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 original C code and the long comment below are
8// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
9// came with this notice. The go code is a simplified
10// version of the original C.
11//
12// ====================================================
13// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
14//
15// Developed at SunPro, a Sun Microsystems, Inc. business.
16// Permission to use, copy, modify, and distribute this
17// software is freely granted, provided that this notice
18// is preserved.
19// ====================================================
20//
21// __ieee754_sqrt(x)
22// Return correctly rounded sqrt.
23//           -----------------------------------------
24//           | Use the hardware sqrt if you have one |
25//           -----------------------------------------
26// Method:
27//   Bit by bit method using integer arithmetic. (Slow, but portable)
28//   1. Normalization
29//      Scale x to y in [1,4) with even powers of 2:
30//      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
31//              sqrt(x) = 2**k * sqrt(y)
32//   2. Bit by bit computation
33//      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
34//           i                                                   0
35//                                     i+1         2
36//          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
37//           i      i            i                 i
38//
39//      To compute q    from q , one checks whether
40//                  i+1       i
41//
42//                            -(i+1) 2
43//                      (q + 2      )  <= y.                     (2)
44//                        i
45//                                                            -(i+1)
46//      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
47//                             i+1   i             i+1   i
48//
49//      With some algebraic manipulation, it is not difficult to see
50//      that (2) is equivalent to
51//                             -(i+1)
52//                      s  +  2       <= y                       (3)
53//                       i                i
54//
55//      The advantage of (3) is that s  and y  can be computed by
56//                                    i      i
57//      the following recurrence formula:
58//          if (3) is false
59//
60//          s     =  s  ,       y    = y   ;                     (4)
61//           i+1      i          i+1    i
62//
63//      otherwise,
64//                         -i                      -(i+1)
65//          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
66//           i+1      i          i+1    i     i
67//
68//      One may easily use induction to prove (4) and (5).
69//      Note. Since the left hand side of (3) contain only i+2 bits,
70//            it is not necessary to do a full (53-bit) comparison
71//            in (3).
72//   3. Final rounding
73//      After generating the 53 bits result, we compute one more bit.
74//      Together with the remainder, we can decide whether the
75//      result is exact, bigger than 1/2ulp, or less than 1/2ulp
76//      (it will never equal to 1/2ulp).
77//      The rounding mode can be detected by checking whether
78//      huge + tiny is equal to huge, and whether huge - tiny is
79//      equal to huge for some floating point number "huge" and "tiny".
80//
81//
82// Notes:  Rounding mode detection omitted. The constants "mask", "shift",
83// and "bias" are found in src/math/bits.go
84
85// Sqrt returns the square root of x.
86//
87// Special cases are:
88//
89//	Sqrt(+Inf) = +Inf
90//	Sqrt(±0) = ±0
91//	Sqrt(x < 0) = NaN
92//	Sqrt(NaN) = NaN
93func Sqrt(x float64) float64 {
94	return sqrt(x)
95}
96
97// Note: On systems where Sqrt is a single instruction, the compiler
98// may turn a direct call into a direct use of that instruction instead.
99
100func sqrt(x float64) float64 {
101	// special cases
102	switch {
103	case x == 0 || IsNaN(x) || IsInf(x, 1):
104		return x
105	case x < 0:
106		return NaN()
107	}
108	ix := Float64bits(x)
109	// normalize x
110	exp := int((ix >> shift) & mask)
111	if exp == 0 { // subnormal x
112		for ix&(1<<shift) == 0 {
113			ix <<= 1
114			exp--
115		}
116		exp++
117	}
118	exp -= bias // unbias exponent
119	ix &^= mask << shift
120	ix |= 1 << shift
121	if exp&1 == 1 { // odd exp, double x to make it even
122		ix <<= 1
123	}
124	exp >>= 1 // exp = exp/2, exponent of square root
125	// generate sqrt(x) bit by bit
126	ix <<= 1
127	var q, s uint64               // q = sqrt(x)
128	r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
129	for r != 0 {
130		t := s + r
131		if t <= ix {
132			s = t + r
133			ix -= t
134			q += r
135		}
136		ix <<= 1
137		r >>= 1
138	}
139	// final rounding
140	if ix != 0 { // remainder, result not exact
141		q += q & 1 // round according to extra bit
142	}
143	ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
144	return Float64frombits(ix)
145}
146