gitee.com/quant1x/num@v0.3.2/math32/log.go (about)

     1  package math32
     2  
     3  /*
     4  	Floating-point logarithm.
     5  */
     6  
     7  // The original C code, the long comment, and the constants
     8  // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
     9  // and came with this notice. The go code is a simpler
    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_log(x)
    22  // Return the logarithm of x
    23  //
    24  // Method :
    25  //   1. Argument Reduction: find k and f such that
    26  //			x = 2**k * (1+f),
    27  //	   where  sqrt(2)/2 < 1+f < sqrt(2) .
    28  //
    29  //   2. Approximation of log(1+f).
    30  //	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    31  //		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    32  //	     	 = 2s + s*R
    33  //      We use a special Reme algorithm on [0,0.1716] to generate
    34  //	a polynomial of degree 14 to approximate R.  The maximum error
    35  //	of this polynomial approximation is bounded by 2**-58.45. In
    36  //	other words,
    37  //		        2      4      6      8      10      12      14
    38  //	    R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s  +L6*s  +L7*s
    39  //	(the values of L1 to L7 are listed in the program) and
    40  //	    |      2          14          |     -58.45
    41  //	    | L1*s +...+L7*s    -  R(z) | <= 2
    42  //	    |                             |
    43  //	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    44  //	In order to guarantee error in log below 1ulp, we compute log by
    45  //		log(1+f) = f - s*(f - R)		(if f is not too large)
    46  //		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
    47  //
    48  //	3. Finally,  log(x) = k*Ln2 + log(1+f).
    49  //			    = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
    50  //	   Here Ln2 is split into two floating point number:
    51  //			Ln2_hi + Ln2_lo,
    52  //	   where n*Ln2_hi is always exact for |n| < 2000.
    53  //
    54  // Special cases:
    55  //	log(x) is NaN with signal if x < 0 (including -INF) ;
    56  //	log(+INF) is +INF; log(0) is -INF with signal;
    57  //	log(NaN) is that NaN with no signal.
    58  //
    59  // Accuracy:
    60  //	according to an error analysis, the error is always less than
    61  //	1 ulp (unit in the last place).
    62  //
    63  // Constants:
    64  // The hexadecimal values are the intended ones for the following
    65  // constants. The decimal values may be used, provided that the
    66  // compiler will convert from decimal to binary accurately enough
    67  // to produce the hexadecimal values shown.
    68  
    69  // Log returns the natural logarithm of x.
    70  //
    71  // Special cases are:
    72  //
    73  //	Log(+Inf) = +Inf
    74  //	Log(0) = -Inf
    75  //	Log(x < 0) = NaN
    76  //	Log(NaN) = NaN
    77  func Log(x float32) float32
    78  
    79  func log(x float32) float32 {
    80  	const (
    81  		Ln2Hi = 6.9313812256e-01 /* 0x3f317180 */
    82  		Ln2Lo = 9.0580006145e-06 /* 0x3717f7d1 */
    83  		L1    = 6.6666668653e-01 /* 0x3f2aaaab */
    84  		L2    = 4.0000000596e-01 /* 0x3ecccccd */
    85  		L3    = 2.8571429849e-01 /* 0x3e924925 */
    86  		L4    = 2.2222198546e-01 /* 0x3e638e29 */
    87  		L5    = 1.8183572590e-01 /* 0x3e3a3325 */
    88  		L6    = 1.5313838422e-01 /* 0x3e1cd04f */
    89  		L7    = 1.4798198640e-01 /* 0x3e178897 */
    90  	)
    91  
    92  	// special cases
    93  	switch {
    94  	case IsNaN(x) || IsInf(x, 1):
    95  		return x
    96  	case x < 0:
    97  		return NaN()
    98  	case x == 0:
    99  		return Inf(-1)
   100  	}
   101  
   102  	// reduce
   103  	f1, ki := Frexp(x)
   104  	if f1 < Sqrt2/2 {
   105  		f1 *= 2
   106  		ki--
   107  	}
   108  	f := f1 - 1
   109  	k := float32(ki)
   110  
   111  	// compute
   112  	s := f / (2 + f)
   113  	s2 := s * s
   114  	s4 := s2 * s2
   115  	t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
   116  	t2 := s4 * (L2 + s4*(L4+s4*L6))
   117  	R := t1 + t2
   118  	hfsq := 0.5 * f * f
   119  	return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
   120  }