aboutsummaryrefslogtreecommitdiffstats
path: root/contrib/go/_std_1.21/src/math/big/sqrt.go
blob: b4b03743f4da71310bbaef1bcc16877183cec401 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
// 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
}