1// Copyright 2009 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package math
6
7// The go code is a modified version of the original C code from
8// http://www.netlib.org/fdlibm/s_cbrt.c and came with this notice.
9//
10// ====================================================
11// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
12//
13// Developed at SunSoft, a Sun Microsystems, Inc. business.
14// Permission to use, copy, modify, and distribute this
15// software is freely granted, provided that this notice
16// is preserved.
17// ====================================================
18
19// Cbrt returns the cube root of x.
20//
21// Special cases are:
22//
23//	Cbrt(±0) = ±0
24//	Cbrt(±Inf) = ±Inf
25//	Cbrt(NaN) = NaN
26func Cbrt(x float64) float64 {
27	if haveArchCbrt {
28		return archCbrt(x)
29	}
30	return cbrt(x)
31}
32
33func cbrt(x float64) float64 {
34	const (
35		B1             = 715094163                   // (682-0.03306235651)*2**20
36		B2             = 696219795                   // (664-0.03306235651)*2**20
37		C              = 5.42857142857142815906e-01  // 19/35     = 0x3FE15F15F15F15F1
38		D              = -7.05306122448979611050e-01 // -864/1225 = 0xBFE691DE2532C834
39		E              = 1.41428571428571436819e+00  // 99/70     = 0x3FF6A0EA0EA0EA0F
40		F              = 1.60714285714285720630e+00  // 45/28     = 0x3FF9B6DB6DB6DB6E
41		G              = 3.57142857142857150787e-01  // 5/14      = 0x3FD6DB6DB6DB6DB7
42		SmallestNormal = 2.22507385850720138309e-308 // 2**-1022  = 0x0010000000000000
43	)
44	// special cases
45	switch {
46	case x == 0 || IsNaN(x) || IsInf(x, 0):
47		return x
48	}
49
50	sign := false
51	if x < 0 {
52		x = -x
53		sign = true
54	}
55
56	// rough cbrt to 5 bits
57	t := Float64frombits(Float64bits(x)/3 + B1<<32)
58	if x < SmallestNormal {
59		// subnormal number
60		t = float64(1 << 54) // set t= 2**54
61		t *= x
62		t = Float64frombits(Float64bits(t)/3 + B2<<32)
63	}
64
65	// new cbrt to 23 bits
66	r := t * t / x
67	s := C + r*t
68	t *= G + F/(s+E+D/s)
69
70	// chop to 22 bits, make larger than cbrt(x)
71	t = Float64frombits(Float64bits(t)&(0xFFFFFFFFC<<28) + 1<<30)
72
73	// one step newton iteration to 53 bits with error less than 0.667ulps
74	s = t * t // t*t is exact
75	r = x / s
76	w := t + t
77	r = (r - t) / (w + r) // r-s is exact
78	t = t + t*r
79
80	// restore the sign bit
81	if sign {
82		t = -t
83	}
84	return t
85}
86