mirror of
git://sourceware.org/git/newlib-cygwin.git
synced 2025-03-04 14:06:13 +08:00
1b7c72fdcc
Fixed another precision bug in powf(). This one is in the computation
[t=p_l+p_h High]. We multiply t by lg2_h, and want the result to be
exact. For the bogus float case of the high-low decomposition trick, we
normally discard the lowest 12 bits of the fraction for the high part,
keeping 12 bits of precision. That was used for t here, but it doesnt't
work because for some reason we only discard the lowest 9 bits in the
fraction for lg2_h. Discard another 3 bits of the fraction for t to
compensate.
This bug gave wrong results like:
powf(0.9999999, -2.9999995) = 1.0000002 (should be 1.0000001)
hex values: 3F7FFFFF C03FFFFE 3F800002 3F800001
As explained in the log for the previous commit, the bug is normally
masked by doing float calculations in extra precision on i386's, but is
easily detected by ucbtest on systems that don't have accidental extra
precision.
Reference: 5f20e5ce7f
Original Author: Bruce Evans
261 lines
7.6 KiB
C
261 lines
7.6 KiB
C
/* ef_pow.c -- float version of e_pow.c.
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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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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#include "fdlibm.h"
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#include "math_config.h"
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#if __OBSOLETE_MATH
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#ifdef __v810__
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#define const
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#endif
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#ifdef __STDC__
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static const float
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#else
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static float
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#endif
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bp[] = {1.0, 1.5,},
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dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
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dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
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zero = 0.0,
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one = 1.0,
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two = 2.0,
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two24 = 16777216.0, /* 0x4b800000 */
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/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
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L1 = 6.0000002384e-01, /* 0x3f19999a */
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L2 = 4.2857143283e-01, /* 0x3edb6db7 */
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L3 = 3.3333334327e-01, /* 0x3eaaaaab */
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L4 = 2.7272811532e-01, /* 0x3e8ba305 */
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L5 = 2.3066075146e-01, /* 0x3e6c3255 */
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L6 = 2.0697501302e-01, /* 0x3e53f142 */
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P1 = 1.6666667163e-01, /* 0x3e2aaaab */
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P2 = -2.7777778450e-03, /* 0xbb360b61 */
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P3 = 6.6137559770e-05, /* 0x388ab355 */
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P4 = -1.6533901999e-06, /* 0xb5ddea0e */
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P5 = 4.1381369442e-08, /* 0x3331bb4c */
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lg2 = 6.9314718246e-01, /* 0x3f317218 */
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lg2_h = 6.93145752e-01, /* 0x3f317200 */
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lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
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ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
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cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
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cp_h = 9.6191406250e-01f, /* 0x3f764000 =12b of cp */
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cp_l = -1.1736857402e-04f, /* 0xb8f623c6 =tail of cp_h */
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ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
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ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
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ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
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#ifdef __STDC__
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float __ieee754_powf(float x, float y)
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#else
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float __ieee754_powf(x,y)
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float x, y;
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#endif
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{
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float z,ax,z_h,z_l,p_h,p_l;
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float y1,t1,t2,r,s,t,u,v,w;
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__int32_t i,j,k,yisint,n;
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__int32_t hx,hy,ix,iy,is;
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GET_FLOAT_WORD(hx,x);
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GET_FLOAT_WORD(hy,y);
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ix = hx&0x7fffffff; iy = hy&0x7fffffff;
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/* y==zero: x**0 = 1 */
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if(FLT_UWORD_IS_ZERO(iy)) {
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if (issignalingf_inline(x)) return x + y;
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return one;
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}
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/* x|y==NaN return NaN unless x==1 then return 1 */
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if(FLT_UWORD_IS_NAN(ix) ||
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FLT_UWORD_IS_NAN(iy)) {
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if(hx==0x3f800000 && !issignalingf_inline(y)) return one;
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else return x + y;
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}
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/* determine if y is an odd int when x < 0
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* yisint = 0 ... y is not an integer
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* yisint = 1 ... y is an odd int
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* yisint = 2 ... y is an even int
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*/
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yisint = 0;
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if(hx<0) {
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if(iy>=0x4b800000) yisint = 2; /* even integer y */
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else if(iy>=0x3f800000) {
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k = (iy>>23)-0x7f; /* exponent */
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j = iy>>(23-k);
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if((j<<(23-k))==iy) yisint = 2-(j&1);
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}
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}
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/* special value of y */
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if (FLT_UWORD_IS_INFINITE(iy)) { /* y is +-inf */
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if (ix==0x3f800000)
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return one; /* +-1**+-inf = 1 */
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else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
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return (hy>=0)? y: zero;
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else /* (|x|<1)**-,+inf = inf,0 */
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return (hy<0)?-y: zero;
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}
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if(iy==0x3f800000) { /* y is +-1 */
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if(hy<0) return one/x; else return x;
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}
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if(hy==0x40000000) return x*x; /* y is 2 */
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if(hy==0x3f000000) { /* y is 0.5 */
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if(hx>=0) /* x >= +0 */
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return __ieee754_sqrtf(x);
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}
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ax = fabsf(x);
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/* special value of x */
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if(FLT_UWORD_IS_INFINITE(ix)||FLT_UWORD_IS_ZERO(ix)||ix==0x3f800000){
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z = ax; /*x is +-0,+-inf,+-1*/
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if(hy<0) z = one/z; /* z = (1/|x|) */
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if(hx<0) {
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if(((ix-0x3f800000)|yisint)==0) {
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z = (z-z)/(z-z); /* (-1)**non-int is NaN */
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} else if(yisint==1)
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z = -z; /* (x<0)**odd = -(|x|**odd) */
