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 /// The natural logarithm of 1+`x` (f64).
69 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
log1p(x: f64) -> f6470 pub fn log1p(x: f64) -> f64 {
71 let mut ui: u64 = x.to_bits();
72 let hfsq: f64;
73 let mut f: f64 = 0.;
74 let mut c: f64 = 0.;
75 let s: f64;
76 let z: f64;
77 let r: f64;
78 let w: f64;
79 let t1: f64;
80 let t2: f64;
81 let dk: f64;
82 let hx: u32;
83 let mut hu: u32;
84 let mut k: i32;
85
86 hx = (ui >> 32) as u32;
87 k = 1;
88 if hx < 0x3fda827a || (hx >> 31) > 0 {
89 /* 1+x < sqrt(2)+ */
90 if hx >= 0xbff00000 {
91 /* x <= -1.0 */
92 if x == -1. {
93 return x / 0.0; /* log1p(-1) = -inf */
94 }
95 return (x - x) / 0.0; /* log1p(x<-1) = NaN */
96 }
97 if hx << 1 < 0x3ca00000 << 1 {
98 /* |x| < 2**-53 */
99 /* underflow if subnormal */
100 if (hx & 0x7ff00000) == 0 {
101 force_eval!(x as f32);
102 }
103 return x;
104 }
105 if hx <= 0xbfd2bec4 {
106 /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
107 k = 0;
108 c = 0.;
109 f = x;
110 }
111 } else if hx >= 0x7ff00000 {
112 return x;
113 }
114 if k > 0 {
115 ui = (1. + x).to_bits();
116 hu = (ui >> 32) as u32;
117 hu += 0x3ff00000 - 0x3fe6a09e;
118 k = (hu >> 20) as i32 - 0x3ff;
119 /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
120 if k < 54 {
121 c = if k >= 2 { 1. - (f64::from_bits(ui) - x) } else { x - (f64::from_bits(ui) - 1.) };
122 c /= f64::from_bits(ui);
123 } else {
124 c = 0.;
125 }
126 /* reduce u into [sqrt(2)/2, sqrt(2)] */
127 hu = (hu & 0x000fffff) + 0x3fe6a09e;
128 ui = (hu as u64) << 32 | (ui & 0xffffffff);
129 f = f64::from_bits(ui) - 1.;
130 }
131 hfsq = 0.5 * f * f;
132 s = f / (2.0 + f);
133 z = s * s;
134 w = z * z;
135 t1 = w * (LG2 + w * (LG4 + w * LG6));
136 t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
137 r = t2 + t1;
138 dk = k as f64;
139 s * (hfsq + r) + (dk * LN2_LO + c) - hfsq + f + dk * LN2_HI
140 }
141