mirror of git://sourceware.org/git/glibc.git
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
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/* e_hypotl.c -- long double version of e_hypot.c.
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/* Euclidean distance function. Long Double/Binary96 version.
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*/
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Copyright (C) 2021 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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/*
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The GNU C Library is free software; you can redistribute it and/or
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* ====================================================
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modify it under the terms of the GNU Lesser General Public
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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License as published by the Free Software Foundation; either
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*
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version 2.1 of the License, or (at your option) any later version.
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* __ieee754_hypotl(x,y)
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The GNU C Library is distributed in the hope that it will be useful,
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*
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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* Method :
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* If (assume round-to-nearest) z=x*x+y*y
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Lesser General Public License for more details.
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* has error less than sqrt(2)/2 ulp, than
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* sqrt(z) has error less than 1 ulp (exercise).
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You should have received a copy of the GNU Lesser General Public
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*
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License along with the GNU C Library; if not, see
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* So, compute sqrt(x*x+y*y) with some care as
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<https://www.gnu.org/licenses/>. */
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* follows to get the error below 1 ulp:
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*
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/* This implementation is based on 'An Improved Algorithm for hypot(a,b)' by
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* Assume x>y>0;
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Carlos F. Borges [1] using the MyHypot3 with the following changes:
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* (if possible, set rounding to round-to-nearest)
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* 1. if x > 2y use
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- Handle qNaN and sNaN.
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* x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
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- Tune the 'widely varying operands' to avoid spurious underflow
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* where x1 = x with lower 32 bits cleared, x2 = x-x1; else
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due the multiplication and fix the return value for upwards
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* 2. if x <= 2y use
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rounding mode.
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* t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
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- Handle required underflow exception for subnormal results.
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* where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
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* y1= y with lower 32 bits chopped, y2 = y-y1.
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[1] https://arxiv.org/pdf/1904.09481.pdf */
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*
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* NOTE: scaling may be necessary if some argument is too
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* large or too tiny
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*
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* Special cases:
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* hypot(x,y) is INF if x or y is +INF or -INF; else
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* hypot(x,y) is NAN if x or y is NAN.
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*
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* Accuracy:
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* hypot(x,y) returns sqrt(x^2+y^2) with error less
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* than 1 ulps (units in the last place)
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*/
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#include <math.h>
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#include <math.h>
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#include <math_private.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <math-underflow.h>
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#include <libm-alias-finite.h>
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#include <libm-alias-finite.h>
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long double __ieee754_hypotl(long double x, long double y)
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#define SCALE 0x8p-8257L
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{
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#define LARGE_VAL 0xb.504f333f9de6484p+8188L
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long double a,b,t1,t2,y1,y2,w;
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#define TINY_VAL 0x8p-8194L
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uint32_t j,k,ea,eb;
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#define EPS 0x8p-68L
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GET_LDOUBLE_EXP(ea,x);
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/* Hypot kernel. The inputs must be adjusted so that ax >= ay >= 0
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ea &= 0x7fff;
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and squaring ax, ay and (ax - ay) does not overflow or underflow. */
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GET_LDOUBLE_EXP(eb,y);
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static inline long double
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eb &= 0x7fff;
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kernel (long double ax, long double ay)
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if(eb > ea) {a=y;b=x;j=ea; ea=eb;eb=j;} else {a=x;b=y;}
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{
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SET_LDOUBLE_EXP(a,ea); /* a <- |a| */
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long double t1, t2;
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SET_LDOUBLE_EXP(b,eb); /* b <- |b| */
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long double h = sqrtl (ax * ax + ay * ay);
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if((ea-eb)>0x46) {return a+b;} /* x/y > 2**70 */
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if (h <= 2.0L * ay)
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k=0;
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{
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if(__builtin_expect(ea > 0x5f3f,0)) { /* a>2**8000 */
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long double delta = h - ay;
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if(ea == 0x7fff) { /* Inf or NaN */
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t1 = ax * (2.0L * delta - ax);
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uint32_t exp __attribute__ ((unused));
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t2 = (delta - 2.0L * (ax - ay)) * delta;
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uint32_t high,low;
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}
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w = a+b; /* for sNaN */
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else
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if (issignaling (a) || issignaling (b))
