Copyright 2010 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 error function and complementary error function.
The original C code and the long comment below are from FreeBSD's /usr/src/lib/msun/src/s_erf.c and came with this notice. The go code is a simplified 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. ==================================================== double erf(double x) double erfc(double x) x 2 |\ erf(x) = --------- | exp(-t*t)dt sqrt(pi) \| 0 erfc(x) = 1-erf(x) Note that erf(-x) = -erf(x) erfc(-x) = 2 - erfc(x) Method: 1. For |x| in [0, 0.84375] erf(x) = x + x*R(x**2) erfc(x) = 1 - erf(x) if x in [-.84375,0.25] = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] where R = P/Q where P is an odd poly of degree 8 and Q is an odd poly of degree 10. -57.90 | R - (erf(x)-x)/x | <= 2 Remark. The formula is derived by noting erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....) and that 2/sqrt(pi) = 1.128379167095512573896158903121545171688 is close to one. The interval is chosen because the fix point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is near 0.6174), and by some experiment, 0.84375 is chosen to guarantee the error is less than one ulp for erf. 2. For |x| in [0.84375,1.25], let s = |x| - 1, and c = 0.84506291151 rounded to single (24 bits) erf(x) = sign(x) * (c + P1(s)/Q1(s)) erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 1+(c+P1(s)/Q1(s)) if x < 0 |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 Remark: here we use the taylor series expansion at x=1. erf(1+s) = erf(1) + s*Poly(s) = 0.845.. + P1(s)/Q1(s) That is, we use rational approximation to approximate erf(1+s) - (c = (single)0.84506291151) Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] where P1(s) = degree 6 poly in s Q1(s) = degree 6 poly in s 3. For x in [1.25,1/0.35(~2.857143)], erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) erf(x) = 1 - erfc(x) where R1(z) = degree 7 poly in z, (z=1/x**2) S1(z) = degree 8 poly in z 4. For x in [1/0.35,28] erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 = 2.0 - tiny (if x <= -6) erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else erf(x) = sign(x)*(1.0 - tiny) where R2(z) = degree 6 poly in z, (z=1/x**2) S2(z) = degree 7 poly in z Note1: To compute exp(-x*x-0.5625+R/S), let s be a single precision number and s := x; then -x*x = -s*s + (s-x)*(s+x) exp(-x*x-0.5626+R/S) = exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); Note2: Here 4 and 5 make use of the asymptotic series exp(-x*x) erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) ) x*sqrt(pi) We use rational approximation to approximate g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625 Here is the error bound for R1/S1 and R2/S2 |R1/S1 - f(x)| < 2**(-62.57) |R2/S2 - f(x)| < 2**(-61.52) 5. For inf > x >= 28 erf(x) = sign(x) *(1 - tiny) (raise inexact) erfc(x) = tiny*tiny (raise underflow) if x > 0 = 2 - tiny if x<0 7. Special case: erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, erfc/erf(NaN) is NaN

const (
Coefficients for approximation to erf in [0, 0.84375]
	efx  = 1.28379167095512586316e-01  // 0x3FC06EBA8214DB69
	efx8 = 1.02703333676410069053e+00  // 0x3FF06EBA8214DB69
	pp0  = 1.28379167095512558561e-01  // 0x3FC06EBA8214DB68
	pp1  = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
	pp2  = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
	pp3  = -5.77027029648944159157e-03 // 0xBF77A291236668E4
	pp4  = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
	qq1  = 3.97917223959155352819e-01  // 0x3FD97779CDDADC09
	qq2  = 6.50222499887672944485e-02  // 0x3FB0A54C5536CEBA
	qq3  = 5.08130628187576562776e-03  // 0x3F74D022C4D36B0F
	qq4  = 1.32494738004321644526e-04  // 0x3F215DC9221C1A10
Coefficients for approximation to erf in [0.84375, 1.25]
	pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
	pa1 = 4.14856118683748331666e-01  // 0x3FDA8D00AD92B34D
	pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
	pa3 = 3.18346619901161753674e-01  // 0x3FD45FCA805120E4
	pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
	pa5 = 3.54783043256182359371e-02  // 0x3FA22A36599795EB
	pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
	qa1 = 1.06420880400844228286e-01  // 0x3FBB3E6618EEE323
	qa2 = 5.40397917702171048937e-01  // 0x3FE14AF092EB6F33
	qa3 = 7.18286544141962662868e-02  // 0x3FB2635CD99FE9A7
	qa4 = 1.26171219808761642112e-01  // 0x3FC02660E763351F
	qa5 = 1.36370839120290507362e-02  // 0x3F8BEDC26B51DD1C
Coefficients for approximation to erfc in [1.25, 1/0.35]
	ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
	ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
	ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
	ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
	ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
	ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
	ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
	ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
	sa1 = 1.96512716674392571292e+01  // 0x4033A6B9BD707687
	sa2 = 1.37657754143519042600e+02  // 0x4061350C526AE721
	sa3 = 4.34565877475229228821e+02  // 0x407B290DD58A1A71
	sa4 = 6.45387271733267880336e+02  // 0x40842B1921EC2868
	sa5 = 4.29008140027567833386e+02  // 0x407AD02157700314
	sa6 = 1.08635005541779435134e+02  // 0x405B28A3EE48AE2C
	sa7 = 6.57024977031928170135e+00  // 0x401A47EF8E484A93
Coefficients for approximation to erfc in [1/.35, 28]
	rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
	rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
	rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
	rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
	rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
	rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
	rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
	sb1 = 3.03380607434824582924e+01  // 0x403E568B261D5190
	sb2 = 3.25792512996573918826e+02  // 0x40745CAE221B9F0A
	sb3 = 1.53672958608443695994e+03  // 0x409802EB189D5118
	sb4 = 3.19985821950859553908e+03  // 0x40A8FFB7688C246A
	sb5 = 2.55305040643316442583e+03  // 0x40A3F219CEDF3BE6
	sb6 = 4.74528541206955367215e+02  // 0x407DA874E79FE763
	sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
)
Erf returns the error function of x. Special cases are: Erf(+Inf) = 1 Erf(-Inf) = -1 Erf(NaN) = NaN
func ( float64) float64

