erfinv.go

     1  // Copyright 2017 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  	Inverse of the floating-point error function.
     9  */
    10  
    11  // This implementation is based on the rational approximation
    12  // of percentage points of normal distribution available from
    13  // https://www.jstor.org/stable/2347330.
    14  
    15  const (
    16  	// Coefficients for approximation to erf in |x| <= 0.85
    17  	a0 = 1.1975323115670912564578e0
    18  	a1 = 4.7072688112383978012285e1
    19  	a2 = 6.9706266534389598238465e2
    20  	a3 = 4.8548868893843886794648e3
    21  	a4 = 1.6235862515167575384252e4
    22  	a5 = 2.3782041382114385731252e4
    23  	a6 = 1.1819493347062294404278e4
    24  	a7 = 8.8709406962545514830200e2
    25  	b0 = 1.0000000000000000000e0
    26  	b1 = 4.2313330701600911252e1
    27  	b2 = 6.8718700749205790830e2
    28  	b3 = 5.3941960214247511077e3
    29  	b4 = 2.1213794301586595867e4
    30  	b5 = 3.9307895800092710610e4
    31  	b6 = 2.8729085735721942674e4
    32  	b7 = 5.2264952788528545610e3
    33  	// Coefficients for approximation to erf in 0.85 < |x| <= 1-2*exp(-25)
    34  	c0 = 1.42343711074968357734e0
    35  	c1 = 4.63033784615654529590e0
    36  	c2 = 5.76949722146069140550e0
    37  	c3 = 3.64784832476320460504e0
    38  	c4 = 1.27045825245236838258e0
    39  	c5 = 2.41780725177450611770e-1
    40  	c6 = 2.27238449892691845833e-2
    41  	c7 = 7.74545014278341407640e-4
    42  	d0 = 1.4142135623730950488016887e0
    43  	d1 = 2.9036514445419946173133295e0
    44  	d2 = 2.3707661626024532365971225e0
    45  	d3 = 9.7547832001787427186894837e-1
    46  	d4 = 2.0945065210512749128288442e-1
    47  	d5 = 2.1494160384252876777097297e-2
    48  	d6 = 7.7441459065157709165577218e-4
    49  	d7 = 1.4859850019840355905497876e-9
    50  	// Coefficients for approximation to erf in 1-2*exp(-25) < |x| < 1
    51  	e0 = 6.65790464350110377720e0
    52  	e1 = 5.46378491116411436990e0
    53  	e2 = 1.78482653991729133580e0
    54  	e3 = 2.96560571828504891230e-1
    55  	e4 = 2.65321895265761230930e-2
    56  	e5 = 1.24266094738807843860e-3
    57  	e6 = 2.71155556874348757815e-5
    58  	e7 = 2.01033439929228813265e-7
    59  	f0 = 1.414213562373095048801689e0
    60  	f1 = 8.482908416595164588112026e-1
    61  	f2 = 1.936480946950659106176712e-1
    62  	f3 = 2.103693768272068968719679e-2
    63  	f4 = 1.112800997078859844711555e-3
    64  	f5 = 2.611088405080593625138020e-5
    65  	f6 = 2.010321207683943062279931e-7
    66  	f7 = 2.891024605872965461538222e-15
    67  )
    68  
    69  // Erfinv returns the inverse error function of x.
    70  //
    71  // Special cases are:
    72  //
    73  //	Erfinv(1) = +Inf
    74  //	Erfinv(-1) = -Inf
    75  //	Erfinv(x) = NaN if x < -1 or x > 1
    76  //	Erfinv(NaN) = NaN
    77  func Erfinv(x float64) float64 {
    78  	// special cases
    79  	if IsNaN(x) || x <= -1 || x >= 1 {
    80  		if x == -1 || x == 1 {
    81  			return Inf(int(x))
    82  		}
    83  		return NaN()
    84  	}
    85  
    86  	sign := false
    87  	if x < 0 {
    88  		x = -x
    89  		sign = true
    90  	}
    91  
    92  	var ans float64
    93  	if x <= 0.85 { // |x| <= 0.85
    94  		r := 0.180625 - 0.25*x*x
    95  		z1 := ((((((a7*r+a6)*r+a5)*r+a4)*r+a3)*r+a2)*r+a1)*r + a0
    96  		z2 := ((((((b7*r+b6)*r+b5)*r+b4)*r+b3)*r+b2)*r+b1)*r + b0
    97  		ans = (x * z1) / z2
    98  	} else {
    99  		var z1, z2 float64
   100  		r := Sqrt(Ln2 - Log(1.0-x))
   101  		if r <= 5.0 {
   102  			r -= 1.6
   103  			z1 = ((((((c7*r+c6)*r+c5)*r+c4)*r+c3)*r+c2)*r+c1)*r + c0
   104  			z2 = ((((((d7*r+d6)*r+d5)*r+d4)*r+d3)*r+d2)*r+d1)*r + d0
   105  		} else {
   106  			r -= 5.0
   107  			z1 = ((((((e7*r+e6)*r+e5)*r+e4)*r+e3)*r+e2)*r+e1)*r + e0
   108  			z2 = ((((((f7*r+f6)*r+f5)*r+f4)*r+f3)*r+f2)*r+f1)*r + f0
   109  		}
   110  		ans = z1 / z2
   111  	}
   112  
   113  	if sign {
   114  		return -ans
   115  	}
   116  	return ans
   117  }
   118  
   119  // Erfcinv returns the inverse of [Erfc](x).
   120  //
   121  // Special cases are:
   122  //
   123  //	Erfcinv(0) = +Inf
   124  //	Erfcinv(2) = -Inf
   125  //	Erfcinv(x) = NaN if x < 0 or x > 2
   126  //	Erfcinv(NaN) = NaN
   127  func Erfcinv(x float64) float64 {
   128  	return Erfinv(1 - x)
   129  }
   130
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