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{
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return w;
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long double delta = h - ax;
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GET_LDOUBLE_WORDS(exp,high,low,a);
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t1 = 2.0L * delta * (ax - 2.0L * ay);
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if(((high&0x7fffffff)|low)==0) w = a;
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t2 = (4.0L * delta - ay) * ay + delta * delta;
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GET_LDOUBLE_WORDS(exp,high,low,b);
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}
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if(((eb^0x7fff)|(high&0x7fffffff)|low)==0) w = b;
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return w;
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h -= (t1 + t2) / (2.0L * h);
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}
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return h;
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/* scale a and b by 2**-9600 */
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}
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ea -= 0x2580; eb -= 0x2580; k += 9600;
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SET_LDOUBLE_EXP(a,ea);
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long double
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SET_LDOUBLE_EXP(b,eb);
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__ieee754_hypotl (long double x, long double y)
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}
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{
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if(__builtin_expect(eb < 0x20bf, 0)) { /* b < 2**-8000 */
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if (!isfinite(x) || !isfinite(y))
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if(eb == 0) { /* subnormal b or 0 */
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{
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uint32_t exp __attribute__ ((unused));
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if ((isinf (x) || isinf (y))
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uint32_t high,low;
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&& !issignaling (x) && !issignaling (y))
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GET_LDOUBLE_WORDS(exp,high,low,b);
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return INFINITY;
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if((high|low)==0) return a;
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return x + y;
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SET_LDOUBLE_WORDS(t1, 0x7ffd, 0x80000000, 0); /* t1=2^16382 */
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}
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b *= t1;
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a *= t1;
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x = fabsl (x);
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k -= 16382;
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y = fabsl (y);
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GET_LDOUBLE_EXP (ea, a);
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GET_LDOUBLE_EXP (eb, b);
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long double ax = x < y ? y : x;
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if (eb > ea)
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long double ay = x < y ? x : y;
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{
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t1 = a;
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/* If ax is huge, scale both inputs down. */
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a = b;
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if (__glibc_unlikely (ax > LARGE_VAL))
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b = t1;
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{
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j = ea;
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if (__glibc_unlikely (ay <= ax * EPS))
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ea = eb;
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return ax + ay;
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eb = j;
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}
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return kernel (ax * SCALE, ay * SCALE) / SCALE;
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} else { /* scale a and b by 2^9600 */
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}
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ea += 0x2580; /* a *= 2^9600 */
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eb += 0x2580; /* b *= 2^9600 */
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/* If ay is tiny, scale both inputs up. */
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k -= 9600;
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if (__glibc_unlikely (ay < TINY_VAL))
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SET_LDOUBLE_EXP(a,ea);
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{
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SET_LDOUBLE_EXP(b,eb);
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if (__glibc_unlikely (ax >= ay / EPS))
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}
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return ax + ay;
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}
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/* medium size a and b */
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ax = kernel (ax / SCALE, ay / SCALE) * SCALE;
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w = a-b;
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math_check_force_underflow_nonneg (ax);
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if (w>b) {
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return ax;
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uint32_t high;
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}
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GET_LDOUBLE_MSW(high,a);
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SET_LDOUBLE_WORDS(t1,ea,high,0);
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/* Common case: ax is not huge and ay is not tiny. */
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t2 = a-t1;
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if (__glibc_unlikely (ay <= ax * EPS))
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w = sqrtl(t1*t1-(b*(-b)-t2*(a+t1)));
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return ax + ay;
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} else {
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uint32_t high;
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return kernel (ax, ay);
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GET_LDOUBLE_MSW(high,b);
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a = a+a;
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SET_LDOUBLE_WORDS(y1,eb,high,0);
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y2 = b - y1;
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GET_LDOUBLE_MSW(high,a);
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SET_LDOUBLE_WORDS(t1,ea+1,high,0);
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t2 = a - t1;
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w = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b)));
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}
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if(k!=0) {
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uint32_t exp;
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t1 = 1.0;
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GET_LDOUBLE_EXP(exp,t1);
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SET_LDOUBLE_EXP(t1,exp+k);
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w *= t1;
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math_check_force_underflow_nonneg (w);
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return w;
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} else return w;
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}
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}
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libm_alias_finite (__ieee754_hypotl, __hypotl)
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libm_alias_finite (__ieee754_hypotl, __hypotl)
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