s_log1p.c 5.44 KB
 Raymond Toy committed Aug 01, 2014 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 `````` /* @(#)s_log1p.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* double log1p(double x) * * Method : * 1. Argument Reduction: find k and f such that * 1+x = 2^k * (1+f), * where sqrt(2)/2 < 1+f < sqrt(2) . * * Note. If k=0, then f=x is exact. However, if k!=0, then f * may not be representable exactly. In that case, a correction * term is need. Let u=1+x rounded. Let c = (1+x)-u, then * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), * and add back the correction term c/u. * (Note: when x > 2**53, one can simply return log(x)) * * 2. Approximation of log1p(f). * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) * = 2s + 2/3 s**3 + 2/5 s**5 + ....., * = 2s + s*R * We use a special Reme algorithm on [0,0.1716] to generate * a polynomial of degree 14 to approximate R The maximum error * of this polynomial approximation is bounded by 2**-58.45. In * other words, * 2 4 6 8 10 12 14 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s * (the values of Lp1 to Lp7 are listed in the program) * and * | 2 14 | -58.45 * | Lp1*s +...+Lp7*s - R(z) | <= 2 * | | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. * In order to guarantee error in log below 1ulp, we compute log * by * log1p(f) = f - (hfsq - s*(hfsq+R)). * * 3. Finally, log1p(x) = k*ln2 + log1p(f). * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) * Here ln2 is split into two floating point number: * ln2_hi + ln2_lo, * where n*ln2_hi is always exact for |n| < 2000. * * Special cases: * log1p(x) is NaN with signal if x < -1 (including -INF) ; * log1p(+INF) is +INF; log1p(-1) is -INF with signal; * log1p(NaN) is that NaN with no signal. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. * * Note: Assuming log() return accurate answer, the following * algorithm can be used to compute log1p(x) to within a few ULP: * * u = 1+x; * if(u==1.0) return x ; else * return log(u)*(x/(u-1.0)); * * See HP-15C Advanced Functions Handbook, p.193. */ #include "fdlibm.h" #ifdef __STDC__ static const double #else static double #endif ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ static double zero = 0.0; #ifdef __STDC__ `````` Raymond Toy committed Aug 01, 2014 100 `````` double fdlibm_log1p(double x) `````` Raymond Toy committed Aug 01, 2014 101 ``````#else `````` Raymond Toy committed Aug 01, 2014 102 `````` double fdlibm_log1p(x) `````` Raymond Toy committed Aug 01, 2014 103 104 105 106 107 `````` double x; #endif { double hfsq,f,c,s,z,R,u; int k,hx,hu,ax; `````` Raymond Toy committed Aug 01, 2014 108 `````` union { int i[2]; double d; } ux; `````` Raymond Toy committed Aug 01, 2014 109 `````` `````` Raymond Toy committed Aug 01, 2014 110 111 `````` ux.d = x; hx = ux.i[HIWORD]; /* high word of x */ `````` Raymond Toy committed Aug 01, 2014 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 `````` ax = hx&0x7fffffff; k = 1; if (hx < 0x3FDA827A) { /* x < 0.41422 */ if(ax>=0x3ff00000) { /* x <= -1.0 */ if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */ else return (x-x)/(x-x); /* log1p(x<-1)=NaN */ } if(ax<0x3e200000) { /* |x| < 2**-29 */ if(two54+x>zero /* raise inexact */ &&ax<0x3c900000) /* |x| < 2**-54 */ return x; else return x - x*x*0.5; } if(hx>0||hx<=((int)0xbfd2bec3)) { k=0;f=x;hu=1;} /* -0.2929= 0x7ff00000) return x+x; if(k!=0) { if(hx<0x43400000) { `````` Raymond Toy committed Aug 01, 2014 133 134 135 `````` u = 1.0+x; ux.d = u; hu = ux.i[HIWORD]; /* high word of u */ `````` Raymond Toy committed Aug 01, 2014 136 137 138 139 140 `````` k = (hu>>20)-1023; c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */ c /= u; } else { u = x; `````` Raymond Toy committed Aug 01, 2014 141 142 `````` ux.d = u; hu = ux.i[HIWORD]; /* high word of u */ `````` Raymond Toy committed Aug 01, 2014 143 144 145 146 147 `````` k = (hu>>20)-1023; c = 0; } hu &= 0x000fffff; if(hu<0x6a09e) { `````` Raymond Toy committed Aug 01, 2014 148 149 150 `````` ux.d = u; ux.i[HIWORD] = hu|0x3ff00000; /* normalize u */ u = ux.d; `````` Raymond Toy committed Aug 01, 2014 151 `````` } else { `````` Raymond Toy committed Aug 01, 2014 152 153 154 155 `````` k += 1; ux.d = u; ux.i[HIWORD] = hu|0x3fe00000; /* normalize u/2 */ u = ux.d; `````` Raymond Toy committed Aug 01, 2014 156 157 158 159 160 161 `````` hu = (0x00100000-hu)>>2; } f = u-1.0; } hfsq=0.5*f*f; if(hu==0) { /* |f| < 2**-20 */ `````` Raymond Toy committed Aug 02, 2014 162 163 164 165 166 167 168 169 `````` if(f==zero) { if(k==0) return zero; else { c += k*ln2_lo; return k*ln2_hi+c; } } `````` Raymond Toy committed Aug 01, 2014 170 171 172 173 174 175 176 177 178 179 `````` R = hfsq*(1.0-0.66666666666666666*f); if(k==0) return f-R; else return k*ln2_hi-((R-(k*ln2_lo+c))-f); } s = f/(2.0+f); z = s*s; R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7)))))); if(k==0) return f-(hfsq-s*(hfsq+R)); else return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f); }``````