s_log1p.c revision 251292
1219820Sjeff/* @(#)s_log1p.c 5.1 93/09/24 */ 2219820Sjeff/* 3219820Sjeff * ==================================================== 4219820Sjeff * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5219820Sjeff * 6219820Sjeff * Developed at SunPro, a Sun Microsystems, Inc. business. 7219820Sjeff * Permission to use, copy, modify, and distribute this 8219820Sjeff * software is freely granted, provided that this notice 9219820Sjeff * is preserved. 10219820Sjeff * ==================================================== 11219820Sjeff */ 12219820Sjeff 13219820Sjeff#include <sys/cdefs.h> 14219820Sjeff__FBSDID("$FreeBSD: head/lib/msun/src/s_log1p.c 251292 2013-06-03 09:14:31Z das $"); 15219820Sjeff 16219820Sjeff/* double log1p(double x) 17219820Sjeff * 18219820Sjeff * Method : 19219820Sjeff * 1. Argument Reduction: find k and f such that 20219820Sjeff * 1+x = 2^k * (1+f), 21219820Sjeff * where sqrt(2)/2 < 1+f < sqrt(2) . 22219820Sjeff * 23219820Sjeff * Note. If k=0, then f=x is exact. However, if k!=0, then f 24219820Sjeff * may not be representable exactly. In that case, a correction 25219820Sjeff * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 26219820Sjeff * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 27219820Sjeff * and add back the correction term c/u. 28219820Sjeff * (Note: when x > 2**53, one can simply return log(x)) 29219820Sjeff * 30219820Sjeff * 2. Approximation of log1p(f). 31219820Sjeff * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 32219820Sjeff * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 33219820Sjeff * = 2s + s*R 34219820Sjeff * We use a special Reme algorithm on [0,0.1716] to generate 35219820Sjeff * a polynomial of degree 14 to approximate R The maximum error 36219820Sjeff * of this polynomial approximation is bounded by 2**-58.45. In 37219820Sjeff * other words, 38219820Sjeff * 2 4 6 8 10 12 14 39219820Sjeff * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 40219820Sjeff * (the values of Lp1 to Lp7 are listed in the program) 41219820Sjeff * and 42219820Sjeff * | 2 14 | -58.45 43219820Sjeff * | Lp1*s +...+Lp7*s - R(z) | <= 2 44219820Sjeff * | | 45219820Sjeff * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 46219820Sjeff * In order to guarantee error in log below 1ulp, we compute log 47219820Sjeff * by 48219820Sjeff * log1p(f) = f - (hfsq - s*(hfsq+R)). 49219820Sjeff * 50219820Sjeff * 3. Finally, log1p(x) = k*ln2 + log1p(f). 51219820Sjeff * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 52219820Sjeff * Here ln2 is split into two floating point number: 53219820Sjeff * ln2_hi + ln2_lo, 54219820Sjeff * where n*ln2_hi is always exact for |n| < 2000. 55219820Sjeff * 56219820Sjeff * Special cases: 57219820Sjeff * log1p(x) is NaN with signal if x < -1 (including -INF) ; 58219820Sjeff * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 59219820Sjeff * log1p(NaN) is that NaN with no signal. 60219820Sjeff * 61219820Sjeff * Accuracy: 62219820Sjeff * according to an error analysis, the error is always less than 63219820Sjeff * 1 ulp (unit in the last place). 64219820Sjeff * 65219820Sjeff * Constants: 66219820Sjeff * The hexadecimal values are the intended ones for the following 67219820Sjeff * constants. The decimal values may be used, provided that the 68219820Sjeff * compiler will convert from decimal to binary accurately enough 69219820Sjeff * to produce the hexadecimal values shown. 70219820Sjeff * 71219820Sjeff * Note: Assuming log() return accurate answer, the following 72219820Sjeff * algorithm can be used to compute log1p(x) to within a few ULP: 73219820Sjeff * 74219820Sjeff * u = 1+x; 75219820Sjeff * if(u==1.0) return x ; else 76219820Sjeff * return log(u)*(x/(u-1.0)); 77219820Sjeff * 78219820Sjeff * See HP-15C Advanced Functions Handbook, p.193. 