erf.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  /*
		 8  	Floating-point error function and complementary error function.
		 9  */
		10  
		11  // The original C code and the long comment below are
		12  // from FreeBSD's /usr/src/lib/msun/src/s_erf.c and
		13  // came with this notice. The go code is a simplified
		14  // version of the original C.
		15  //
		16  // ====================================================
		17  // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
		18  //
		19  // Developed at SunPro, a Sun Microsystems, Inc. business.
		20  // Permission to use, copy, modify, and distribute this
		21  // software is freely granted, provided that this notice
		22  // is preserved.
		23  // ====================================================
		24  //
		25  //
		26  // double erf(double x)
		27  // double erfc(double x)
		28  //													 x
		29  //										2			|\
		30  //		 erf(x)	=	---------	| exp(-t*t)dt
		31  //								 sqrt(pi) \|
		32  //													 0
		33  //
		34  //		 erfc(x) =	1-erf(x)
		35  //	Note that
		36  //							erf(-x) = -erf(x)
		37  //							erfc(-x) = 2 - erfc(x)
		38  //
		39  // Method:
		40  //			1. For |x| in [0, 0.84375]
		41  //					erf(x)	= x + x*R(x**2)
		42  //					erfc(x) = 1 - erf(x)					 if x in [-.84375,0.25]
		43  //									= 0.5 + ((0.5-x)-x*R)	if x in [0.25,0.84375]
		44  //				 where R = P/Q where P is an odd poly of degree 8 and
		45  //				 Q is an odd poly of degree 10.
		46  //																							 -57.90
		47  //											| R - (erf(x)-x)/x | <= 2
		48  //
		49  //
		50  //				 Remark. The formula is derived by noting
		51  //					erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
		52  //				 and that
		53  //					2/sqrt(pi) = 1.128379167095512573896158903121545171688
		54  //				 is close to one. The interval is chosen because the fix
		55  //				 point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
		56  //				 near 0.6174), and by some experiment, 0.84375 is chosen to
		57  //				 guarantee the error is less than one ulp for erf.
		58  //
		59  //			2. For |x| in [0.84375,1.25], let s = |x| - 1, and
		60  //				 c = 0.84506291151 rounded to single (24 bits)
		61  //							erf(x)	= sign(x) * (c	+ P1(s)/Q1(s))
		62  //							erfc(x) = (1-c)	- P1(s)/Q1(s) if x > 0
		63  //												1+(c+P1(s)/Q1(s))		if x < 0
		64  //							|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
		65  //				 Remark: here we use the taylor series expansion at x=1.
		66  //							erf(1+s) = erf(1) + s*Poly(s)
		67  //											 = 0.845.. + P1(s)/Q1(s)
		68  //				 That is, we use rational approximation to approximate
		69  //											erf(1+s) - (c = (single)0.84506291151)
		70  //				 Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
		71  //				 where
		72  //							P1(s) = degree 6 poly in s
		73  //							Q1(s) = degree 6 poly in s
		74  //
		75  //			3. For x in [1.25,1/0.35(~2.857143)],
		76  //							erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
		77  //							erf(x)	= 1 - erfc(x)
		78  //				 where
		79  //							R1(z) = degree 7 poly in z, (z=1/x**2)
		80  //							S1(z) = degree 8 poly in z
		81  //
		82  //			4. For x in [1/0.35,28]
		83  //							erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
		84  //											= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
		85  //											= 2.0 - tiny						(if x <= -6)
		86  //							erf(x)	= sign(x)*(1.0 - erfc(x)) if x < 6, else
		87  //							erf(x)	= sign(x)*(1.0 - tiny)
		88  //				 where
		89  //							R2(z) = degree 6 poly in z, (z=1/x**2)
		90  //							S2(z) = degree 7 poly in z
		91  //
		92  //			Note1:
		93  //				 To compute exp(-x*x-0.5625+R/S), let s be a single
		94  //				 precision number and s := x; then
		95  //							-x*x = -s*s + (s-x)*(s+x)
		96  //							exp(-x*x-0.5626+R/S) =
		97  //											exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
		98  //			Note2:
		99  //				 Here 4 and 5 make use of the asymptotic series
	 100  //												exp(-x*x)
	 101  //							erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
	 102  //												x*sqrt(pi)
	 103  //				 We use rational approximation to approximate
	 104  //							g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
	 105  //				 Here is the error bound for R1/S1 and R2/S2
	 106  //							|R1/S1 - f(x)|	< 2**(-62.57)
	 107  //							|R2/S2 - f(x)|	< 2**(-61.52)
	 108  //
	 109  //			5. For inf > x >= 28
	 110  //							erf(x)	= sign(x) *(1 - tiny)	(raise inexact)
	 111  //							erfc(x) = tiny*tiny (raise underflow) if x > 0
	 112  //											= 2 - tiny if x<0
	 113  //
	 114  //			7. Special case:
	 115  //							erf(0)	= 0, erf(inf)	= 1, erf(-inf) = -1,
