1
2 /* @(#)e_log.c 1.3 95/01/18 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
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
10 * is preserved.
11 * ====================================================
12 */
13
14 #include <sys/cdefs.h>
15 __FBSDID("$FreeBSD: stable/9/lib/msun/src/k_log.h 216210 2010-12-05 22:11:03Z das $");
16
17 /* __kernel_log(x)
18 * Return log(x) - (x-1) for x in ~[sqrt(2)/2, sqrt(2)].
19 *
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.
24 *
25 * Method :
26 * 1. Argument Reduction: find k and f such that
27 * x = 2^k * (1+f),
28 * where sqrt(2)/2 < 1+f < sqrt(2) .
29 *
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 + .....,
33 * = 2s + s*R
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
37 * other words,
38 * 2 4 6 8 10 12 14
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)
41 * and
42 * | 2 14 | -58.45
43 * | Lg1*s +...+Lg7*s - R(z) | <= 2
44 * | |
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
47 * by
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)
50 *
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:
54 * ln2_hi + ln2_lo,
55 * where n*ln2_hi is always exact for |n| < 2000.
56 *
57 * Special cases:
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.
61 *
62 * Accuracy:
63 * according to an error analysis, the error is always less than
64 * 1 ulp (unit in the last place).
65 *
66 * Constants:
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.
71 */
72
73 static const double
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 */
81
82 /*
83 * We always inline __kernel_log(), since doing so produces a
84 * substantial performance improvement (~40% on amd64).
85 */
86 static inline double
__kernel_log(double x)87 __kernel_log(double x)
88 {
89 double hfsq,f,s,z,R,w,t1,t2;
90 int32_t hx,i,j;
91 u_int32_t lx;
92
93 EXTRACT_WORDS(hx,lx,x);
94
95 f = x-1.0;
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);
99 }
100 s = f/(2.0+f);
101 z = s*s;
102 hx &= 0x000fffff;
103 i = hx-0x6147a;
104 w = z*z;
105 j = 0x6b851-hx;
106 t1= w*(Lg2+w*(Lg4+w*Lg6));
107 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
108 i |= j;
109 R = t2+t1;
110 if (i>0) {
111 hfsq=0.5*f*f;
112 return s*(hfsq+R) - hfsq;
113 } else {
114 return s*(R-f);
115 }
116 }
117