forked from KolibriOS/kolibrios
359 lines
10 KiB
C
359 lines
10 KiB
C
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/* @(#)e_pow.c 5.1 93/09/24 */
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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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/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
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for performance improvement on pipelined processors.
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*/
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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
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#endif
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/* __ieee754_pow(x,y) return x**y
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*
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* n
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* Method: Let x = 2 * (1+f)
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* 1. Compute and return log2(x) in two pieces:
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* log2(x) = w1 + w2,
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* where w1 has 53-24 = 29 bit trailing zeros.
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* 2. Perform y*log2(x) = n+y' by simulating muti-precision
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* arithmetic, where |y'|<=0.5.
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* 3. Return x**y = 2**n*exp(y'*log2)
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*
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* Special cases:
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* 1. (anything) ** 0 is 1
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* 2. (anything) ** 1 is itself
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* 3. (anything) ** NAN is NAN
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* 4. NAN ** (anything except 0) is NAN
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* 5. +-(|x| > 1) ** +INF is +INF
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* 6. +-(|x| > 1) ** -INF is +0
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* 7. +-(|x| < 1) ** +INF is +0
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* 8. +-(|x| < 1) ** -INF is +INF
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* 9. +-1 ** +-INF is NAN
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* 10. +0 ** (+anything except 0, NAN) is +0
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* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
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* 12. +0 ** (-anything except 0, NAN) is +INF
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* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
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* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
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* 15. +INF ** (+anything except 0,NAN) is +INF
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* 16. +INF ** (-anything except 0,NAN) is +0
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* 17. -INF ** (anything) = -0 ** (-anything)
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* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
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* 19. (-anything except 0 and inf) ** (non-integer) is NAN
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*
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* Accuracy:
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* pow(x,y) returns x**y nearly rounded. In particular
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* pow(integer,integer)
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* always returns the correct integer provided it is
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* representable.
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*
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* Constants :
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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*/
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#include "math.h"
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#include "math_private.h"
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#define zero C[0]
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#define one C[1]
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#define two C[2]
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#define two53 C[3]
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#define huge C[4]
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#define tiny C[5]
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#define L1 C[6]
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#define L2 C[7]
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#define L3 C[8]
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#define L4 C[9]
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#define L5 C[10]
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#define L6 C[11]
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#define P1 C[12]
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#define P2 C[13]
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#define P3 C[14]
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#define P4 C[15]
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#define P5 C[16]
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#define lg2 C[17]
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#define lg2_h C[18]
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#define lg2_l C[19]
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#define ovt C[20]
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#define cp C[21]
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#define cp_h C[22]
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#define cp_l C[23]
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#define ivln2 C[24]
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#define ivln2_h C[25]
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#define ivln2_l C[26]
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double scalbn(double,int);
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#define EXTRACT_WORDS(ix0,ix1,d) \
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do { \
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ieee_double_shape_type ew_u; \
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ew_u.value = (d); \
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(ix0) = ew_u.parts.msw; \
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(ix1) = ew_u.parts.lsw; \
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} while (0)
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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bp[] = {1.0, 1.5,},
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dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
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dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
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C[] = {
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0.0,
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1.0,
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2.0,
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9007199254740992.0 ,
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1.0e300,
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1.0e-300,
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5.99999999999994648725e-01 ,
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4.28571428578550184252e-01 ,
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3.33333329818377432918e-01 ,
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2.72728123808534006489e-01 ,
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2.30660745775561754067e-01 ,
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2.06975017800338417784e-01 ,
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1.66666666666666019037e-01 ,
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-2.77777777770155933842e-03 ,
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6.61375632143793436117e-05 ,
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-1.65339022054652515390e-06 ,
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4.13813679705723846039e-08 ,
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6.93147180559945286227e-01 ,
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6.93147182464599609375e-01 ,
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-1.90465429995776804525e-09 ,
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8.0085662595372944372e-0017 ,
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9.61796693925975554329e-01 ,
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9.61796700954437255859e-01 ,
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-7.02846165095275826516e-09 ,
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1.44269504088896338700e+00 ,
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1.44269502162933349609e+00 ,
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1.92596299112661746887e-08 };
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double pow_test(x,y)
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double x, y;
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{
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double z,ax,z_h,z_l,p_h,p_l;
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double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3;
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int32_t i,j,k,yisint,n;
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int32_t hx,hy,ix,iy;
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u_int32_t lx,ly;
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EXTRACT_WORDS(hx,lx,x);
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EXTRACT_WORDS(hy,ly,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((iy|ly)==0) return C[1];
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/* +-NaN return x+y */
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if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
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iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
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return x+y;
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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>=0x43400000) yisint = 2; /* even integer y */
