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Source file src/math/erfinv.go

Documentation: math

  // Copyright 2017 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
  
  /*
  	Inverse of the floating-point error function.
  */
  
  // This implementation is based on the rational approximation
  // of percentage points of normal distribution available from
  // https://www.jstor.org/stable/2347330.
  
  const (
  	// Coefficients for approximation to erf in |x| <= 0.85
  	a0 = 1.1975323115670912564578e0
  	a1 = 4.7072688112383978012285e1
  	a2 = 6.9706266534389598238465e2
  	a3 = 4.8548868893843886794648e3
  	a4 = 1.6235862515167575384252e4
  	a5 = 2.3782041382114385731252e4
  	a6 = 1.1819493347062294404278e4
  	a7 = 8.8709406962545514830200e2
  	b0 = 1.0000000000000000000e0
  	b1 = 4.2313330701600911252e1
  	b2 = 6.8718700749205790830e2
  	b3 = 5.3941960214247511077e3
  	b4 = 2.1213794301586595867e4
  	b5 = 3.9307895800092710610e4
  	b6 = 2.8729085735721942674e4
  	b7 = 5.2264952788528545610e3
  	// Coefficients for approximation to erf in 0.85 < |x| <= 1-2*exp(-25)
  	c0 = 1.42343711074968357734e0
  	c1 = 4.63033784615654529590e0
  	c2 = 5.76949722146069140550e0
  	c3 = 3.64784832476320460504e0
  	c4 = 1.27045825245236838258e0
  	c5 = 2.41780725177450611770e-1
  	c6 = 2.27238449892691845833e-2
  	c7 = 7.74545014278341407640e-4
  	d0 = 1.4142135623730950488016887e0
  	d1 = 2.9036514445419946173133295e0
  	d2 = 2.3707661626024532365971225e0
  	d3 = 9.7547832001787427186894837e-1
  	d4 = 2.0945065210512749128288442e-1
  	d5 = 2.1494160384252876777097297e-2
  	d6 = 7.7441459065157709165577218e-4
  	d7 = 1.4859850019840355905497876e-9
  	// Coefficients for approximation to erf in 1-2*exp(-25) < |x| < 1
  	e0 = 6.65790464350110377720e0
  	e1 = 5.46378491116411436990e0
  	e2 = 1.78482653991729133580e0
  	e3 = 2.96560571828504891230e-1
  	e4 = 2.65321895265761230930e-2
  	e5 = 1.24266094738807843860e-3
  	e6 = 2.71155556874348757815e-5
  	e7 = 2.01033439929228813265e-7
  	f0 = 1.414213562373095048801689e0
  	f1 = 8.482908416595164588112026e-1
  	f2 = 1.936480946950659106176712e-1
  	f3 = 2.103693768272068968719679e-2
  	f4 = 1.112800997078859844711555e-3
  	f5 = 2.611088405080593625138020e-5
  	f6 = 2.010321207683943062279931e-7
  	f7 = 2.891024605872965461538222e-15
  )
  
  // Erfinv returns the inverse error function of x.
  //
  // Special cases are:
  //	Erfinv(1) = +Inf
  //	Erfinv(-1) = -Inf
  //	Erfinv(x) = NaN if x < -1 or x > 1
  //	Erfinv(NaN) = NaN
  func Erfinv(x float64) float64 {
  	// special cases
  	if IsNaN(x) || x <= -1 || x >= 1 {
  		if x == -1 || x == 1 {
  			return Inf(int(x))
  		}
  		return NaN()
  	}
  
  	sign := false
  	if x < 0 {
  		x = -x
  		sign = true
  	}
  
  	var ans float64
  	if x <= 0.85 { // |x| <= 0.85
  		r := 0.180625 - 0.25*x*x
  		z1 := ((((((a7*r+a6)*r+a5)*r+a4)*r+a3)*r+a2)*r+a1)*r + a0
  		z2 := ((((((b7*r+b6)*r+b5)*r+b4)*r+b3)*r+b2)*r+b1)*r + b0
  		ans = (x * z1) / z2
  	} else {
  		var z1, z2 float64
  		r := Sqrt(Ln2 - Log(1.0-x))
  		if r <= 5.0 {
  			r -= 1.6
  			z1 = ((((((c7*r+c6)*r+c5)*r+c4)*r+c3)*r+c2)*r+c1)*r + c0
  			z2 = ((((((d7*r+d6)*r+d5)*r+d4)*r+d3)*r+d2)*r+d1)*r + d0
  		} else {
  			r -= 5.0
  			z1 = ((((((e7*r+e6)*r+e5)*r+e4)*r+e3)*r+e2)*r+e1)*r + e0
  			z2 = ((((((f7*r+f6)*r+f5)*r+f4)*r+f3)*r+f2)*r+f1)*r + f0
  		}
  		ans = z1 / z2
  	}
  
  	if sign {
  		return -ans
  	}
  	return ans
  }
  
  // Erfcinv returns the inverse of Erfc(x).
  //
  // Special cases are:
  //	Erfcinv(0) = +Inf
  //	Erfcinv(2) = -Inf
  //	Erfcinv(x) = NaN if x < 0 or x > 2
  //	Erfcinv(NaN) = NaN
  func Erfcinv(x float64) float64 {
  	return Erfinv(1 - x)
  }
  

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