/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, 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) * Return the natural logarithm of 1+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 log(1+f): See log.c * * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c * * 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 "libm.h" static const double ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ double log1p(double x) { union {double f; uint64_t i;} u = {x}; double_t hfsq,f,c,s,z,R,w,t1,t2,dk; uint32_t hx,hu; int k; hx = u.i>>32; k = 1; if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */ if (hx >= 0xbff00000) { /* x <= -1.0 */ if (x == -1) return x/0.0; /* log1p(-1) = -inf */ return (x-x)/0.0; /* log1p(x<-1) = NaN */ } if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */ /* underflow if subnormal */ if ((hx&0x7ff00000) == 0) FORCE_EVAL((float)x); return x; } if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ k = 0; c = 0; f = x; } } else if (hx >= 0x7ff00000) return x; if (k) { u.f = 1 + x; hu = u.i>>32; hu += 0x3ff00000 - 0x3fe6a09e; k = (int)(hu>>20) - 0x3ff; /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */ if (k < 54) { c = k >= 2 ? 1-(u.f-x) : x-(u.f-1); c /= u.f; } else c = 0; /* reduce u into [sqrt(2)/2, sqrt(2)] */ hu = (hu&0x000fffff) + 0x3fe6a09e; u.i = (uint64_t)hu<<32 | (u.i&0xffffffff); f = u.f - 1; } hfsq = 0.5*f*f; s = f/(2.0+f); z = s*s; w = z*z; t1 = w*(Lg2+w*(Lg4+w*Lg6)); t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); R = t2 + t1; dk = k; return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi; }