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else if(iy>=0x3ff00000) {
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k = (iy>>20)-0x3ff; /* exponent */
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if(k>20) {
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j = ly>>(52-k);
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if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
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} else if(ly==0) {
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j = iy>>(20-k);
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if((int32_t)(j<<(20-k))==iy) yisint = 2-(j&1);
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}
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}
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}
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/* special value of y */
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if(ly==0) {
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if (iy==0x7ff00000) { /* y is +-inf */
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if(((ix-0x3ff00000)|lx)==0)
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return y - y; /* inf**+-1 is NaN */
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else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
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return (hy>=0)? y: C[0];
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else /* (|x|<1)**-,+inf = inf,0 */
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return (hy<0)?-y: C[0];
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}
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if(iy==0x3ff00000) { /* y is +-1 */
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if(hy<0) return C[1]/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==0x3fe00000) { /* y is 0.5 */
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if(hx>=0) /* x >= +0 */
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return sqrt(x);
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}
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}
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ax = fabs(x);
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/* special value of x */
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if(lx==0) {
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if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
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z = ax; /*x is +-0,+-inf,+-1*/
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if(hy<0) z = C[1]/z; /* z = (1/|x|) */
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if(hx<0) {
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if(((ix-0x3ff00000)|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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}
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/* (x<0)**(non-int) is NaN */
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if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
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/* |y| is huge */
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if(iy>0x41e00000) { /* if |y| > 2**31 */
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if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
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if(ix<=0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5];
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if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
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}
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/* over/underflow if x is not close to one */
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if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5];
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if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
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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 = x-1; /* t has 20 trailing zeros */
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w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
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u = C[25]*t; /* ivln2_h has 21 sig. bits */
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v = t*C[26]-w*C[24];
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t1 = u+v;
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SET_LOW_WORD(t1,0);
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t2 = v-(t1-u);
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} else {
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double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3;
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n = 0;
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/* take care subnormal number */
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if(ix<0x00100000)
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{ax *= C[3]; n -= 53; GET_HIGH_WORD(ix,ax); }
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n += ((ix)>>20)-0x3ff;
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j = ix&0x000fffff;
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/* determine interval */
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ix = j|0x3ff00000; /* normalize ix */
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if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
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else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
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else {k=0;n+=1;ix -= 0x00100000;}
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SET_HIGH_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 = C[1]/(ax+bp[k]);
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s = u*v;
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s_h = s;
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SET_LOW_WORD(s_h,0);
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/* t_h=ax+bp[k] High */
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t_h = C[0];
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SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
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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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#ifdef DO_NOT_USE_THIS
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r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
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#else
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r1 = C[10]+s2*C[11]; s22=s2*s2;
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r2 = C[8]+s2*C[9]; s24=s22*s22;
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r3 = C[6]+s2*C[7]; s26=s24*s22;
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r = r3*s22 + r2*s24 + r1*s26;
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#endif
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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 = 3.0+s2+r;
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SET_LOW_WORD(t_h,0);
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t_l = r-((t_h-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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SET_LOW_WORD(p_h,0);
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p_l = v-(p_h-u);
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z_h = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = C[23]*p_h+p_l*C[21]+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 = (double)n;
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t1 = (((z_h+z_l)+dp_h[k])+t);
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SET_LOW_WORD(t1,0);
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t2 = z_l-(((t1-t)-dp_h[k])-z_h);
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}
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s = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */
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if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
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s = -C[1];/* (-ve)**(odd int) */
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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y1 = y;
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SET_LOW_WORD(y1,0);
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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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EXTRACT_WORDS(j,i,z);
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if (j>=0x40900000) { /* z >= 1024 */
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if(((j-0x40900000)|i)!=0) /* if z > 1024 */
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return s*C[4]*C[4]; /* overflow */
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else {
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if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */
|
||
|
|
}
|
||
|
|
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
|
||
|
|
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
|
||
|
|
return s*C[5]*C[5]; /* underflow */
|
||
|
|
else {
|
||
|
|
if(p_l<=z-p_h) return s*C[5]*C[5]; /* underflow */
|
||
|
|
}
|
||
|
|
}
|
||
|
|
/*
|
||
|
|
* compute 2**(p_h+p_l)
|
||
|
|
*/
|
||
|
|
i = j&0x7fffffff;
|
||
|
|
k = (i>>20)-0x3ff;
|
||
|
|
n = 0;
|
||
|
|
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||
|
|
n = j+(0x00100000>>(k+1));
|
||
|
|
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
|
||
|
|
t = C[0];
|
||
|
|
SET_HIGH_WORD(t,n&~(0x000fffff>>k));
|
||
|
|
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
||
|
|
if(j<0) n = -n;
|
||
|
|
p_h -= t;
|
||
|
|
}
|
||
|
|
t = p_l+p_h;
|
||
|
|
SET_LOW_WORD(t,0);
|
||
|
|
u = t*C[18];
|
||
|
|
v = (p_l-(t-p_h))*C[17]+t*C[19];
|
||
|
|
z = u+v;
|
||
|
|
w = v-(z-u);
|
||
|
|
t = z*z;
|
||
|
|
#ifdef DO_NOT_USE_THIS
|
||
|
|
t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16]))));
|
||
|
|
#else
|
||
|
|
r_1 = C[15]+t*C[16]; t12 = t*t;
|
||
|
|
r_2 = C[13]+t*C[14]; t14 = t12*t12;
|
||
|
|
r_3 = t*C[12];
|
||
|
|
t1 = z - r_3 - t12*r_2 - t14*r_1;
|
||
|
|
#endif
|
||
|
|
r = (z*t1)/(t1-C[2])-(w+z*w);
|
||
|
|
z = C[1]-(r-z);
|
||
|
|
GET_HIGH_WORD(j,z);
|
||
|
|
j += (n<<20);
|
||
|
|
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
|
||
|
|
else SET_HIGH_WORD(z,j);
|
||
|
|
return s*z;
|
||
|
|
}
|