79219820Sjeff */ 80219820Sjeff 81219820Sjeff#include <float.h> 82219820Sjeff 83219820Sjeff#include "math.h" 84219820Sjeff#include "math_private.h" 85219820Sjeff 86219820Sjeffstatic const double 87219820Sjeffln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 88219820Sjeffln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 89219820Sjefftwo54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 90219820SjeffLp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 91219820SjeffLp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 92219820SjeffLp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 93219820SjeffLp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 94219820SjeffLp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 95219820SjeffLp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 96219820SjeffLp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 97219820Sjeff 98219820Sjeffstatic const double zero = 0.0; 99219820Sjeffstatic volatile double vzero = 0.0; 100219820Sjeff 101219820Sjeffdouble 102219820Sjefflog1p(double x) 103219820Sjeff{ 104219820Sjeff double hfsq,f,c,s,z,R,u; 105219820Sjeff int32_t k,hx,hu,ax; 106219820Sjeff 107219820Sjeff GET_HIGH_WORD(hx,x); 108219820Sjeff ax = hx&0x7fffffff; 109219820Sjeff 110219820Sjeff k = 1; 111219820Sjeff if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ 112219820Sjeff if(ax>=0x3ff00000) { /* x <= -1.0 */ 113219820Sjeff if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */ 114219820Sjeff else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 115219820Sjeff } 116219820Sjeff if(ax<0x3e200000) { /* |x| < 2**-29 */ 117219820Sjeff if(two54+x>zero /* raise inexact */ 118219820Sjeff &&ax<0x3c900000) /* |x| < 2**-54 */ 119219820Sjeff return x; 120219820Sjeff else 121219820Sjeff return x - x*x*0.5; 122219820Sjeff } 123219820Sjeff if(hx>0||hx<=((int32_t)0xbfd2bec4)) { 124219820Sjeff k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 125219820Sjeff } 126219820Sjeff if (hx >= 0x7ff00000) return x+x; 127219820Sjeff if(k!=0) { 128219820Sjeff if(hx<0x43400000) { 129219820Sjeff STRICT_ASSIGN(double,u,1.0+x); 130219820Sjeff GET_HIGH_WORD(hu,u); 131219820Sjeff k = (hu>>20)-1023; 132219820Sjeff c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 133219820Sjeff c /= u; 134219820Sjeff } else { 135219820Sjeff u = x; 136219820Sjeff GET_HIGH_WORD(hu,u); 137219820Sjeff k = (hu>>20)-1023; 138219820Sjeff c = 0; 139219820Sjeff } 140219820Sjeff hu &= 0x000fffff; 141219820Sjeff /* 142219820Sjeff * The approximation to sqrt(2) used in thresholds is not 143 * critical. However, the ones used above must give less 144 * strict bounds than the one here so that the k==0 case is 145 * never reached from here, since here we have committed to 146 * using the correction term but don't use it if k==0. 147 */ 148 if(hu<0x6a09e) { /* u ~< sqrt(2) */ 149 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 150 } else { 151 k += 1; 152 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 153 hu = (0x00100000-hu)>>2; 154 } 155 f = u-1.0; 156 } 157 hfsq=0.5*f*f; 158 if(hu==0) { /* |f| < 2**-20 */ 159 if(f==zero) { 160 if(k==0) { 161 return zero; 162 } else { 163 c += k*ln2_lo; 164 return k*ln2_hi+c; 165 } 166 } 167 R = hfsq*(1.0-0.66666666666666666*f); 168 if(k==0) return f-R; else 169 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 170 } 171 s = f/(2.0+f); 172 z = s*s; 173 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 174 if(k==0) return f-(hfsq-s*(hfsq+R)); else 175 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 176} 177 178#if (LDBL_MANT_DIG == 53) 179__weak_reference(log1p, log1pl); 180#endif 181