Adding upstream version 1.16.10.upstream/1.16.10upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 128 insertions, 0 deletions

diff --git a/src/math/big/sqrt.go b/src/math/big/sqrt.go
new file mode 100644
index 0000000..0d50164
--- /dev/null
+++ b/src/math/big/sqrt.go
@@ -0,0 +1,128 @@
+// Copyright 2017 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 big
+
+import (
+	"math"
+	"sync"
+)
+
+var threeOnce struct {
+	sync.Once
+	v *Float
+}
+
+func three() *Float {
+	threeOnce.Do(func() {
+		threeOnce.v = NewFloat(3.0)
+	})
+	return threeOnce.v
+}
+
+// Sqrt sets z to the rounded square root of x, and returns it.
+//
+// If z's precision is 0, it is changed to x's precision before the
+// operation. Rounding is performed according to z's precision and
+// rounding mode, but z's accuracy is not computed. Specifically, the
+// result of z.Acc() is undefined.
+//
+// The function panics if z < 0. The value of z is undefined in that
+// case.
+func (z *Float) Sqrt(x *Float) *Float {
+	if debugFloat {
+		x.validate()
+	}
+
+	if z.prec == 0 {
+		z.prec = x.prec
+	}
+
+	if x.Sign() == -1 {
+		// following IEEE754-2008 (section 7.2)
+		panic(ErrNaN{"square root of negative operand"})
+	}
+
+	// handle ±0 and +∞
+	if x.form != finite {
+		z.acc = Exact
+		z.form = x.form
+		z.neg = x.neg // IEEE754-2008 requires √±0 = ±0
+		return z
+	}
+
+	// MantExp sets the argument's precision to the receiver's, and
+	// when z.prec > x.prec this will lower z.prec. Restore it after
+	// the MantExp call.
+	prec := z.prec
+	b := x.MantExp(z)
+	z.prec = prec
+
+	// Compute √(z·2**b) as
+	//   √( z)·2**(½b)     if b is even
+	//   √(2z)·2**(⌊½b⌋)   if b > 0 is odd
+	//   √(½z)·2**(⌈½b⌉)   if b < 0 is odd
+	switch b % 2 {
+	case 0:
+		// nothing to do
+	case 1:
+		z.exp++
+	case -1:
+		z.exp--
+	}
+	// 0.25 <= z < 2.0
+
+	// Solving 1/x² - z = 0 avoids Quo calls and is faster, especially
+	// for high precisions.
+	z.sqrtInverse(z)
+
+	// re-attach halved exponent
+	return z.SetMantExp(z, b/2)
+}
+
+// Compute √x (to z.prec precision) by solving
+//   1/t² - x = 0
+// for t (using Newton's method), and then inverting.
+func (z *Float) sqrtInverse(x *Float) {
+	// let
+	//   f(t) = 1/t² - x
+	// then
+	//   g(t) = f(t)/f'(t) = -½t(1 - xt²)
+	// and the next guess is given by
+	//   t2 = t - g(t) = ½t(3 - xt²)
+	u := newFloat(z.prec)
+	v := newFloat(z.prec)
+	three := three()
+	ng := func(t *Float) *Float {
+		u.prec = t.prec
+		v.prec = t.prec
+		u.Mul(t, t)     // u = t²
+		u.Mul(x, u)     //   = xt²
+		v.Sub(three, u) // v = 3 - xt²
+		u.Mul(t, v)     // u = t(3 - xt²)
+		u.exp--         //   = ½t(3 - xt²)
+		return t.Set(u)
+	}
+
+	xf, _ := x.Float64()
+	sqi := newFloat(z.prec)
+	sqi.SetFloat64(1 / math.Sqrt(xf))
+	for prec := z.prec + 32; sqi.prec < prec; {
+		sqi.prec *= 2
+		sqi = ng(sqi)
+	}
+	// sqi = 1/√x
+
+	// x/√x = √x
+	z.Mul(x, sqi)
+}
+
+// newFloat returns a new *Float with space for twice the given
+// precision.
+func newFloat(prec2 uint32) *Float {
+	z := new(Float)
+	// nat.make ensures the slice length is > 0
+	z.mant = z.mant.make(int(prec2/_W) * 2)
+	return z
+}