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math: Use an improved algorithm for hypotl (ldbl-128)

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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 1e9 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc
current implementation shows around 0.05% results with an error of
1 ulp (453266 results) while the new implementation only shows
0.0001% of total (1280).

Checked on aarch64-linux-gnu and x86_64-linux-gnu.

[1] https://arxiv.org/pdf/1904.09481.pdf
This commit is contained in:
Adhemerval Zanella 2021-04-05 23:55:55 -03:00
parent aa9c28cde3
commit c212d6397e
1 changed files with 93 additions and 127 deletions
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220
sysdeps/ieee754/ldbl-128/e_hypotl.c
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/* e_hypotl.c -- long double version of e_hypot.c.
*/
/* Euclidean distance function. Long Double/Binary128 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 sqrtl(2)/2 ulp, than
* sqrtl(z) has error less than 1 ulp (exercise).
*
* So, compute sqrtl(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 64 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 64 bits cleared, t2 = 2x-t1,
* y1= y with lower 64 bits chopped, y2 = y-y1.
*
* NOTE: scaling may be necessary if some argument is too
* large or too tiny
*
* Special cases:
* hypotl(x,y) is INF if x or y is +INF or -INF; else
* hypotl(x,y) is NAN if x or y is NAN.
*
* Accuracy:
* hypotl(x,y) returns sqrtl(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>
#define SCALE L(0x1p-8303)
#define LARGE_VAL L(0x1.6a09e667f3bcc908b2fb1366ea95p+8191)
#define TINY_VAL L(0x1p-8191)
#define EPS L(0x1p-114)
/* 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 _Float128
kernel (_Float128 ax, _Float128 ay)
{
_Float128 t1, t2;
_Float128 h = sqrtl (ax * ax + ay * ay);
if (h <= L(2.0) * ay)
{
_Float128 delta = h - ay;
t1 = ax * (L(2.0) * delta - ax);
t2 = (delta - L(2.0) * (ax - ay)) * delta;
}
else
{
_Float128 delta = h - ax;
t1 = L(2.0) * delta * (ax - L(2.0) * ay);
t2 = (L(4.0) * delta - ay) * ay + delta * delta;
}
h -= (t1 + t2) / (L(2.0) * h);
return h;
}
_Float128
__ieee754_hypotl(_Float128 x, _Float128 y)
{
_Float128 a,b,t1,t2,y1,y2,w;
int64_t j,k,ha,hb;
if (!isfinite(x) || !isfinite(y))
{
if ((isinf (x) || isinf (y))
&& !issignaling (x) && !issignaling (y))
return INFINITY;
return x + y;
}
GET_LDOUBLE_MSW64(ha,x);
ha &= 0x7fffffffffffffffLL;
GET_LDOUBLE_MSW64(hb,y);
hb &= 0x7fffffffffffffffLL;
if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
SET_LDOUBLE_MSW64(a,ha); /* a <- |a| */
SET_LDOUBLE_MSW64(b,hb); /* b <- |b| */
if((ha-hb)>0x78000000000000LL) {return a+b;} /* x/y > 2**120 */
k=0;
if(ha > 0x5f3f000000000000LL) { /* a>2**8000 */
if(ha >= 0x7fff000000000000LL) { /* Inf or NaN */
uint64_t low;
w = a+b; /* for sNaN */
if (issignaling (a) || issignaling (b))
return w;
GET_LDOUBLE_LSW64(low,a);
if(((ha&0xffffffffffffLL)|low)==0) w = a;
GET_LDOUBLE_LSW64(low,b);
if(((hb^0x7fff000000000000LL)|low)==0) w = b;
return w;
}
/* scale a and b by 2**-9600 */
ha -= 0x2580000000000000LL;
hb -= 0x2580000000000000LL; k += 9600;
SET_LDOUBLE_MSW64(a,ha);
SET_LDOUBLE_MSW64(b,hb);
}
if(hb < 0x20bf000000000000LL) { /* b < 2**-8000 */
if(hb <= 0x0000ffffffffffffLL) { /* subnormal b or 0 */
uint64_t low;
GET_LDOUBLE_LSW64(low,b);
if((hb|low)==0) return a;
t1=0;
SET_LDOUBLE_MSW64(t1,0x7ffd000000000000LL); /* t1=2^16382 */
b *= t1;
a *= t1;
k -= 16382;
GET_LDOUBLE_MSW64 (ha, a);
GET_LDOUBLE_MSW64 (hb, b);
if (hb > ha)
{
t1 = a;
a = b;
b = t1;
j = ha;
ha = hb;
hb = j;
}
} else { /* scale a and b by 2^9600 */
ha += 0x2580000000000000LL; /* a *= 2^9600 */
hb += 0x2580000000000000LL; /* b *= 2^9600 */
k -= 9600;
SET_LDOUBLE_MSW64(a,ha);
SET_LDOUBLE_MSW64(b,hb);
}
}
/* medium size a and b */
w = a-b;
if (w>b) {
t1 = 0;
SET_LDOUBLE_MSW64(t1,ha);
t2 = a-t1;
w = sqrtl(t1*t1-(b*(-b)-t2*(a+t1)));
} else {
a = a+a;
y1 = 0;
SET_LDOUBLE_MSW64(y1,hb);
y2 = b - y1;
t1 = 0;
SET_LDOUBLE_MSW64(t1,ha+0x0001000000000000LL);
t2 = a - t1;
w = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b)));
}
if(k!=0) {
uint64_t high;
t1 = 1;
GET_LDOUBLE_MSW64(high,t1);
SET_LDOUBLE_MSW64(t1,high+(k<<48));
w *= t1;
math_check_force_underflow_nonneg (w);
return w;
} else return w;
x = fabsl (x);
y = fabsl (y);
_Float128 ax = x < y ? y : x;
_Float128 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)