1 /* @(#)s_log1p.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: s_log1p.c,v 1.12 2002/05/26 22:01:57 wiz Exp $"); 16 #endif 17 18 /* double log1p(double x) 19 * 20 * Method : 21 * 1. Argument Reduction: find k and f such that 22 * 1+x = 2^k * (1+f), 23 * where sqrt(2)/2 < 1+f < sqrt(2) . 24 * 25 * Note. If k=0, then f=x is exact. However, if k!=0, then f 26 * may not be representable exactly. In that case, a correction 27 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 28 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 29 * and add back the correction term c/u. 30 * (Note: when x > 2**53, one can simply return log(x)) 31 * 32 * 2. Approximation of log1p(f). 33 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 34 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 35 * = 2s + s*R 36 * We use a special Reme algorithm on [0,0.1716] to generate 37 * a polynomial of degree 14 to approximate R The maximum error 38 * of this polynomial approximation is bounded by 2**-58.45. In 39 * other words, 40 * 2 4 6 8 10 12 14 41 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 42 * (the values of Lp1 to Lp7 are listed in the program) 43 * and 44 * | 2 14 | -58.45 45 * | Lp1*s +...+Lp7*s - R(z) | <= 2 46 * | | 47 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 48 * In order to guarantee error in log below 1ulp, we compute log 49 * by 50 * log1p(f) = f - (hfsq - s*(hfsq+R)). 51 * 52 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 53 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 54 * Here ln2 is split into two floating point number: 55 * ln2_hi + ln2_lo, 56 * where n*ln2_hi is always exact for |n| < 2000. 57 * 58 * Special cases: 59 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 60 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 61 * log1p(NaN) is that NaN with no signal. 62 * 63 * Accuracy: 64 * according to an error analysis, the error is always less than 65 * 1 ulp (unit in the last place). 66 * 67 * Constants: 68 * The hexadecimal values are the intended ones for the following 69 * constants. The decimal values may be used, provided that the 70 * compiler will convert from decimal to binary accurately enough 71 * to produce the hexadecimal values shown. 72 * 73 * Note: Assuming log() return accurate answer, the following 74 * algorithm can be used to compute log1p(x) to within a few ULP: 75 * 76 * u = 1+x; 77 * if(u==1.0) return x ; else 78 * return log(u)*(x/(u-1.0)); 79 * 80 * See HP-15C Advanced Functions Handbook, p.193. 81 */ 82 83 #include "math.h" 84 #include "math_private.h" 85 86 static const double 87 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 88 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 89 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 90 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 91 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 92 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 93 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 94 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 95 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 96 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 97 98 static const double zero = 0.0; 99 100 double 101 log1p(double x) 102 { 103 double hfsq,f,c,s,z,R,u; 104 int32_t k,hx,hu,ax; 105 106 f = c = 0; 107 hu = 0; 108 GET_HIGH_WORD(hx,x); 109 ax = hx&0x7fffffff; 110 111 k = 1; 112 if (hx < 0x3FDA827A) { /* x < 0.41422 */ 113 if(ax>=0x3ff00000) { /* x <= -1.0 */ 114 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ 115 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 116 } 117 if(ax<0x3e200000) { /* |x| < 2**-29 */ 118 if(two54+x>zero /* raise inexact */ 119 &&ax<0x3c900000) /* |x| < 2**-54 */ 120 return x; 121 else 122 return x - x*x*0.5; 123 } 124 if(hx>0||hx<=((int32_t)0xbfd2bec3)) { 125 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ 126 } 127 if (hx >= 0x7ff00000) return x+x; 128 if(k!=0) { 129 if(hx<0x43400000) { 130 u = 1.0+x; 131 GET_HIGH_WORD(hu,u); 132 k = (hu>>20)-1023; 133 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 134 c /= u; 135 } else { 136 u = x; 137 GET_HIGH_WORD(hu,u); 138 k = (hu>>20)-1023; 139 c = 0; 140 } 141 hu &= 0x000fffff; 142 if(hu<0x6a09e) { 143 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */ 144 } else { 145 k += 1; 146 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */ 147 hu = (0x00100000-hu)>>2; 148 } 149 f = u-1.0; 150 } 151 hfsq=0.5*f*f; 152 if(hu==0) { /* |f| < 2**-20 */ 153 if(f==zero) { if(k==0) return zero; 154 else {c += k*ln2_lo; return k*ln2_hi+c;} 155 } 156 R = hfsq*(1.0-0.66666666666666666*f); 157 if(k==0) return f-R; else 158 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 159 } 160 s = f/(2.0+f); 161 z = s*s; 162 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 163 if(k==0) return f-(hfsq-s*(hfsq+R)); else 164 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 165 } 166