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//	Cbrt(±0) = ±0
23//	Cbrt(±Inf) = ±Inf
24//	Cbrt(NaN) = NaN
25func Cbrt(x float64) float64 {
26	const (
27		B1             = 715094163                   // (682-0.03306235651)*2**20
28		B2             = 696219795                   // (664-0.03306235651)*2**20
29		C              = 5.42857142857142815906e-01  // 19/35     = 0x3FE15F15F15F15F1
30		D              = -7.05306122448979611050e-01 // -864/1225 = 0xBFE691DE2532C834
31		E              = 1.41428571428571436819e+00  // 99/70     = 0x3FF6A0EA0EA0EA0F
32		F              = 1.60714285714285720630e+00  // 45/28     = 0x3FF9B6DB6DB6DB6E
33		G              = 3.57142857142857150787e-01  // 5/14      = 0x3FD6DB6DB6DB6DB7
34		SmallestNormal = 2.22507385850720138309e-308 // 2**-1022  = 0x0010000000000000
35	)
36	// special cases
37	switch {
38	case x == 0 || IsNaN(x) || IsInf(x, 0):
39		return x
40	}
41
42	sign := false
43	if x < 0 {
44		x = -x
45		sign = true
46	}
47
48	// rough cbrt to 5 bits
49	t := Float64frombits(Float64bits(x)/3 + B1<<32)
50	if x < SmallestNormal {
51		// subnormal number
52		t = float64(1 << 54) // set t= 2**54
53		t *= x
54		t = Float64frombits(Float64bits(t)/3 + B2<<32)
55	}
56
57	// new cbrt to 23 bits
58	r := t * t / x
59	s := C + r*t
60	t *= G + F/(s+E+D/s)
61
62	// chop to 22 bits, make larger than cbrt(x)
63	t = Float64frombits(Float64bits(t)&(0xFFFFFFFFC<<28) + 1<<30)
64
65	// one step newton iteration to 53 bits with error less than 0.667ulps
66	s = t * t // t*t is exact
67	r = x / s
68	w := t + t
69	r = (r - t) / (w + r) // r-s is exact
70	t = t + t*r
71
72	// restore the sign bit
73	if sign {
74		t = -t
75	}
76	return t
77}
78