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Source file src/math/expm1.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_expm1.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  // expm1(x)
    22  // Returns exp(x)-1, the exponential of x minus 1.
    23  //
    24  // Method
    25  //   1. Argument reduction:
    26  //      Given x, find r and integer k such that
    27  //
    28  //               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
    29  //
    30  //      Here a correction term c will be computed to compensate
    31  //      the error in r when rounded to a floating-point number.
    32  //
    33  //   2. Approximating expm1(r) by a special rational function on
    34  //      the interval [0,0.34658]:
    35  //      Since
    36  //          r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
    37  //      we define R1(r*r) by
    38  //          r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
    39  //      That is,
    40  //          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
    41  //                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
    42  //                   = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
    43  //      We use a special Reme algorithm on [0,0.347] to generate
    44  //      a polynomial of degree 5 in r*r to approximate R1. The
    45  //      maximum error of this polynomial approximation is bounded
    46  //      by 2**-61. In other words,
    47  //          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
    48  //      where   Q1  =  -1.6666666666666567384E-2,
    49  //              Q2  =   3.9682539681370365873E-4,
    50  //              Q3  =  -9.9206344733435987357E-6,
    51  //              Q4  =   2.5051361420808517002E-7,
    52  //              Q5  =  -6.2843505682382617102E-9;
    53  //      (where z=r*r, and the values of Q1 to Q5 are listed below)
    54  //      with error bounded by
    55  //          |                  5           |     -61
    56  //          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
    57  //          |                              |
    58  //
    59  //      expm1(r) = exp(r)-1 is then computed by the following
    60  //      specific way which minimize the accumulation rounding error:
    61  //                             2     3
    62  //                            r     r    [ 3 - (R1 + R1*r/2)  ]
    63  //            expm1(r) = r + --- + --- * [--------------------]
    64  //                            2     2    [ 6 - r*(3 - R1*r/2) ]
    65  //
    66  //      To compensate the error in the argument reduction, we use
    67  //              expm1(r+c) = expm1(r) + c + expm1(r)*c
    68  //                         ~ expm1(r) + c + r*c
    69  //      Thus c+r*c will be added in as the correction terms for
    70  //      expm1(r+c). Now rearrange the term to avoid optimization
    71  //      screw up:
    72  //                      (      2                                    2 )
    73  //                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
    74  //       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
    75  //                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
    76  //                      (                                             )
    77  //
    78  //                 = r - E
    79  //   3. Scale back to obtain expm1(x):
    80  //      From step 1, we have
    81  //         expm1(x) = either 2**k*[expm1(r)+1] - 1
    82  //                  = or     2**k*[expm1(r) + (1-2**-k)]
    83  //   4. Implementation notes:
    84  //      (A). To save one multiplication, we scale the coefficient Qi
    85  //           to Qi*2**i, and replace z by (x**2)/2.
    86  //      (B). To achieve maximum accuracy, we compute expm1(x) by
    87  //        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
    88  //        (ii)  if k=0, return r-E
    89  //        (iii) if k=-1, return 0.5*(r-E)-0.5
    90  //        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
    91  //                     else          return  1.0+2.0*(r-E);
    92  //        (v)   if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
    93  //        (vi)  if k <= 20, return 2**k((1-2**-k)-(E-r)), else
    94  //        (vii) return 2**k(1-((E+2**-k)-r))
    95  //
    96  // Special cases:
    97  //      expm1(INF) is INF, expm1(NaN) is NaN;
    98  //      expm1(-INF) is -1, and
    99  //      for finite argument, only expm1(0)=0 is exact.
   100  //
   101  // Accuracy:
   102  //      according to an error analysis, the error is always less than
   103  //      1 ulp (unit in the last place).
   104  //
   105  // Misc. info.
   106  //      For IEEE double
   107  //          if x >  7.09782712893383973096e+02 then expm1(x) overflow
   108  //
   109  // Constants:
   110  // The hexadecimal values are the intended ones for the following
   111  // constants. The decimal values may be used, provided that the
   112  // compiler will convert from decimal to binary accurately enough
   113  // to produce the hexadecimal values shown.
