1 2 /* @(#)s_log1p.c 1.3 95/01/18 */ 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 ieee_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 ieee_sqrt(2)/2 < 1+f < ieee_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) - ieee_log(u) ~ c/u. Thus, we proceed to compute ieee_log(u), 25 * and add back the correction term c/u. 26 * (Note: when x > 2**53, one can simply return ieee_log(x)) 27 * 28 * 2. Approximation of ieee_log1p(f). 29 * Let s = f/(2+f) ; based on ieee_log(1+f) = ieee_log(1+s) - ieee_log(1-s) 30 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 31 * = 2s + s*R 32 * We use a special Reme 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, ieee_log1p(x) = k*ln2 + ieee_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; ieee_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 ieee_log() return accurate answer, the following 70 * algorithm can be used to compute ieee_log1p(x) to within a few ULP: 71 * 72 * u = 1+x; 73 * if(u==1.0) return x ; else 74 * return ieee_log(u)*(x/(u-1.0)); 75 * 76 * See HP-15C Advanced Functions Handbook, p.193. 77 */ 78 79 #include "fdlibm.h" 80 81 #ifdef __STDC__ 82 static const double 83 #else 84 static double 85 #endif 86 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 87 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 88 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 89 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 90 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 91 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 92 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 93 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 94 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 95 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 96 97 static double zero = 0.0; 98 99 #ifdef __STDC__ 100 double ieee_log1p(double x) 101 #else 102 double ieee_log1p(x) 103 double x; 104 #endif 105 { 106 double hfsq,f,c,s,z,R,u; 107 int k,hx,hu,ax; 108 109 hx = __HI(x); /* high word of x */ 110 ax = hx&0x7fffffff; 111 112 k = 1; 113 if (hx < 0x3FDA827A) { /* x < 0.41422 */ 114 if(ax>=0x3ff00000) { /* x <= -1.0 */ 115 if(x==-1.0) return -two54/zero; /* ieee_log1p(-1)=+inf */ 116 else return (x-x)/(x-x); /* ieee_log1p(x<-1)=NaN */ 117 } 118 if(ax<0x3e200000) { /* |x| < 2**-29 */ 119 if(two54+x>zero /* raise inexact */ 120 &&ax<0x3c900000) /* |x| < 2**-54 */ 121 return x; 122 else 123 return x - x*x*0.5; 124 } 125 if(hx>0||hx<=((int)0xbfd2bec3)) { 126 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */ 127 } 128 if (hx >= 0x7ff00000) return x+x; 129 if(k!=0) { 130 if(hx<0x43400000) { 131 u = 1.0+x; 132 hu = __HI(u); /* high word of u */ 133 k = (hu>>20)-1023; 134 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ 135 c /= u; 136 } else { 137 u = x; 138 hu = __HI(u); /* high word of u */ 139 k = (hu>>20)-1023; 140 c = 0; 141 } 142 hu &= 0x000fffff; 143 if(hu<0x6a09e) { 144 __HI(u) = hu|0x3ff00000; /* normalize u */ 145 } else { 146 k += 1; 147 __HI(u) = hu|0x3fe00000; /* normalize u/2 */ 148 hu = (0x00100000-hu)>>2; 149 } 150 f = u-1.0; 151 } 152 hfsq=0.5*f*f; 153 if(hu==0) { /* |f| < 2**-20 */ 154 if(f==zero) if(k==0) return zero; 155 else {c += k*ln2_lo; return k*ln2_hi+c;} 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
