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 | |
68 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
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; |
142 | s * (hfsq + r) + (dk * LN2_LO + c) - hfsq + f + dk * LN2_HI |
143 | } |
144 | |