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+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ The algorithm is based in part on "Optimal Partitioning of
+ Newton's Method for Calculating Roots", by Gunter Meinardus
+ and G. D. Taylor, Mathematics of Computation © 1980 American
+ Mathematical Society.
+ (http://www.jstor.org/stable/2006387?seq=9, accessed 11-Feb-2010)
+*/
+
+// Cbrt returns the cube root of x.
+//
+// Special cases are:
+// Cbrt(±0) = ±0
+// Cbrt(±Inf) = ±Inf
+// Cbrt(NaN) = NaN
+func Cbrt(x float64) float64 {
+ const (
+ A1 = 1.662848358e-01
+ A2 = 1.096040958e+00
+ A3 = 4.105032829e-01
+ A4 = 5.649335816e-01
+ B1 = 2.639607233e-01
+ B2 = 8.699282849e-01
+ B3 = 1.629083358e-01
+ B4 = 2.824667908e-01
+ C1 = 4.190115298e-01
+ C2 = 6.904625373e-01
+ C3 = 6.46502159e-02
+ C4 = 1.412333954e-01
+ )
+ // special cases
+ switch {
+ case x == 0 || IsNaN(x) || IsInf(x, 0):
+ return x
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ // Reduce argument and estimate cube root
+ f, e := Frexp(x) // 0.5 <= f < 1.0
+ m := e % 3
+ if m > 0 {
+ m -= 3
+ e -= m // e is multiple of 3
+ }
+ switch m {
+ case 0: // 0.5 <= f < 1.0
+ f = A1*f + A2 - A3/(A4+f)
+ case -1:
+ f *= 0.5 // 0.25 <= f < 0.5
+ f = B1*f + B2 - B3/(B4+f)
+ default: // m == -2
+ f *= 0.25 // 0.125 <= f < 0.25
+ f = C1*f + C2 - C3/(C4+f)
+ }
+ y := Ldexp(f, e/3) // e/3 = exponent of cube root
+
+ // Iterate
+ s := y * y * y
+ t := s + x
+ y *= (t + x) / (s + t)
+ // Reiterate
+ s = (y*y*y - x) / x
+ y -= y * (((14.0/81.0)*s-(2.0/9.0))*s + (1.0 / 3.0)) * s
+ if sign {
+ y = -y
+ }
+ return y
+}