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
---
sysdeps/ieee754/ldbl-128/e_hypotl.c | 226 ++++++++++++----------------
1 file changed, 96 insertions(+), 130 deletions(-)
@@ -1,141 +1,107 @@
-/* e_hypotl.c -- long double version of e_hypot.c.
- */
-
-/*
- * ====================================================
- * 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.
- * ====================================================
- */
-
-/* __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)
- */
+/* Euclidean distance function. Long Double/Binary128 version.
+ Copyright (C) 2021 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ 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/>. */
+
+/* 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;
-
- 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;
+ 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);
+
+ _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)