1 /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */ 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 /* double log1p(double x) 13 * Return the natural logarithm of 1+x. 14 * 15 * Method : 16 * 1. Argument Reduction: find k and f such that 17 * 1+x = 2^k * (1+f), 18 * where sqrt(2)/2 < 1+f < sqrt(2) . 19 * 20 * Note. If k=0, then f=x is exact. However, if k!=0, then f 21 * may not be representable exactly. In that case, a correction 22 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then 23 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), 24 * and add back the correction term c/u. 25 * (Note: when x > 2**53, one can simply return log(x)) 26 * 27 * 2. Approximation of log(1+f): See log.c 28 * 29 * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c 30 * 31 * Special cases: 32 * log1p(x) is NaN with signal if x < -1 (including -INF) ; 33 * log1p(+INF) is +INF; log1p(-1) is -INF with signal; 34 * log1p(NaN) is that NaN with no signal. 35 * 36 * Accuracy: 37 * according to an error analysis, the error is always less than 38 * 1 ulp (unit in the last place). 39 * 40 * Constants: 41 * The hexadecimal values are the intended ones for the following 42 * constants. The decimal values may be used, provided that the 43 * compiler will convert from decimal to binary accurately enough 44 * to produce the hexadecimal values shown. 45 * 46 * Note: Assuming log() return accurate answer, the following 47 * algorithm can be used to compute log1p(x) to within a few ULP: 48 * 49 * u = 1+x; 50 * if(u==1.0) return x ; else 51 * return log(u)*(x/(u-1.0)); 52 * 53 * See HP-15C Advanced Functions Handbook, p.193. 54 */ 55 56 #include "libm.h" 57 58 static const double 59 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ 60 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 61 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 62 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 63 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 64 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 65 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 66 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 67 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 68 69 double log1p(double x) 70 { 71 union {double f; uint64_t i;} u = {x}; 72 double_t hfsq,f,c,s,z,R,w,t1,t2,dk; 73 uint32_t hx,hu; 74 int k; 75 76 hx = u.i>>32; 77 k = 1; 78 if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */ 79 if (hx >= 0xbff00000) { /* x <= -1.0 */ 80 if (x == -1) 81 return x/0.0; /* log1p(-1) = -inf */ 82 return (x-x)/0.0; /* log1p(x<-1) = NaN */ 83 } 84 if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */ 85 /* underflow if subnormal */ 86 if ((hx&0x7ff00000) == 0) 87 FORCE_EVAL((float)x); 88 return x; 89 } 90 if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ 91 k = 0; 92 c = 0; 93 f = x; 94 } 95 } else if (hx >= 0x7ff00000) 96 return x; 97 if (k) { 98 u.f = 1 + x; 99 hu = u.i>>32; 100 hu += 0x3ff00000 - 0x3fe6a09e; 101 k = (int)(hu>>20) - 0x3ff; 102 /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */ 103 if (k < 54) { 104 c = k >= 2 ? 1-(u.f-x) : x-(u.f-1); 105 c /= u.f; 106 } else 107 c = 0; 108 /* reduce u into [sqrt(2)/2, sqrt(2)] */ 109 hu = (hu&0x000fffff) + 0x3fe6a09e; 110 u.i = (uint64_t)hu<<32 | (u.i&0xffffffff); 111 f = u.f - 1; 112 } 113 hfsq = 0.5*f*f; 114 s = f/(2.0+f); 115 z = s*s; 116 w = z*z; 117 t1 = w*(Lg2+w*(Lg4+w*Lg6)); 118 t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 119 R = t2 + t1; 120 dk = k; 121 return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi; 122 } 123