/* 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; }