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 ```/* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */ ``` ```/* ``` ``` * ==================================================== ``` ``` * 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. ``` ``` * ==================================================== ``` ``` */ ``` ```/* log(x) ``` ``` * Return the logarithm of x ``` ``` * ``` ``` * Method : ``` ``` * 1. Argument Reduction: find k and f such that ``` ``` * x = 2^k * (1+f), ``` ``` * where sqrt(2)/2 < 1+f < sqrt(2) . ``` ``` * ``` ``` * 2. Approximation of log(1+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 Remez 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) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s ``` ``` * (the values of Lg1 to Lg7 are listed in the program) ``` ``` * and ``` ``` * | 2 14 | -58.45 ``` ``` * | Lg1*s +...+Lg7*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 ``` ``` * log(1+f) = f - s*(f - R) (if f is not too large) ``` ``` * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) ``` ``` * ``` ``` * 3. Finally, log(x) = k*ln2 + log(1+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: ``` ``` * log(x) is NaN with signal if x < 0 (including -INF) ; ``` ``` * log(+INF) is +INF; log(0) is -INF with signal; ``` ``` * log(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. ``` ``` */ ``` ``` ``` ```#include ``` ```#include ``` ``` ``` ```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 log(double x) ``` ```{ ``` ``` union {double f; uint64_t i;} u = {x}; ``` ``` double_t hfsq,f,s,z,R,w,t1,t2,dk; ``` ``` uint32_t hx; ``` ``` int k; ``` ``` ``` ``` hx = u.i>>32; ``` ``` k = 0; ``` ``` if (hx < 0x00100000 || hx>>31) { ``` ``` if (u.i<<1 == 0) ``` ``` return -1/(x*x); /* log(+-0)=-inf */ ``` ``` if (hx>>31) ``` ``` return (x-x)/0.0; /* log(-#) = NaN */ ``` ``` /* subnormal number, scale x up */ ``` ``` k -= 54; ``` ``` x *= 0x1p54; ``` ``` u.f = x; ``` ``` hx = u.i>>32; ``` ``` } else if (hx >= 0x7ff00000) { ``` ``` return x; ``` ``` } else if (hx == 0x3ff00000 && u.i<<32 == 0) ``` ``` return 0; ``` ``` ``` ``` /* reduce x into [sqrt(2)/2, sqrt(2)] */ ``` ``` hx += 0x3ff00000 - 0x3fe6a09e; ``` ``` k += (int)(hx>>20) - 0x3ff; ``` ``` hx = (hx&0x000fffff) + 0x3fe6a09e; ``` ``` u.i = (uint64_t)hx<<32 | (u.i&0xffffffff); ``` ``` x = u.f; ``` ``` ``` ``` f = x - 1.0; ``` ``` 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 - hfsq + f + dk*ln2_hi; ``` ```} ``` ``` ```