1 /* origin: FreeBSD /usr/src/lib/msun/src/k_log.h */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
14 * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
16 * The following describes the overall strategy for computing
17 * logarithms in base e. The argument reduction and adding the final
18 * term of the polynomial are done by the caller for increased accuracy
19 * when different bases are used.
22 * 1. Argument Reduction: find k and f such that
24 * where sqrt(2)/2 < 1+f < sqrt(2) .
26 * 2. Approximation of log(1+f).
27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
28 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
30 * We use a special Reme algorithm on [0,0.1716] to generate
31 * a polynomial of degree 14 to approximate R The maximum error
32 * of this polynomial approximation is bounded by 2**-58.45. In
35 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
36 * (the values of Lg1 to Lg7 are listed in the program)
39 * | Lg1*s +...+Lg7*s - R(z) | <= 2
41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
42 * In order to guarantee error in log below 1ulp, we compute log
44 * log(1+f) = f - s*(f - R) (if f is not too large)
45 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
47 * 3. Finally, log(x) = k*ln2 + log(1+f).
48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
49 * Here ln2 is split into two floating point number:
51 * where n*ln2_hi is always exact for |n| < 2000.
54 * log(x) is NaN with signal if x < 0 (including -INF) ;
55 * log(+INF) is +INF; log(0) is -INF with signal;
56 * log(NaN) is that NaN with no signal.
59 * according to an error analysis, the error is always less than
60 * 1 ulp (unit in the last place).
63 * The hexadecimal values are the intended ones for the following
64 * constants. The decimal values may be used, provided that the
65 * compiler will convert from decimal to binary accurately enough
66 * to produce the hexadecimal values shown.
70 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
71 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
72 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
73 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
74 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
75 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
76 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
79 * We always inline __log1p(), since doing so produces a
80 * substantial performance improvement (~40% on amd64).
82 static inline double __log1p(double f)
84 double hfsq,s,z,R,w,t1,t2;
89 t1= w*(Lg2+w*(Lg4+w*Lg6));
90 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));