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 ```/* 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; ``` ```} ``` ``` ```