# Source file src/math/sqrt.go

```     1  // Copyright 2009 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
4
5  package 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  // ====================================================
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
93  func 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
100  func 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
```

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