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Source file src/math/log1p.go

Documentation: math

     1  // Copyright 2010 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, the long comment, and the constants
     8  // below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
     9  // and 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  //
    22  // double log1p(double x)
    23  //
    24  // Method :
    25  //   1. Argument Reduction: find k and f such that
    26  //                      1+x = 2**k * (1+f),
    27  //         where  sqrt(2)/2 < 1+f < sqrt(2) .
    28  //
    29  //      Note. If k=0, then f=x is exact. However, if k!=0, then f
    30  //      may not be representable exactly. In that case, a correction
    31  //      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
    32  //      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
    33  //      and add back the correction term c/u.
    34  //      (Note: when x > 2**53, one can simply return log(x))
    35  //
    36  //   2. Approximation of log1p(f).
    37  //      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    38  //               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    39  //               = 2s + s*R
    40  //      We use a special Reme algorithm on [0,0.1716] to generate
    41  //      a polynomial of degree 14 to approximate R The maximum error
    42  //      of this polynomial approximation is bounded by 2**-58.45. In
    43  //      other words,
    44  //                      2      4      6      8      10      12      14
    45  //          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
    46  //      (the values of Lp1 to Lp7 are listed in the program)
    47  //      and
    48  //          |      2          14          |     -58.45
    49  //          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
    50  //          |                             |
    51  //      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    52  //      In order to guarantee error in log below 1ulp, we compute log
    53  //      by
    54  //              log1p(f) = f - (hfsq - s*(hfsq+R)).
    55  //
    56  //   3. Finally, log1p(x) = k*ln2 + log1p(f).
    57  //                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    58  //      Here ln2 is split into two floating point number:
    59  //                   ln2_hi + ln2_lo,
    60  //      where n*ln2_hi is always exact for |n| < 2000.
    61  //
    62  // Special cases:
    63  //      log1p(x) is NaN with signal if x < -1 (including -INF) ;
    64  //      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
    65  //      log1p(NaN) is that NaN with no signal.
    66  //
    67  // Accuracy:
    68  //      according to an error analysis, the error is always less than
    69  //      1 ulp (unit in the last place).
    70  //
    71  // Constants:
    72  // The hexadecimal values are the intended ones for the following
    73  // constants. The decimal values may be used, provided that the
    74  // compiler will convert from decimal to binary accurately enough
    75  // to produce the hexadecimal values shown.
    76  //
    77  // Note: Assuming log() return accurate answer, the following
    78  //       algorithm can be used to compute log1p(x) to within a few ULP:
    79  //
    80  //              u = 1+x;
    81  //              if(u==1.0) return x ; else
    82  //                         return log(u)*(x/(u-1.0));
    83  //
    84  //       See HP-15C Advanced Functions Handbook, p.193.
    85  
    86  // Log1p returns the natural logarithm of 1 plus its argument x.
    87  // It is more accurate than Log(1 + x) when x is near zero.
    88  //
    89  // Special cases are:
    90  //	Log1p(+Inf) = +Inf
    91  //	Log1p(±0) = ±0
    92  //	Log1p(-1) = -Inf
    93  //	Log1p(x < -1) = NaN
    94  //	Log1p(NaN) = NaN
    95  func Log1p(x float64) float64
    96  
    97  func log1p(x float64) float64 {
    98  	const (
    99  		Sqrt2M1     = 4.142135623730950488017e-01  // Sqrt(2)-1 = 0x3fda827999fcef34
   100  		Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
   101  		Small       = 1.0 / (1 << 29)              // 2**-29 = 0x3e20000000000000
   102  		Tiny        = 1.0 / (1 << 54)              // 2**-54
   103  		Two53       = 1 << 53                      // 2**53
   104  		Ln2Hi       = 6.93147180369123816490e-01   // 3fe62e42fee00000
   105  		Ln2Lo       = 1.90821492927058770002e-10   // 3dea39ef35793c76
   106  		Lp1         = 6.666666666666735130e-01     // 3FE5555555555593
   107  		Lp2         = 3.999999999940941908e-01     // 3FD999999997FA04
   108  		Lp3         = 2.857142874366239149e-01     // 3FD2492494229359
   109  		Lp4         = 2.222219843214978396e-01     // 3FCC71C51D8E78AF
   110  		Lp5         = 1.818357216161805012e-01     // 3FC7466496CB03DE
   111  		Lp6         = 1.531383769920937332e-01     // 3FC39A09D078C69F
   112  		Lp7         = 1.479819860511658591e-01     // 3FC2F112DF3E5244
   113  	)
   114  
   115  	// special cases
   116  	switch {
   117  	case x < -1 || IsNaN(x): // includes -Inf
   118  		return NaN()
   119  	case x == -1:
   120  		return Inf(-1)
   121  	case IsInf(x, 1):
   122  		return Inf(1)
   123  	}
   124  
   125  	absx := x
   126  	if absx < 0 {
   127  		absx = -absx
   128  	}
   129  
   130  	var f float64
   131  	var iu uint64
   132  	k := 1
   133  	if absx < Sqrt2M1 { //  |x| < Sqrt(2)-1
   134  		if absx < Small { // |x| < 2**-29
   135  			if absx < Tiny { // |x| < 2**-54
   136  				return x
   137  			}
   138  			return x - x*x*0.5
   139  		}
   140  		if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
   141  			// (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
   142  			k = 0
   143  			f = x
   144  			iu = 1
   145  		}
   146  	}
   147  	var c float64
   148  	if k != 0 {
   149  		var u float64
   150  		if absx < Two53 { // 1<<53
   151  			u = 1.0 + x
   152  			iu = Float64bits(u)
   153  			k = int((iu >> 52) - 1023)
   154  			if k > 0 {
   155  				c = 1.0 - (u - x)
   156  			} else {
   157  				c = x - (u - 1.0) // correction term
   158  				c /= u
   159  			}
   160  		} else {
   161  			u = x
   162  			iu = Float64bits(u)
   163  			k = int((iu >> 52) - 1023)
   164  			c = 0
   165  		}
   166  		iu &= 0x000fffffffffffff
   167  		if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
   168  			u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
   169  		} else {
   170  			k++
   171  			u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
   172  			iu = (0x0010000000000000 - iu) >> 2
   173  		}
   174  		f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
   175  	}
   176  	hfsq := 0.5 * f * f
   177  	var s, R, z float64
   178  	if iu == 0 { // |f| < 2**-20
   179  		if f == 0 {
   180  			if k == 0 {
   181  				return 0
   182  			}
   183  			c += float64(k) * Ln2Lo
   184  			return float64(k)*Ln2Hi + c
   185  		}
   186  		R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
   187  		if k == 0 {
   188  			return f - R
   189  		}
   190  		return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
   191  	}
   192  	s = f / (2.0 + f)
   193  	z = s * s
   194  	R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
   195  	if k == 0 {
   196  		return f - (hfsq - s*(hfsq+R))
   197  	}
   198  	return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
   199  }
   200  

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