trig_const/
exp.rs

1/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
2/*
3 * ====================================================
4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11/* exp(x)
12 * Returns the exponential of x.
13 *
14 * Method
15 *   1. Argument reduction:
16 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
17 *      Given x, find r and integer k such that
18 *
19 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
20 *
21 *      Here r will be represented as r = hi-lo for better
22 *      accuracy.
23 *
24 *   2. Approximation of exp(r) by a special rational function on
25 *      the interval [0,0.34658]:
26 *      Write
27 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
28 *      We use a special Remez algorithm on [0,0.34658] to generate
29 *      a polynomial of degree 5 to approximate R. The maximum error
30 *      of this polynomial approximation is bounded by 2**-59. In
31 *      other words,
32 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
33 *      (where z=r*r, and the values of P1 to P5 are listed below)
34 *      and
35 *          |                  5          |     -59
36 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
37 *          |                             |
38 *      The computation of exp(r) thus becomes
39 *                              2*r
40 *              exp(r) = 1 + ----------
41 *                            R(r) - r
42 *                                 r*c(r)
43 *                     = 1 + r + ----------- (for better accuracy)
44 *                                2 - c(r)
45 *      where
46 *                              2       4             10
47 *              c(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
48 *
49 *   3. Scale back to obtain exp(x):
50 *      From step 1, we have
51 *         exp(x) = 2^k * exp(r)
52 *
53 * Special cases:
54 *      exp(INF) is INF, exp(NaN) is NaN;
55 *      exp(-INF) is 0, and
56 *      for finite argument, only exp(0)=1 is exact.
57 *
58 * Accuracy:
59 *      according to an error analysis, the error is always less than
60 *      1 ulp (unit in the last place).
61 *
62 * Misc. info.
63 *      For IEEE double
64 *          if x >  709.782712893383973096 then exp(x) overflows
65 *          if x < -745.133219101941108420 then exp(x) underflows
66 */
67
68use super::scalbn;
69
70const HALF: [f64; 2] = [0.5, -0.5];
71const LN2HI: f64 = 6.93147180369123816490e-01; /* 0x3fe62e42, 0xfee00000 */
72const LN2LO: f64 = 1.90821492927058770002e-10; /* 0x3dea39ef, 0x35793c76 */
73const INVLN2: f64 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
74const P1: f64 = 1.66666666666666019037e-01; /* 0x3FC55555, 0x5555553E */
75const P2: f64 = -2.77777777770155933842e-03; /* 0xBF66C16C, 0x16BEBD93 */
76const P3: f64 = 6.61375632143793436117e-05; /* 0x3F11566A, 0xAF25DE2C */
77const P4: f64 = -1.65339022054652515390e-06; /* 0xBEBBBD41, 0xC5D26BF1 */
78const P5: f64 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
79
80/// Exponential, base *e* (f64)
81///
82/// Calculate the exponential of `x`, that is, *e* raised to the power `x`
83/// (where *e* is the base of the natural system of logarithms, approximately 2.71828).
84pub const fn exp(x: f64) -> f64 {
85    nightly_exp!(exp, exp_inner, x)
86}
87
88const fn exp_inner(mut x: f64) -> f64 {
89    let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 === 2 ^ 1023
90
91    let hi: f64;
92    let lo: f64;
93    let k: i32;
94    let mut hx: u32;
95
96    hx = (x.to_bits() >> 32) as u32;
97    let sign = (hx >> 31) as i32;
98    hx &= 0x7fffffff; /* high word of |x| */
99
100    /* special cases */
101    if hx >= 0x4086232b {
102        /* if |x| >= 708.39... */
103        if x.is_nan() {
104            return x;
105        }
106        if x > 709.782712893383973096 {
107            /* overflow if x!=inf */
108            x *= x1p1023;
109            return x;
110        }
111        if x < -708.39641853226410622 {
112            /* underflow if x!=-inf */
113            if x < -745.13321910194110842 {
114                return 0.;
115            }
116        }
117    }
118
119    /* argument reduction */
120    if hx > 0x3fd62e42 {
121        /* if |x| > 0.5 ln2 */
122        if hx >= 0x3ff0a2b2 {
123            /* if |x| >= 1.5 ln2 */
124            k = (INVLN2 * x + HALF[sign as usize]) as i32;
125        } else {
126            k = 1 - sign - sign;
127        }
128        hi = x - k as f64 * LN2HI; /* k*ln2hi is exact here */
129        lo = k as f64 * LN2LO;
130        x = hi - lo;
131    } else if hx > 0x3e300000 {
132        /* if |x| > 2**-28 */
133        k = 0;
134        hi = x;
135        lo = 0.;
136    } else {
137        /* inexact if x!=0 */
138        return 1. + x;
139    }
140
141    /* x is now in primary range */
142    let xx = x * x;
143    let c = x - xx * (P1 + xx * (P2 + xx * (P3 + xx * (P4 + xx * P5))));
144    let y = 1. + (x * c / (2. - c) - lo + hi);
145    if k == 0 {
146        y
147    } else {
148        scalbn::scalbn(y, k)
149    }
150}
trig_const/exp.rs

trig_const/
exp.rs
