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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
     3	// license that can be found in the LICENSE file.
     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	// ====================================================
    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 does 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	//	Sqrt(+Inf) = +Inf
    89	//	Sqrt(±0) = ±0
    90	//	Sqrt(x < 0) = NaN
    91	//	Sqrt(NaN) = NaN
    92	func Sqrt(x float64) float64
    93	
    94	// Note: Sqrt is implemented in assembly on some systems.
    95	// Others have assembly stubs that jump to func sqrt below.
    96	// On systems where Sqrt is a single instruction, the compiler
    97	// may turn a direct call into a direct use of that instruction instead.
    98	
    99	func sqrt(x float64) float64 {
   100		// special cases
   101		switch {
   102		case x == 0 || IsNaN(x) || IsInf(x, 1):
   103			return x
   104		case x < 0:
   105			return NaN()
   106		}
   107		ix := Float64bits(x)
   108		// normalize x
   109		exp := int((ix >> shift) & mask)
   110		if exp == 0 { // subnormal x
   111			for ix&(1<<shift) == 0 {
   112				ix <<= 1
   113				exp--
   114			}
   115			exp++
   116		}
   117		exp -= bias // unbias exponent
   118		ix &^= mask << shift
   119		ix |= 1 << shift
   120		if exp&1 == 1 { // odd exp, double x to make it even
   121			ix <<= 1
   122		}
   123		exp >>= 1 // exp = exp/2, exponent of square root
   124		// generate sqrt(x) bit by bit
   125		ix <<= 1
   126		var q, s uint64               // q = sqrt(x)
   127		r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
   128		for r != 0 {
   129			t := s + r
   130			if t <= ix {
   131				s = t + r
   132				ix -= t
   133				q += r
   134			}
   135			ix <<= 1
   136			r >>= 1
   137		}
   138		// final rounding
   139		if ix != 0 { // remainder, result not exact
   140			q += q & 1 // round according to extra bit
   141		}
   142		ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
   143		return Float64frombits(ix)
   144	}
   145	

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