libm/math/
log1p.rs

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
56use core::f64;
57
58const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
59const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
60const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
61const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
62const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
63const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
64const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
65const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
66const 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)]
70pub 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}
libm/math/log1p.rs

libm/math/
log1p.rs
