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Source file src/math/log.go

  // Copyright 2009 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  package math
  
  /*
  	Floating-point logarithm.
  */
  
  // The original C code, the long comment, and the constants
  // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
  // and came with this notice. The go code is a simpler
  // version of the original C.
  //
  // ====================================================
  // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  //
  // 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_log(x)
  // Return the logarithm of x
  //
  // Method :
  //   1. Argument Reduction: find k and f such that
  //			x = 2**k * (1+f),
  //	   where  sqrt(2)/2 < 1+f < sqrt(2) .
  //
  //   2. Approximation of log(1+f).
  //	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  //		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  //	     	 = 2s + s*R
  //      We use a special Reme algorithm on [0,0.1716] to generate
  //	a polynomial of degree 14 to approximate R.  The maximum error
  //	of this polynomial approximation is bounded by 2**-58.45. In
  //	other words,
  //		        2      4      6      8      10      12      14
  //	    R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s  +L6*s  +L7*s
  //	(the values of L1 to L7 are listed in the program) and
  //	    |      2          14          |     -58.45
  //	    | L1*s +...+L7*s    -  R(z) | <= 2
  //	    |                             |
  //	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  //	In order to guarantee error in log below 1ulp, we compute log by
  //		log(1+f) = f - s*(f - R)		(if f is not too large)
  //		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
  //
  //	3. Finally,  log(x) = k*Ln2 + log(1+f).
  //			    = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
  //	   Here Ln2 is split into two floating point number:
  //			Ln2_hi + Ln2_lo,
  //	   where n*Ln2_hi is always exact for |n| < 2000.
  //
  // Special cases:
  //	log(x) is NaN with signal if x < 0 (including -INF) ;
  //	log(+INF) is +INF; log(0) is -INF with signal;
  //	log(NaN) is that NaN with no signal.
  //
  // Accuracy:
  //	according to an error analysis, the error is always less than
  //	1 ulp (unit in the last place).
  //
  // Constants:
  // The hexadecimal values are the intended ones for the following
  // constants. The decimal values may be used, provided that the
  // compiler will convert from decimal to binary accurately enough
  // to produce the hexadecimal values shown.
  
  // Log returns the natural logarithm of x.
  //
  // Special cases are:
  //	Log(+Inf) = +Inf
  //	Log(0) = -Inf
  //	Log(x < 0) = NaN
  //	Log(NaN) = NaN
  func Log(x float64) float64
  
  func log(x float64) float64 {
  	const (
  		Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
  		Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
  		L1    = 6.666666666666735130e-01   /* 3FE55555 55555593 */
  		L2    = 3.999999999940941908e-01   /* 3FD99999 9997FA04 */
  		L3    = 2.857142874366239149e-01   /* 3FD24924 94229359 */
  		L4    = 2.222219843214978396e-01   /* 3FCC71C5 1D8E78AF */
  		L5    = 1.818357216161805012e-01   /* 3FC74664 96CB03DE */
  		L6    = 1.531383769920937332e-01   /* 3FC39A09 D078C69F */
  		L7    = 1.479819860511658591e-01   /* 3FC2F112 DF3E5244 */
  	)
  
  	// special cases
  	switch {
  	case IsNaN(x) || IsInf(x, 1):
  		return x
  	case x < 0:
  		return NaN()
  	case x == 0:
  		return Inf(-1)
  	}
  
  	// reduce
  	f1, ki := Frexp(x)
  	if f1 < Sqrt2/2 {
  		f1 *= 2
  		ki--
  	}
  	f := f1 - 1
  	k := float64(ki)
  
  	// compute
  	s := f / (2 + f)
  	s2 := s * s
  	s4 := s2 * s2
  	t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
  	t2 := s4 * (L2 + s4*(L4+s4*L6))
  	R := t1 + t2
  	hfsq := 0.5 * f * f
  	return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
  }
  

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