| /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>. */ |
| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| /* __ieee754_log2(x) |
| * Return the logarithm to base 2 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 + log(1+f). |
| * = k+(f-(hfsq-(s*(hfsq+R)))) |
| * |
| * Special cases: |
| * log2(x) is NaN with signal if x < 0 (including -INF) ; |
| * log2(+INF) is +INF; log(0) is -INF with signal; |
| * log2(NaN) is that NaN with no signal. |
| * |
| * 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.h> |
| #include <math_private.h> |
| |
| static const double ln2 = 0.69314718055994530942; |
| static const double two54 = 1.80143985094819840000e+16; /* 43500000 00000000 */ |
| static const double Lg1 = 6.666666666666735130e-01; /* 3FE55555 55555593 */ |
| static const double Lg2 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */ |
| static const double Lg3 = 2.857142874366239149e-01; /* 3FD24924 94229359 */ |
| static const double Lg4 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */ |
| static const double Lg5 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */ |
| static const double Lg6 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */ |
| static const double Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
| |
| static const double zero = 0.0; |
| |
| double |
| __ieee754_log2 (double x) |
| { |
| 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 (__builtin_expect (((hx & 0x7fffffff) | lx) == 0, 0)) |
| return -two54 / (x - x); /* log(+-0)=-inf */ |
| if (__builtin_expect (hx < 0, 0)) |
| return (x - x) / (x - x); /* log(-#) = NaN */ |
| k -= 54; |
| x *= two54; /* subnormal number, scale up x */ |
| GET_HIGH_WORD (hx, x); |
| } |
| if (__builtin_expect (hx >= 0x7ff00000, 0)) |
| 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); |
| dk = (double) k; |
| f = x - 1.0; |
| if ((0x000fffff & (2 + hx)) < 3) |
| { /* |f| < 2**-20 */ |
| if (f == zero) |
| return dk; |
| R = f * f * (0.5 - 0.33333333333333333 * f); |
| return dk - (R - f) / ln2; |
| } |
| s = f / (2.0 + f); |
| 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; |
| return dk - ((hfsq - (s * (hfsq + R))) - f) / ln2; |
| } |
| else |
| { |
| return dk - ((s * (f - R)) - f) / ln2; |
| } |
| } |
| |
| strong_alias (__ieee754_log2, __log2_finite) |
