny. */
|
||||
if (__glibc_unlikely (ay <= ax * EPS))
|
||||
return ax + ay;
|
||||
|
||||
return kernel (ax, ay);
|
||||
}
|
||||
libm_alias_finite (__ieee754_hypotl, __hypotl)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user