| |
| /* @(#)e_log.c 1.3 95/01/18 */ |
| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| #include <sys/cdefs.h> |
| __FBSDID("$FreeBSD$"); |
| |
| /* |
| * k_log1p(f): |
| * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. |
| * |
| * The following describes the overall strategy for computing |
| * logarithms in base e. The argument reduction and adding the final |
| * term of the polynomial are done by the caller for increased accuracy |
| * when different bases are used. |
| * |
| * 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. |
| */ |
| |
| static const double |
| 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 */ |
| |
| /* |
| * We always inline k_log1p(), since doing so produces a |
| * substantial performance improvement (~40% on amd64). |
| */ |
| static inline double |
| k_log1p(double f) |
| { |
| double hfsq,s,z,R,w,t1,t2; |
| |
| 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; |
| hfsq=0.5*f*f; |
| return s*(hfsq+R); |
| } |