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}
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return z;
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}
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/* (x<0)**(non-int) is NaN */
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if(((((__uint32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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/* |y| is huge */
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if(iy>0x4d000000) { /* if |y| > 2**27 */
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/* over/underflow if x is not close to one */
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if(ix<0x3f7ffff4) return (hy<0)? __math_oflowf(0):__math_uflowf(0);
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if(ix>0x3f800007) return (hy>0)? __math_oflowf(0):__math_uflowf(0);
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/* now |1-x| is tiny <= 2**-20, suffice to compute
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log(x) by x-x^2/2+x^3/3-x^4/4 */
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t = ax-1; /* t has 20 trailing zeros */
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w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
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u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
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v = t*ivln2_l-w*ivln2;
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t1 = u+v;
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GET_FLOAT_WORD(is,t1);
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SET_FLOAT_WORD(t1,is&0xfffff000);
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t2 = v-(t1-u);
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} else {
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float s2,s_h,s_l,t_h,t_l;
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n = 0;
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/* take care subnormal number */
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if(FLT_UWORD_IS_SUBNORMAL(ix))
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{ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
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n += ((ix)>>23)-0x7f;
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j = ix&0x007fffff;
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/* determine interval */
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ix = j|0x3f800000; /* normalize ix */
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if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
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else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
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else {k=0;n+=1;ix -= 0x00800000;}
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SET_FLOAT_WORD(ax,ix);
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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v = one/(ax+bp[k]);
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s = u*v;
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s_h = s;
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GET_FLOAT_WORD(is,s_h);
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SET_FLOAT_WORD(s_h,is&0xfffff000);
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/* t_h=ax+bp[k] High */
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is = ((ix >> 1) & 0xfffff000U) | 0x20000000;
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SET_FLOAT_WORD(t_h, is + 0x00400000 + (k << 21));
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t_l = ax - (t_h-bp[k]);
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s_l = v*((u-s_h*t_h)-s_h*t_l);
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/* compute log(ax) */
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s2 = s*s;
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r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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r += s_l*(s_h+s);
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s2 = s_h*s_h;
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t_h = (float)3.0+s2+r;
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GET_FLOAT_WORD(is,t_h);
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SET_FLOAT_WORD(t_h,is&0xfffff000);
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t_l = r-((t_h-(float)3.0)-s2);
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/* u+v = s*(1+...) */
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u = s_h*t_h;
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v = s_l*t_h+t_l*s;
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/* 2/(3log2)*(s+...) */
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p_h = u+v;
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GET_FLOAT_WORD(is,p_h);
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SET_FLOAT_WORD(p_h,is&0xfffff000);
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p_l = v-(p_h-u);
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z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = cp_l*p_h+p_l*cp+dp_l[k];
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/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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t = (float)n;
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t1 = (((z_h+z_l)+dp_h[k])+t);
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GET_FLOAT_WORD(is,t1);
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SET_FLOAT_WORD(t1,is&0xfffff000);
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t2 = z_l-(((t1-t)-dp_h[k])-z_h);
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}
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s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
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if(((((__uint32_t)hx>>31)-1)|(yisint-1))==0)
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s = -one; /* (-ve)**(odd int) */
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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GET_FLOAT_WORD(is,y);
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SET_FLOAT_WORD(y1,is&0xfffff000);
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p_l = (y-y1)*t1+y*t2;
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p_h = y1*t1;
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z = p_l+p_h;
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GET_FLOAT_WORD(j,z);
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i = j&0x7fffffff;
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if (j>0) {
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if (i>FLT_UWORD_EXP_MAX)
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return __math_oflowf(s<0); /* overflow */
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else if (i==FLT_UWORD_EXP_MAX)
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if(p_l+ovt>z-p_h) return __math_oflowf(s<0); /* overflow */
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} else {
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if (i>FLT_UWORD_EXP_MIN)
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return __math_uflowf(s<0); /* underflow */
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else if (i==FLT_UWORD_EXP_MIN)
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if(p_l<=z-p_h) return __math_uflowf(s<0); /* underflow */
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}
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/*
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* compute 2**(p_h+p_l)
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*/
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k = (i>>23)-0x7f;
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n = 0;
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if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
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n = j+(0x00800000>>(k+1));
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k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
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SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
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n = ((n&0x007fffff)|0x00800000)>>(23-k);
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if(j<0) n = -n;
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p_h -= t;
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}
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t = p_l+p_h;
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GET_FLOAT_WORD(is,t);
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SET_FLOAT_WORD(t,is&0xffff8000);
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u = t*lg2_h;
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v = (p_l-(t-p_h))*lg2+t*lg2_l;
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z = u+v;
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w = v-(z-u);
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t = z*z;
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t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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r = (z*t1)/(t1-two)-(w+z*w);
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z = one-(r-z);
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GET_FLOAT_WORD(j,z);
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j += (n<<23);
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if((j>>23)<=0) z = scalbnf(z,(int)n); /* subnormal output */
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else SET_FLOAT_WORD(z,j);
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return s*z;
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}
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#endif /* __OBSOLETE_MATH */
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