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 | const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */ |
64 | const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */ |
65 | const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */ |
66 | const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */ |
67 | const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */ |
68 | const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */ |
69 | const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */ |
70 | const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */ |
71 | const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
72 | |
73 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
74 | pub fn log(mut x: f64) -> f64 { |
75 | let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54 |
76 | |
77 | let mut ui = x.to_bits(); |
78 | let mut hx: u32 = (ui >> 32) as u32; |
79 | let mut k: i32 = 0; |
80 | |
81 | if (hx < 0x00100000) || ((hx >> 31) != 0) { |
82 | /* x < 2**-126 */ |
83 | if ui << 1 == 0 { |
84 | return -1. / (x * x); /* log(+-0)=-inf */ |
85 | } |
86 | if hx >> 31 != 0 { |
87 | return (x - x) / 0.0; /* log(-#) = NaN */ |
88 | } |
89 | /* subnormal number, scale x up */ |
90 | k -= 54; |
91 | x *= x1p54; |
92 | ui = x.to_bits(); |
93 | hx = (ui >> 32) as u32; |
94 | } else if hx >= 0x7ff00000 { |
95 | return x; |
96 | } else if hx == 0x3ff00000 && ui << 32 == 0 { |
97 | return 0.; |
98 | } |
99 | |
100 | /* reduce x into [sqrt(2)/2, sqrt(2)] */ |
101 | hx += 0x3ff00000 - 0x3fe6a09e; |
102 | k += ((hx >> 20) as i32) - 0x3ff; |
103 | hx = (hx & 0x000fffff) + 0x3fe6a09e; |
104 | ui = ((hx as u64) << 32) | (ui & 0xffffffff); |
105 | x = f64::from_bits(ui); |
106 | |
107 | let f: f64 = x - 1.0; |
108 | let hfsq: f64 = 0.5 * f * f; |
109 | let s: f64 = f / (2.0 + f); |
110 | let z: f64 = s * s; |
111 | let w: f64 = z * z; |
112 | let t1: f64 = w * (LG2 + w * (LG4 + w * LG6)); |
113 | let t2: f64 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7))); |
114 | let r: f64 = t2 + t1; |
115 | let dk: f64 = k as f64; |
116 | s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI |
117 | } |
118 | |