s_log1p.c revision 2116
118316Swollman/* @(#)s_log1p.c 5.1 93/09/24 */ 218316Swollman/* 318316Swollman * ==================================================== 418316Swollman * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 518316Swollman * 618316Swollman * Developed at SunPro, a Sun Microsystems, Inc. business. 718316Swollman * Permission to use, copy, modify, and distribute this 818316Swollman * software is freely granted, provided that this notice 918316Swollman * is preserved. 1018316Swollman * ==================================================== 1118316Swollman */ 1218316Swollman 1318316Swollman#ifndef lint 1418316Swollmanstatic char rcsid[] = "$Id: s_log1p.c,v 1.6 1994/08/18 23:06:59 jtc Exp $"; 1518316Swollman#endif 1618316Swollman 1718316Swollman/* double log1p(double x) 1818316Swollman * 1918316Swollman * Method : 2018316Swollman * 1. Argument Reduction: find k and f such that 2118316Swollman * 1+x = 2^k * (1+f), 2218316Swollman * where sqrt(2)/2 < 1+f < sqrt(2) . 2318316Swollman * 2418316Swollman * Note. If k=0, then f=x is exact. However, if k!=0, then f 2518316Swollman * may not be representable exactly. In that case, a correction 2618316Swollman * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 2718316Swollman * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 2818316Swollman * and add back the correction term c/u. 2918316Swollman * (Note: when x > 2**53, one can simply return log(x)) 3018316Swollman * 3118316Swollman * 2. Approximation of log1p(f). 3218316Swollman * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 3318316Swollman * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 3418316Swollman * = 2s + s*R 3518316Swollman * We use a special Reme algorithm on [0,0.1716] to generate 3618316Swollman * a polynomial of degree 14 to approximate R The maximum error 3718316Swollman * of this polynomial approximation is bounded by 2**-58.45. In 3818316Swollman * other words, 3918316Swollman * 2 4 6 8 10 12 14 4018316Swollman * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 4118316Swollman * (the values of Lp1 to Lp7 are listed in the program) 4218316Swollman * and 4318316Swollman * | 2 14 | -58.45 4418316Swollman * | Lp1*s +...+Lp7*s - R(z) | <= 2 4518316Swollman * | | 4618316Swollman * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 4718316Swollman * In order to guarantee error in log below 1ulp, we compute log 4818316Swollman * by 4918316Swollman * log1p(f) = f - (hfsq - s*(hfsq+R)). 5018316Swollman * 5118316Swollman * 3. Finally, log1p(x) = k*ln2 + log1p(f). 5218316Swollman * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 5318316Swollman * Here ln2 is split into two floating point number: 5418316Swollman * ln2_hi + ln2_lo, 5518316Swollman * where n*ln2_hi is always exact for |n| < 2000. 5618316Swollman * 5718316Swollman * Special cases: 5818316Swollman * log1p(x) is NaN with signal if x < -1 (including -INF) ; 5918316Swollman * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 6018316Swollman * log1p(NaN) is that NaN with no signal. 6118316Swollman * 6218316Swollman * Accuracy: 6318316Swollman * according to an error analysis, the error is always less than 6418316Swollman * 1 ulp (unit in the last place). 6518316Swollman * 6618316Swollman * Constants: 6718316Swollman * The hexadecimal values are the intended ones for the following 6818316Swollman * constants. The decimal values may be used, provided that the 6918316Swollman * compiler will convert from decimal to binary accurately enough 7018316Swollman * to produce the hexadecimal values shown. 7118316Swollman * 7218316Swollman * Note: Assuming log() return accurate answer, the following 7318316Swollman * algorithm can be used to compute log1p(x) to within a few ULP: 7418316Swollman * 7518316Swollman * u = 1+x; 7618316Swollman * if(u==1.0) return x ; else 7718316Swollman * return log(u)*(x/(u-1.0)); 7818316Swollman * 7918316Swollman * See HP-15C Advanced Functions Handbook, p.193. 