2 /* @(#)e_log.c 1.3 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
14 #include <sys/cdefs.h>
15 __FBSDID("$FreeBSD$");
18 * Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)].
20 * The following describes the overall strategy for computing
21 * logarithms in base e. The argument reduction and adding the final
22 * term of the polynomial are done by the caller for increased accuracy
23 * when different bases are used.
26 * 1. Argument Reduction: find k and f such that
28 * where sqrt(2)/2 < 1+f < sqrt(2) .
30 * 2. Approximation of log(1+f).
31 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
32 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
34 * We use a special Reme algorithm on [0,0.1716] to generate
35 * a polynomial of degree 14 to approximate R The maximum error
36 * of this polynomial approximation is bounded by 2**-58.45. In
39 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
40 * (the values of Lg1 to Lg7 are listed in the program)
43 * | Lg1*s +...+Lg7*s - R(z) | <= 2
45 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
46 * In order to guarantee error in log below 1ulp, we compute log
48 * log(1+f) = f - s*(f - R) (if f is not too large)
49 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
51 * 3. Finally, log(x) = k*ln2 + log(1+f).
52 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
53 * Here ln2 is split into two floating point number:
55 * where n*ln2_hi is always exact for |n| < 2000.
58 * log(x) is NaN with signal if x < 0 (including -INF) ;
59 * log(+INF) is +INF; log(0) is -INF with signal;
60 * log(NaN) is that NaN with no signal.
63 * according to an error analysis, the error is always less than
64 * 1 ulp (unit in the last place).
67 * The hexadecimal values are the intended ones for the following
68 * constants. The decimal values may be used, provided that the
69 * compiler will convert from decimal to binary accurately enough
70 * to produce the hexadecimal values shown.
74 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
75 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
76 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
77 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
78 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
79 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
80 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
83 * We always inline __kernel_log(), since doing so produces a
84 * substantial performance improvement (~40% on amd64).
87 __kernel_log(double x)
89 double hfsq,f,s,z,R,w,t1,t2;
93 EXTRACT_WORDS(hx,lx,x);
96 if((0x000fffff&(2+hx))<3) { /* -2**-20 <= f < 2**-20 */
97 if(f==0.0) return 0.0;
98 return f*f*(0.33333333333333333*f-0.5);
106 t1= w*(Lg2+w*(Lg4+w*Lg6));
107 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
112 return s*(hfsq+R) - hfsq;