func ( float64) float64 {
	const (
		 = 2.848094538889218e-306 // 0x0080000000000000
		    = 1.0 / (1 << 28)        // 2**-28
special cases
	switch {
	case IsNaN():
		return NaN()
	case IsInf(, 1):
		return 1
	case IsInf(, -1):
		return -1
	}
	 := false
	if  < 0 {
		 = -
		 = true
	}
	if  < 0.84375 { // |x| < 0.84375
		var  float64
		if  <  { // |x| < 2**-28
			if  <  {
				 = 0.125 * (8.0* + efx8*) // avoid underflow
			} else {
				 =  + efx*
			}
		} else {
			 :=  * 
			 := pp0 + *(pp1+*(pp2+*(pp3+*pp4)))
			 := 1 + *(qq1+*(qq2+*(qq3+*(qq4+*qq5))))
			 :=  / 
			 =  + *
		}
		if  {
			return -
		}
		return 
	}
	if  < 1.25 { // 0.84375 <= |x| < 1.25
		 :=  - 1
		 := pa0 + *(pa1+*(pa2+*(pa3+*(pa4+*(pa5+*pa6)))))
		 := 1 + *(qa1+*(qa2+*(qa3+*(qa4+*(qa5+*qa6)))))
		if  {
			return -erx - /
		}
		return erx + /
	}
	if  >= 6 { // inf > |x| >= 6
		if  {
			return -1
		}
		return 1
	}
	 := 1 / ( * )
	var ,  float64
	if  < 1/0.35 { // |x| < 1 / 0.35  ~ 2.857143
		 = ra0 + *(ra1+*(ra2+*(ra3+*(ra4+*(ra5+*(ra6+*ra7))))))
		 = 1 + *(sa1+*(sa2+*(sa3+*(sa4+*(sa5+*(sa6+*(sa7+*sa8)))))))
	} else { // |x| >= 1 / 0.35  ~ 2.857143
		 = rb0 + *(rb1+*(rb2+*(rb3+*(rb4+*(rb5+*rb6)))))
		 = 1 + *(sb1+*(sb2+*(sb3+*(sb4+*(sb5+*(sb6+*sb7))))))
	}
	 := Float64frombits(Float64bits() & 0xffffffff00000000) // pseudo-single (20-bit) precision x
	 := Exp(-*-0.5625) * Exp((-)*(+)+/)
	if  {
		return / - 1
	}
	return 1 - /
}
Erfc returns the complementary error function of x. Special cases are: Erfc(+Inf) = 0 Erfc(-Inf) = 2 Erfc(NaN) = NaN
func ( float64) float64

func ( float64) float64 {
special cases
	switch {
	case IsNaN():
		return NaN()
	case IsInf(, 1):
		return 0
	case IsInf(, -1):
		return 2
	}
	 := false
	if  < 0 {
		 = -
		 = true
	}
	if  < 0.84375 { // |x| < 0.84375
		var  float64
		if  <  { // |x| < 2**-56
			 = 
		} else {
			 :=  * 
			 := pp0 + *(pp1+*(pp2+*(pp3+*pp4)))
			 := 1 + *(qq1+*(qq2+*(qq3+*(qq4+*qq5))))
			 :=  / 
			if  < 0.25 { // |x| < 1/4
				 =  + *
			} else {
				 = 0.5 + (* + ( - 0.5))
			}
		}
		if  {
			return 1 + 
		}
		return 1 - 
	}
	if  < 1.25 { // 0.84375 <= |x| < 1.25
		 :=  - 1
		 := pa0 + *(pa1+*(pa2+*(pa3+*(pa4+*(pa5+*pa6)))))
		 := 1 + *(qa1+*(qa2+*(qa3+*(qa4+*(qa5+*qa6)))))
		if  {
			return 1 + erx + /
		}
		return 1 - erx - /

	}
	if  < 28 { // |x| < 28
		 := 1 / ( * )
		var ,  float64
		if  < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
			 = ra0 + *(ra1+*(ra2+*(ra3+*(ra4+*(ra5+*(ra6+*ra7))))))
			 = 1 + *(sa1+*(sa2+*(sa3+*(sa4+*(sa5+*(sa6+*(sa7+*sa8)))))))
		} else { // |x| >= 1 / 0.35 ~ 2.857143
			if  &&  > 6 {
				return 2 // x < -6
			}
			 = rb0 + *(rb1+*(rb2+*(rb3+*(rb4+*(rb5+*rb6)))))
			 = 1 + *(sb1+*(sb2+*(sb3+*(sb4+*(sb5+*(sb6+*sb7))))))
		}
		 := Float64frombits(Float64bits() & 0xffffffff00000000) // pseudo-single (20-bit) precision x
		 := Exp(-*-0.5625) * Exp((-)*(+)+/)
		if  {
			return 2 - /
		}
		return  / 
	}
	if  {
		return 2
	}
	return 0