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# Source file src/runtime/sqrt.go

## Documentation: runtime

```  // Copyright 2009 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style

// Copy of math/sqrt.go, here for use by ARM softfloat.
// Modified to not use any floating point arithmetic so
// that we don't clobber any floating-point registers
// while emulating the sqrt instruction.

package runtime

// The original C code and the long comment below are
// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
// came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_sqrt(x)
// Return correctly rounded sqrt.
//           -----------------------------------------
//           | Use the hardware sqrt if you have one |
//           -----------------------------------------
// Method:
//   Bit by bit method using integer arithmetic. (Slow, but portable)
//   1. Normalization
//      Scale x to y in [1,4) with even powers of 2:
//      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
//              sqrt(x) = 2**k * sqrt(y)
//   2. Bit by bit computation
//      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
//           i                                                   0
//                                     i+1         2
//          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
//           i      i            i                 i
//
//      To compute q    from q , one checks whether
//                  i+1       i
//
//                            -(i+1) 2
//                      (q + 2      )  <= y.                     (2)
//                        i
//                                                            -(i+1)
//      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
//                             i+1   i             i+1   i
//
//      With some algebraic manipulation, it is not difficult to see
//      that (2) is equivalent to
//                             -(i+1)
//                      s  +  2       <= y                       (3)
//                       i                i
//
//      The advantage of (3) is that s  and y  can be computed by
//                                    i      i
//      the following recurrence formula:
//          if (3) is false
//
//          s     =  s  ,       y    = y   ;                     (4)
//           i+1      i          i+1    i
//
//      otherwise,
//                         -i                      -(i+1)
//          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
//           i+1      i          i+1    i     i
//
//      One may easily use induction to prove (4) and (5).
//      Note. Since the left hand side of (3) contain only i+2 bits,
//            it does not necessary to do a full (53-bit) comparison
//            in (3).
//   3. Final rounding
//      After generating the 53 bits result, we compute one more bit.
//      Together with the remainder, we can decide whether the
//      result is exact, bigger than 1/2ulp, or less than 1/2ulp
//      (it will never equal to 1/2ulp).
//      The rounding mode can be detected by checking whether
//      huge + tiny is equal to huge, and whether huge - tiny is
//      equal to huge for some floating point number "huge" and "tiny".
//
//
// Notes:  Rounding mode detection omitted.

const (
float64Shift = 64 - 11 - 1
float64Bias  = 1023
float64NaN   = 0x7FF8000000000001
float64Inf   = 0x7FF0000000000000
maxFloat64   = 1.797693134862315708145274237317043567981e+308 // 2**1023 * (2**53 - 1) / 2**52
)

// isnanu returns whether ix represents a NaN floating point number.
func isnanu(ix uint64) bool {
exp := (ix >> float64Shift) & float64Mask
sig := ix << (64 - float64Shift) >> (64 - float64Shift)
return exp == float64Mask && sig != 0
}

func sqrt(ix uint64) uint64 {
// special cases
switch {
case ix == 0 || ix == 1<<63: // x == 0
return ix
case isnanu(ix): // x != x
return ix
case ix&(1<<63) != 0: // x < 0
return float64NaN
case ix == float64Inf: // x > MaxFloat
return ix
}
// normalize x
exp := int((ix >> float64Shift) & float64Mask)
if exp == 0 { // subnormal x
for ix&(1<<float64Shift) == 0 {
ix <<= 1
exp--
}
exp++
}
exp -= float64Bias // unbias exponent
ix |= 1 << float64Shift
if exp&1 == 1 { // odd exp, double x to make it even
ix <<= 1
}
exp >>= 1 // exp = exp/2, exponent of square root
// generate sqrt(x) bit by bit
ix <<= 1
var q, s uint64                      // q = sqrt(x)
r := uint64(1 << (float64Shift + 1)) // r = moving bit from MSB to LSB
for r != 0 {
t := s + r
if t <= ix {
s = t + r
ix -= t
q += r
}
ix <<= 1
r >>= 1
}
// final rounding
if ix != 0 { // remainder, result not exact
q += q & 1 // round according to extra bit
}
ix = q>>1 + uint64(exp-1+float64Bias)<<float64Shift // significand + biased exponent
return ix
}

```

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