Edit
kc3-lang/SDL/src/libm/e_log.c
Gabriel Jacobo
- src/libm/e_log.c
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/* @(#)e_log.c 5.1 93/09/24 */ /* * ==================================================== * 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. * ==================================================== */ #if defined(LIBM_SCCS) && !defined(lint) static const char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $"; #endif /* __ieee754_log(x) * Return the logrithm 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 Reme 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 "math_libm.h" #include "math_private.h" #ifdef __STDC__ static const double #else static double #endif ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ 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 */ #ifdef __STDC__ static const double zero = 0.0; #else static double zero = 0.0; #endif #ifdef __STDC__ double attribute_hidden __ieee754_log(double x) #else double attribute_hidden __ieee754_log(x) double x; #endif { double hfsq, f, s, z, R, w, t1, t2, dk; int32_t k, hx, i, j; u_int32_t lx; EXTRACT_WORDS(hx, lx, x); k = 0; if (hx < 0x00100000) { /* x < 2**-1022 */ if (((hx & 0x7fffffff) | lx) == 0) return -two54 / zero; /* log(+-0)=-inf */ if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */ k -= 54; x *= two54; /* subnormal number, scale up x */ GET_HIGH_WORD(hx, x); } if (hx >= 0x7ff00000) return x + x; k += (hx >> 20) - 1023; hx &= 0x000fffff; i = (hx + 0x95f64) & 0x100000; SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ k += (i >> 20); f = x - 1.0; if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */ if (f == zero) { if (k == 0) return zero; else { dk = (double) k; return dk * ln2_hi + dk * ln2_lo; } } R = f * f * (0.5 - 0.33333333333333333 * f); if (k == 0) return f - R; else { dk = (double) k; return dk * ln2_hi - ((R - dk * ln2_lo) - f); } } s = f / (2.0 + f); dk = (double) k; z = s * s; i = hx - 0x6147a; w = z * z; j = 0x6b851 - hx; t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); i |= j; R = t2 + t1; if (i > 0) { hfsq = 0.5 * f * f; if (k == 0) return f - (hfsq - s * (hfsq + R)); else return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); } else { if (k == 0) return f - s * (f - R); else return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); } }