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1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */ |
2 | /* | |
3 | * ==================================================== | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
5 | * | |
6 | * Developed at SunPro, 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 | /* double log1p(double x) | |
13 | * Return the natural logarithm of 1+x. | |
14 | * | |
15 | * Method : | |
16 | * 1. Argument Reduction: find k and f such that | |
17 | * 1+x = 2^k * (1+f), | |
18 | * where sqrt(2)/2 < 1+f < sqrt(2) . | |
19 | * | |
20 | * Note. If k=0, then f=x is exact. However, if k!=0, then f | |
21 | * may not be representable exactly. In that case, a correction | |
22 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then | |
23 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), | |
24 | * and add back the correction term c/u. | |
25 | * (Note: when x > 2**53, one can simply return log(x)) | |
26 | * | |
27 | * 2. Approximation of log(1+f): See log.c | |
28 | * | |
29 | * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c | |
30 | * | |
31 | * Special cases: | |
32 | * log1p(x) is NaN with signal if x < -1 (including -INF) ; | |
33 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal; | |
34 | * log1p(NaN) is that NaN with no signal. | |
35 | * | |
36 | * Accuracy: | |
37 | * according to an error analysis, the error is always less than | |
38 | * 1 ulp (unit in the last place). | |
39 | * | |
40 | * Constants: | |
41 | * The hexadecimal values are the intended ones for the following | |
42 | * constants. The decimal values may be used, provided that the | |
43 | * compiler will convert from decimal to binary accurately enough | |
44 | * to produce the hexadecimal values shown. | |
45 | * | |
46 | * Note: Assuming log() return accurate answer, the following | |
47 | * algorithm can be used to compute log1p(x) to within a few ULP: | |
48 | * | |
49 | * u = 1+x; | |
50 | * if(u==1.0) return x ; else | |
51 | * return log(u)*(x/(u-1.0)); | |
52 | * | |
53 | * See HP-15C Advanced Functions Handbook, p.193. | |
54 | */ | |
55 | ||
56 | use core::f64; | |
57 | ||
58 | const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */ | |
59 | const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */ | |
60 | const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */ | |
61 | const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */ | |
62 | const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */ | |
63 | const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */ | |
64 | const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */ | |
65 | const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */ | |
66 | const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ | |
67 | ||
48663c56 | 68 | #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] |
8faf50e0 XL |
69 | pub fn log1p(x: f64) -> f64 { |
70 | let mut ui: u64 = x.to_bits(); | |
71 | let hfsq: f64; | |
72 | let mut f: f64 = 0.; | |
73 | let mut c: f64 = 0.; | |
74 | let s: f64; | |
75 | let z: f64; | |
76 | let r: f64; | |
77 | let w: f64; | |
78 | let t1: f64; | |
79 | let t2: f64; | |
80 | let dk: f64; | |
81 | let hx: u32; | |
82 | let mut hu: u32; | |
83 | let mut k: i32; | |
84 | ||
85 | hx = (ui >> 32) as u32; | |
86 | k = 1; | |
87 | if hx < 0x3fda827a || (hx >> 31) > 0 { | |
88 | /* 1+x < sqrt(2)+ */ | |
89 | if hx >= 0xbff00000 { | |
90 | /* x <= -1.0 */ | |
91 | if x == -1. { | |
92 | return x / 0.0; /* log1p(-1) = -inf */ | |
93 | } | |
94 | return (x - x) / 0.0; /* log1p(x<-1) = NaN */ | |
95 | } | |
96 | if hx << 1 < 0x3ca00000 << 1 { | |
97 | /* |x| < 2**-53 */ | |
98 | /* underflow if subnormal */ | |
99 | if (hx & 0x7ff00000) == 0 { | |
100 | force_eval!(x as f32); | |
101 | } | |
102 | return x; | |
103 | } | |
104 | if hx <= 0xbfd2bec4 { | |
105 | /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ | |
106 | k = 0; | |
107 | c = 0.; | |
108 | f = x; | |
109 | } | |
110 | } else if hx >= 0x7ff00000 { | |
111 | return x; | |
112 | } | |
113 | if k > 0 { | |
114 | ui = (1. + x).to_bits(); | |
115 | hu = (ui >> 32) as u32; | |
116 | hu += 0x3ff00000 - 0x3fe6a09e; | |
117 | k = (hu >> 20) as i32 - 0x3ff; | |
118 | /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */ | |
119 | if k < 54 { | |
120 | c = if k >= 2 { | |
121 | 1. - (f64::from_bits(ui) - x) | |
122 | } else { | |
123 | x - (f64::from_bits(ui) - 1.) | |
124 | }; | |
125 | c /= f64::from_bits(ui); | |
126 | } else { | |
127 | c = 0.; | |
128 | } | |
129 | /* reduce u into [sqrt(2)/2, sqrt(2)] */ | |
130 | hu = (hu & 0x000fffff) + 0x3fe6a09e; | |
131 | ui = (hu as u64) << 32 | (ui & 0xffffffff); | |
132 | f = f64::from_bits(ui) - 1.; | |
133 | } | |
134 | hfsq = 0.5 * f * f; | |
135 | s = f / (2.0 + f); | |
136 | z = s * s; | |
137 | w = z * z; | |
138 | t1 = w * (LG2 + w * (LG4 + w * LG6)); | |
139 | t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7))); | |
140 | r = t2 + t1; | |
141 | dk = k as f64; | |
48663c56 | 142 | s * (hfsq + r) + (dk * LN2_LO + c) - hfsq + f + dk * LN2_HI |
8faf50e0 | 143 | } |