1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.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 | * Optimized by Bruce D. Evans. |
13 | */ |
14 | /* cbrt(x) |
15 | * Return cube root of x |
16 | */ |
17 | |
18 | use core::f64; |
19 | |
20 | const B1: u32 = 715094163; /* B1 = (1023-1023/3-0.03306235651)*2**20 */ |
21 | const B2: u32 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ |
22 | |
23 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ |
24 | const P0: f64 = 1.87595182427177009643; /* 0x3ffe03e6, 0x0f61e692 */ |
25 | const P1: f64 = -1.88497979543377169875; /* 0xbffe28e0, 0x92f02420 */ |
26 | const P2: f64 = 1.621429720105354466140; /* 0x3ff9f160, 0x4a49d6c2 */ |
27 | const P3: f64 = -0.758397934778766047437; /* 0xbfe844cb, 0xbee751d9 */ |
28 | const P4: f64 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ |
29 | |
30 | // Cube root (f64) |
31 | /// |
32 | /// Computes the cube root of the argument. |
33 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
34 | pub fn cbrt(x: f64) -> f64 { |
35 | let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54 |
36 | |
37 | let mut ui: u64 = x.to_bits(); |
38 | let mut r: f64; |
39 | let s: f64; |
40 | let mut t: f64; |
41 | let w: f64; |
42 | let mut hx: u32 = (ui >> 32) as u32 & 0x7fffffff; |
43 | |
44 | if hx >= 0x7ff00000 { |
45 | /* cbrt(NaN,INF) is itself */ |
46 | return x + x; |
47 | } |
48 | |
49 | /* |
50 | * Rough cbrt to 5 bits: |
51 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) |
52 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and |
53 | * "%" are integer division and modulus with rounding towards minus |
54 | * infinity. The RHS is always >= the LHS and has a maximum relative |
55 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the |
56 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE |
57 | * floating point representation, for finite positive normal values, |
58 | * ordinary integer divison of the value in bits magically gives |
59 | * almost exactly the RHS of the above provided we first subtract the |
60 | * exponent bias (1023 for doubles) and later add it back. We do the |
61 | * subtraction virtually to keep e >= 0 so that ordinary integer |
62 | * division rounds towards minus infinity; this is also efficient. |
63 | */ |
64 | if hx < 0x00100000 { |
65 | /* zero or subnormal? */ |
66 | ui = (x * x1p54).to_bits(); |
67 | hx = (ui >> 32) as u32 & 0x7fffffff; |
68 | if hx == 0 { |
69 | return x; /* cbrt(0) is itself */ |
70 | } |
71 | hx = hx / 3 + B2; |
72 | } else { |
73 | hx = hx / 3 + B1; |
74 | } |
75 | ui &= 1 << 63; |
76 | ui |= (hx as u64) << 32; |
77 | t = f64::from_bits(ui); |
78 | |
79 | /* |
80 | * New cbrt to 23 bits: |
81 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) |
82 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) |
83 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation |
84 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this |
85 | * gives us bounds for r = t**3/x. |
86 | * |
87 | * Try to optimize for parallel evaluation as in __tanf.c. |
88 | */ |
89 | r = (t * t) * (t / x); |
90 | t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4)); |
91 | |
92 | /* |
93 | * Round t away from zero to 23 bits (sloppily except for ensuring that |
94 | * the result is larger in magnitude than cbrt(x) but not much more than |
95 | * 2 23-bit ulps larger). With rounding towards zero, the error bound |
96 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps |
97 | * in the rounded t, the infinite-precision error in the Newton |
98 | * approximation barely affects third digit in the final error |
99 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps |
100 | * before the final error is larger than 0.667 ulps. |
101 | */ |
102 | ui = t.to_bits(); |
103 | ui = (ui + 0x80000000) & 0xffffffffc0000000; |
104 | t = f64::from_bits(ui); |
105 | |
106 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */ |
107 | s = t * t; /* t*t is exact */ |
108 | r = x / s; /* error <= 0.5 ulps; |r| < |t| */ |
109 | w = t + t; /* t+t is exact */ |
110 | r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */ |
111 | t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */ |
112 | t |
113 | } |
114 | |