8018316Swollman */ 8118316Swollman 8218316Swollman#include "math.h" 8318316Swollman#include "math_private.h" 8418316Swollman 8518316Swollman#ifdef __STDC__ 8618316Swollmanstatic const double 8718316Swollman#else 8818316Swollmanstatic double 8918316Swollman#endif 9018316Swollmanln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 9118316Swollmanln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 9218316Swollmantwo54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 9318316SwollmanLp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 9418316SwollmanLp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 9518316SwollmanLp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 9618316SwollmanLp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 9718316SwollmanLp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 9818316SwollmanLp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 9918316SwollmanLp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 10018316Swollman 10118316Swollman#ifdef __STDC__ 10218316Swollmanstatic const double zero = 0.0; 10318316Swollman#else 10418316Swollmanstatic double zero = 0.0; 10518316Swollman#endif 10618316Swollman 10718316Swollman#ifdef __STDC__ 10818316Swollman double log1p(double x) 10918316Swollman#else 11018316Swollman double log1p(x) 11118316Swollman double x; 11218316Swollman#endif 11318316Swollman{ 11418316Swollman double hfsq,f,c,s,z,R,u; 11518316Swollman int32_t k,hx,hu,ax; 11618316Swollman 11718316Swollman GET_HIGH_WORD(hx,x); 11818316Swollman ax = hx&0x7fffffff; 11918316Swollman 12018316Swollman k = 1; 12118316Swollman if (hx < 0x3FDA827A) { /* x < 0.41422 */ 12218316Swollman if(ax>=0x3ff00000) { /* x <= -1.0 */ 12318316Swollman if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ 12418316Swollman else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 12518316Swollman } 12618316Swollman if(ax<0x3e200000) { /* |x| < 2**-29 */ 12718316Swollman if(two54+x>zero /* raise inexact */ 12818316Swollman &&ax<0x3c900000) /* |x| < 2**-54 */ 12918316Swollman return x; 13018316Swollman else 13118316Swollman return x - x*x*0.5; 13218316Swollman } 13318316Swollman if(hx>0||hx<=((int32_t)0xbfd2bec3)) { 13418316Swollman k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ 13518316Swollman } 13618316Swollman if (hx >= 0x7ff00000) return x+x; 13718316Swollman if(k!=0) { 13818316Swollman if(hx<0x43400000) { 13918316Swollman u = 1.0+x; 14018316Swollman GET_HIGH_WORD(hu,u); 14118316Swollman k = (hu>>20)-1023; 14218316Swollman c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 14318316Swollman c /= u; 14418316Swollman } else { 14518316Swollman u = x; 14618316Swollman GET_HIGH_WORD(hu,u); 14718316Swollman k = (hu>>20)-1023; 14818316Swollman c = 0; 14918316Swollman } 15018316Swollman hu &= 0x000fffff; 15118316Swollman if(hu<0x6a09e) { 15218316Swollman SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 15318316Swollman } else { 15418316Swollman k += 1; 15518316Swollman SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 15618316Swollman hu = (0x00100000-hu)>>2; 15718316Swollman } 15818316Swollman f = u-1.0; 15918316Swollman } 16018316Swollman hfsq=0.5*f*f; 16118316Swollman if(hu==0) { /* |f| < 2**-20 */ 16218316Swollman if(f==zero) if(k==0) return zero; 16318316Swollman else {c += k*ln2_lo; return k*ln2_hi+c;} 16418316Swollman R = hfsq*(1.0-0.66666666666666666*f); 16518316Swollman if(k==0) return f-R; else 16618316Swollman return k*ln2_hi-((R-(k*ln2_lo+c))-f); 16718316Swollman } 16818316Swollman s = f/(2.0+f); 16918316Swollman z = s*s; 17018316Swollman R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 17118316Swollman if(k==0) return f-(hfsq-s*(hfsq+R)); else 17218316Swollman return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 17318316Swollman} 17418316Swollman