log.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  /*
     8  	Floating-point logarithm.
     9  */
    10  
    11  // The original C code, the long comment, and the constants
    12  // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
    13  // and came with this notice. The go code is a simpler
    14  // version of the original C.
    15  //
    16  // ====================================================
    17  // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    18  //
    19  // Developed at SunPro, a Sun Microsystems, Inc. business.
    20  // Permission to use, copy, modify, and distribute this
    21  // software is freely granted, provided that this notice
    22  // is preserved.
    23  // ====================================================
    24  //
    25  // __ieee754_log(x)
    26  // Return the logarithm of x
    27  //
    28  // Method :
    29  //   1. Argument Reduction: find k and f such that
    30  //			x = 2**k * (1+f),
    31  //	   where  sqrt(2)/2 < 1+f < sqrt(2) .
    32  //
    33  //   2. Approximation of log(1+f).
    34  //	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    35  //		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    36  //	     	 = 2s + s*R
    37  //      We use a special Reme algorithm on [0,0.1716] to generate
    38  //	a polynomial of degree 14 to approximate R.  The maximum error
    39  //	of this polynomial approximation is bounded by 2**-58.45. In
    40  //	other words,
    41  //		        2      4      6      8      10      12      14
    42  //	    R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s  +L6*s  +L7*s
    43  //	(the values of L1 to L7 are listed in the program) and
    44  //	    |      2          14          |     -58.45
    45  //	    | L1*s +...+L7*s    -  R(z) | <= 2
    46  //	    |                             |
    47  //	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    48  //	In order to guarantee error in log below 1ulp, we compute log by
    49  //		log(1+f) = f - s*(f - R)		(if f is not too large)
    50  //		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
    51  //
    52  //	3. Finally,  log(x) = k*Ln2 + log(1+f).
    53  //			    = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
    54  //	   Here Ln2 is split into two floating point number:
    55  //			Ln2_hi + Ln2_lo,
    56  //	   where n*Ln2_hi is always exact for |n| < 2000.
    57  //
    58  // Special cases:
    59  //	log(x) is NaN with signal if x < 0 (including -INF) ;
    60  //	log(+INF) is +INF; log(0) is -INF with signal;
    61  //	log(NaN) is that NaN with no signal.
    62  //
    63  // Accuracy:
    64  //	according to an error analysis, the error is always less than
    65  //	1 ulp (unit in the last place).
    66  //
    67  // Constants:
    68  // The hexadecimal values are the intended ones for the following
    69  // constants. The decimal values may be used, provided that the
    70  // compiler will convert from decimal to binary accurately enough
    71  // to produce the hexadecimal values shown.
    72  
    73  // Log returns the natural logarithm of x.
    74  //
    75  // Special cases are:
    76  //
    77  //	Log(+Inf) = +Inf
    78  //	Log(0) = -Inf
    79  //	Log(x < 0) = NaN
    80  //	Log(NaN) = NaN
    81  func Log(x float64) float64 {
    82  	if haveArchLog {
    83  		return archLog(x)
    84  	}
    85  	return log(x)
    86  }
    87  
    88  func log(x float64) float64 {
    89  	const (
    90  		Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
    91  		Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
    92  		L1    = 6.666666666666735130e-01   /* 3FE55555 55555593 */
    93  		L2    = 3.999999999940941908e-01   /* 3FD99999 9997FA04 */
    94  		L3    = 2.857142874366239149e-01   /* 3FD24924 94229359 */
    95  		L4    = 2.222219843214978396e-01   /* 3FCC71C5 1D8E78AF */
    96  		L5    = 1.818357216161805012e-01   /* 3FC74664 96CB03DE */
    97  		L6    = 1.531383769920937332e-01   /* 3FC39A09 D078C69F */
    98  		L7    = 1.479819860511658591e-01   /* 3FC2F112 DF3E5244 */
    99  	)
   100  
   101  	// special cases
   102  	switch {
   103  	case IsNaN(x) || IsInf(x, 1):
   104  		return x
   105  	case x < 0:
   106  		return NaN()
   107  	case x == 0:
   108  		return Inf(-1)
   109  	}
   110  
   111  	// reduce
   112  	f1, ki := Frexp(x)
   113  	if f1 < Sqrt2/2 {
   114  		f1 *= 2
   115  		ki--
   116  	}
   117  	f := f1 - 1
   118  	k := float64(ki)
   119  
   120  	// compute
   121  	s := f / (2 + f)
   122  	s2 := s * s
   123  	s4 := s2 * s2
   124  	t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
   125  	t2 := s4 * (L2 + s4*(L4+s4*L6))
   126  	R := t1 + t2
   127  	hfsq := 0.5 * f * f
   128  	return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
   129  }
   130
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