1 2 /* @(#)s_log1p.c 1.4 96/03/07 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 /* double log1p(double x) 15 * 16 * Method : 17 * 1. Argument Reduction: find k and f such that 18 * 1+x = 2^k * (1+f), 19 * where sqrt(2)/2 < 1+f < sqrt(2) . 20 * 21 * Note. If k=0, then f=x is exact. However, if k!=0, then f 22 * may not be representable exactly. In that case, a correction 23 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 24 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 25 * and add back the correction term c/u. 26 * (Note: when x > 2**53, one can simply return log(x)) 27 * 28 * 2. Approximation of log1p(f). 29 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) 30 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 31 * = 2s + s*R 32 * We use a special Remes algorithm on [0,0.1716] to generate 33 * a polynomial of degree 14 to approximate R The maximum error 34 * of this polynomial approximation is bounded by 2**-58.45. In 35 * other words, 36 * 2 4 6 8 10 12 14 37 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s 38 * (the values of Lp1 to Lp7 are listed in the program) 39 * and 40 * | 2 14 | -58.45 41 * | Lp1*s +...+Lp7*s - R(z) | <= 2 42 * | | 43 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. 44 * In order to guarantee error in log below 1ulp, we compute log 45 * by 46 * log1p(f) = f - (hfsq - s*(hfsq+R)). 47 * 48 * 3. Finally, log1p(x) = k*ln2 + log1p(f). 49 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) 50 * Here ln2 is split into two floating point number: 51 * ln2_hi + ln2_lo, 52 * where n*ln2_hi is always exact for |n| < 2000. 53 * 54 * Special cases: 55 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 56 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 57 * log1p(NaN) is that NaN with no signal. 58 * 59 * Accuracy: 60 * according to an error analysis, the error is always less than 61 * 1 ulp (unit in the last place). 62 * 63 * Constants: 64 * The hexadecimal values are the intended ones for the following 65 * constants. The decimal values may be used, provided that the 66 * compiler will convert from decimal to binary accurately enough 67 * to produce the hexadecimal values shown. 68 * 69 * Note: Assuming log() return accurate answer, the following 70 * algorithm can be used to compute log1p(x) to within a few ULP: 71 * 72 * u = 1+x; 73 * if(u==1.0) return x ; else 74 * return log(u)*(x/(u-1.0)); 75 * 76 * See HP-15C Advanced Functions Handbook, p.193. 77 */ 78 79 #include "fdlibm.h" 80 81 #ifndef _DOUBLE_IS_32BITS 82 83 #ifdef __STDC__ 84 static const double 85 #else 86 static double 87 #endif 88 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 89 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 90 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 91 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 92 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 93 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 94 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 95 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 96 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 97 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 98 99 static double zero = 0.0; 100 101 #ifdef __STDC__ log1p(double x)102 double log1p(double x) 103 #else 104 double log1p(x) 105 double x; 106 #endif 107 { 108 double hfsq,f,c,s,z,R,u; 109 int32_t k,hx,hu,ax; 110 111 GET_HIGH_WORD(hx,x); /* high word of x */ 112 ax = hx&0x7fffffff; 113 114 k = 1; 115 if (hx < 0x3FDA827A) { /* x < 0.41422 */ 116 if(ax>=0x3ff00000) { /* x <= -1.0 */ 117 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ 118 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ 119 } 120 if(ax<0x3e200000) { /* |x| < 2**-29 */ 121 if(two54+x>zero /* raise inexact */ 122 &&ax<0x3c900000) /* |x| < 2**-54 */ 123 return x; 124 else 125 return x - x*x*0.5; 126 } 127 if(hx>0||hx<=((int)0xbfd2bec3)) { 128 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ 129 } 130 if (hx >= 0x7ff00000) return x+x; 131 if(k!=0) { 132 if(hx<0x43400000) { 133 u = 1.0+x; 134 GET_HIGH_WORD(hu,u); /* high word of u */ 135 k = (hu>>20)-1023; 136 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 137 c /= u; 138 } else { 139 u = x; 140 GET_HIGH_WORD(hu,u); /* high word of u */ 141 k = (hu>>20)-1023; 142 c = 0; 143 } 144 hu &= 0x000fffff; 145 if(hu<0x6a09e) { 146 SET_HIGH_WORD(u, hu|0x3ff00000); /* normalize u */ 147 } else { 148 k += 1; 149 SET_HIGH_WORD(u, hu|0x3fe00000); /* normalize u/2 */ 150 hu = (0x00100000-hu)>>2; 151 } 152 f = u-1.0; 153 } 154 hfsq=0.5*f*f; 155 if(hu==0) { /* |f| < 2**-20 */ 156 if(f==zero) if(k==0) return zero; 157 else {c += k*ln2_lo; return k*ln2_hi+c;} 158 R = hfsq*(1.0-0.66666666666666666*f); 159 if(k==0) return f-R; else 160 return k*ln2_hi-((R-(k*ln2_lo+c))-f); 161 } 162 s = f/(2.0+f); 163 z = s*s; 164 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); 165 if(k==0) return f-(hfsq-s*(hfsq+R)); else 166 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); 167 } 168 #endif /* _DOUBLE_IS_32BITS */ 169