1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
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
9 * is preserved.
10 * ====================================================
11 */
12 /* log(x)
13 * Return the logarithm of x
14 *
15 * Method :
16 * 1. Argument Reduction: find k and f such that
17 * x = 2^k * (1+f),
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
19 *
20 * 2. Approximation of log(1+f).
21 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
22 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
23 * = 2s + s*R
24 * We use a special Remez algorithm on [0,0.1716] to generate
25 * a polynomial of degree 14 to approximate R The maximum error
26 * of this polynomial approximation is bounded by 2**-58.45. In
27 * other words,
28 * 2 4 6 8 10 12 14
29 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
30 * (the values of Lg1 to Lg7 are listed in the program)
31 * and
32 * | 2 14 | -58.45
33 * | Lg1*s +...+Lg7*s - R(z) | <= 2
34 * | |
35 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
36 * In order to guarantee error in log below 1ulp, we compute log
37 * by
38 * log(1+f) = f - s*(f - R) (if f is not too large)
39 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
40 *
41 * 3. Finally, log(x) = k*ln2 + log(1+f).
42 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
43 * Here ln2 is split into two floating point number:
44 * ln2_hi + ln2_lo,
45 * where n*ln2_hi is always exact for |n| < 2000.
46 *
47 * Special cases:
48 * log(x) is NaN with signal if x < 0 (including -INF) ;
49 * log(+INF) is +INF; log(0) is -INF with signal;
50 * log(NaN) is that NaN with no signal.
51 *
52 * Accuracy:
53 * according to an error analysis, the error is always less than
54 * 1 ulp (unit in the last place).
55 *
56 * Constants:
57 * The hexadecimal values are the intended ones for the following
58 * constants. The decimal values may be used, provided that the
59 * compiler will convert from decimal to binary accurately enough
60 * to produce the hexadecimal values shown.
61 */
62
63 #include <math.h>
64 #include <stdint.h>
65
66 static const double
67 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
68 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
69 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
70 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
71 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
72 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
73 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
74 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
75 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
76
log(double x)77 double log(double x)
78 {
79 union {double f; uint64_t i;} u = {x};
80 double_t hfsq,f,s,z,R,w,t1,t2,dk;
81 uint32_t hx;
82 int k;
83
84 hx = u.i>>32;
85 k = 0;
86 if (hx < 0x00100000 || hx>>31) {
87 if (u.i<<1 == 0)
88 return -1/(x*x); /* log(+-0)=-inf */
89 if (hx>>31)
90 return (x-x)/0.0; /* log(-#) = NaN */
91 /* subnormal number, scale x up */
92 k -= 54;
93 x *= 0x1p54;
94 u.f = x;
95 hx = u.i>>32;
96 } else if (hx >= 0x7ff00000) {
97 return x;
98 } else if (hx == 0x3ff00000 && u.i<<32 == 0)
99 return 0;
100
101 /* reduce x into [sqrt(2)/2, sqrt(2)] */
102 hx += 0x3ff00000 - 0x3fe6a09e;
103 k += (int)(hx>>20) - 0x3ff;
104 hx = (hx&0x000fffff) + 0x3fe6a09e;
105 u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
106 x = u.f;
107
108 f = x - 1.0;
109 hfsq = 0.5*f*f;
110 s = f/(2.0+f);
111 z = s*s;
112 w = z*z;
113 t1 = w*(Lg2+w*(Lg4+w*Lg6));
114 t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
115 R = t2 + t1;
116 dk = k;
117 return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
118 }
119