   114  //
   115  
   116  // Expm1 returns e**x - 1, the base-e exponential of x minus 1.
   117  // It is more accurate than Exp(x) - 1 when x is near zero.
   118  //
   119  // Special cases are:
   120  //	Expm1(+Inf) = +Inf
   121  //	Expm1(-Inf) = -1
   122  //	Expm1(NaN) = NaN
   123  // Very large values overflow to -1 or +Inf.
   124  func Expm1(x float64) float64
   125  
   126  func expm1(x float64) float64 {
   127  	const (
   128  		Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
   129  		Ln2X56     = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
   130  		Ln2HalfX3  = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
   131  		Ln2Half    = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
   132  		Ln2Hi      = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
   133  		Ln2Lo      = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
   134  		InvLn2     = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
   135  		Tiny       = 1.0 / (1 << 54)            // 2**-54 = 0x3c90000000000000
   136  		// scaled coefficients related to expm1
   137  		Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
   138  		Q2 = 1.58730158725481460165e-03  // 0x3F5A01A019FE5585
   139  		Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
   140  		Q4 = 4.00821782732936239552e-06  // 0x3ED0CFCA86E65239
   141  		Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
   142  	)
   143  
   144  	// special cases
   145  	switch {
   146  	case IsInf(x, 1) || IsNaN(x):
   147  		return x
   148  	case IsInf(x, -1):
   149  		return -1
   150  	}
   151  
   152  	absx := x
   153  	sign := false
   154  	if x < 0 {
   155  		absx = -absx
   156  		sign = true
   157  	}
   158  
   159  	// filter out huge argument
   160  	if absx >= Ln2X56 { // if |x| >= 56 * ln2
   161  		if sign {
   162  			return -1 // x < -56*ln2, return -1
   163  		}
   164  		if absx >= Othreshold { // if |x| >= 709.78...
   165  			return Inf(1)
   166  		}
   167  	}
   168  
   169  	// argument reduction
   170  	var c float64
   171  	var k int
   172  	if absx > Ln2Half { // if  |x| > 0.5 * ln2
   173  		var hi, lo float64
   174  		if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
   175  			if !sign {
   176  				hi = x - Ln2Hi
   177  				lo = Ln2Lo
   178  				k = 1
   179  			} else {
   180  				hi = x + Ln2Hi
   181  				lo = -Ln2Lo
   182  				k = -1
   183  			}
   184  		} else {
   185  			if !sign {
   186  				k = int(InvLn2*x + 0.5)
   187  			} else {
   188  				k = int(InvLn2*x - 0.5)
   189  			}
   190  			t := float64(k)
   191  			hi = x - t*Ln2Hi // t * Ln2Hi is exact here
   192  			lo = t * Ln2Lo
   193  		}
   194  		x = hi - lo
   195  		c = (hi - x) - lo
   196  	} else if absx < Tiny { // when |x| < 2**-54, return x
   197  		return x
   198  	} else {
   199  		k = 0
   200  	}
   201  
   202  	// x is now in primary range
   203  	hfx := 0.5 * x
   204  	hxs := x * hfx
   205  	r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
   206  	t := 3 - r1*hfx
   207  	e := hxs * ((r1 - t) / (6.0 - x*t))
   208  	if k == 0 {
   209  		return x - (x*e - hxs) // c is 0
   210  	}
   211  	e = (x*(e-c) - c)
   212  	e -= hxs
   213  	switch {
   214  	case k == -1:
   215  		return 0.5*(x-e) - 0.5
   216  	case k == 1:
   217  		if x < -0.25 {
   218  			return -2 * (e - (x + 0.5))
   219  		}
   220  		return 1 + 2*(x-e)
   221  	case k <= -2 || k > 56: // suffice to return exp(x)-1
   222  		y := 1 - (e - x)
   223  		y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
   224  		return y - 1
   225  	}
   226  	if k < 20 {
   227  		t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
   228  		y := t - (e - x)
   229  		y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
   230  		return y
   231  	}
   232  	t = Float64frombits(uint64(0x3ff-k) << 52) // 2**-k
   233  	y := x - (e + t)
   234  	y++
   235  	y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
   236  	return y
   237  }
   238  

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