	 116  //							erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
	 117  //							erfc/erf(NaN) is NaN
	 118  
	 119  const (
	 120  	erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
	 121  	// Coefficients for approximation to	erf in [0, 0.84375]
	 122  	efx	= 1.28379167095512586316e-01	// 0x3FC06EBA8214DB69
	 123  	efx8 = 1.02703333676410069053e+00	// 0x3FF06EBA8214DB69
	 124  	pp0	= 1.28379167095512558561e-01	// 0x3FC06EBA8214DB68
	 125  	pp1	= -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
	 126  	pp2	= -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
	 127  	pp3	= -5.77027029648944159157e-03 // 0xBF77A291236668E4
	 128  	pp4	= -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
	 129  	qq1	= 3.97917223959155352819e-01	// 0x3FD97779CDDADC09
	 130  	qq2	= 6.50222499887672944485e-02	// 0x3FB0A54C5536CEBA
	 131  	qq3	= 5.08130628187576562776e-03	// 0x3F74D022C4D36B0F
	 132  	qq4	= 1.32494738004321644526e-04	// 0x3F215DC9221C1A10
	 133  	qq5	= -3.96022827877536812320e-06 // 0xBED09C4342A26120
	 134  	// Coefficients for approximation to	erf	in [0.84375, 1.25]
	 135  	pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
	 136  	pa1 = 4.14856118683748331666e-01	// 0x3FDA8D00AD92B34D
	 137  	pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
	 138  	pa3 = 3.18346619901161753674e-01	// 0x3FD45FCA805120E4
	 139  	pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
	 140  	pa5 = 3.54783043256182359371e-02	// 0x3FA22A36599795EB
	 141  	pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
	 142  	qa1 = 1.06420880400844228286e-01	// 0x3FBB3E6618EEE323
	 143  	qa2 = 5.40397917702171048937e-01	// 0x3FE14AF092EB6F33
	 144  	qa3 = 7.18286544141962662868e-02	// 0x3FB2635CD99FE9A7
	 145  	qa4 = 1.26171219808761642112e-01	// 0x3FC02660E763351F
	 146  	qa5 = 1.36370839120290507362e-02	// 0x3F8BEDC26B51DD1C
	 147  	qa6 = 1.19844998467991074170e-02	// 0x3F888B545735151D
	 148  	// Coefficients for approximation to	erfc in [1.25, 1/0.35]
	 149  	ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
	 150  	ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
	 151  	ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
	 152  	ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
	 153  	ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
	 154  	ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
	 155  	ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
	 156  	ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
	 157  	sa1 = 1.96512716674392571292e+01	// 0x4033A6B9BD707687
	 158  	sa2 = 1.37657754143519042600e+02	// 0x4061350C526AE721
	 159  	sa3 = 4.34565877475229228821e+02	// 0x407B290DD58A1A71
	 160  	sa4 = 6.45387271733267880336e+02	// 0x40842B1921EC2868
	 161  	sa5 = 4.29008140027567833386e+02	// 0x407AD02157700314
	 162  	sa6 = 1.08635005541779435134e+02	// 0x405B28A3EE48AE2C
	 163  	sa7 = 6.57024977031928170135e+00	// 0x401A47EF8E484A93
	 164  	sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
	 165  	// Coefficients for approximation to	erfc in [1/.35, 28]
	 166  	rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
	 167  	rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
	 168  	rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
	 169  	rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
	 170  	rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
	 171  	rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
	 172  	rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
	 173  	sb1 = 3.03380607434824582924e+01	// 0x403E568B261D5190
	 174  	sb2 = 3.25792512996573918826e+02	// 0x40745CAE221B9F0A
	 175  	sb3 = 1.53672958608443695994e+03	// 0x409802EB189D5118
	 176  	sb4 = 3.19985821950859553908e+03	// 0x40A8FFB7688C246A
	 177  	sb5 = 2.55305040643316442583e+03	// 0x40A3F219CEDF3BE6
	 178  	sb6 = 4.74528541206955367215e+02	// 0x407DA874E79FE763
	 179  	sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
	 180  )
	 181  
	 182  // Erf returns the error function of x.
	 183  //
	 184  // Special cases are:
	 185  //	Erf(+Inf) = 1
	 186  //	Erf(-Inf) = -1
	 187  //	Erf(NaN) = NaN
	 188  func Erf(x float64) float64 {
	 189  	if haveArchErf {
	 190  		return archErf(x)
	 191  	}
	 192  	return erf(x)
	 193  }
	 194  
	 195  func erf(x float64) float64 {
	 196  	const (
	 197  		VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
	 198  		Small		= 1.0 / (1 << 28)				// 2**-28
	 199  	)
	 200  	// special cases
	 201  	switch {
	 202  	case IsNaN(x):
	 203  		return NaN()
	 204  	case IsInf(x, 1):
	 205  		return 1
	 206  	case IsInf(x, -1):
	 207  		return -1
	 208  	}
	 209  	sign := false
	 210  	if x < 0 {
	 211  		x = -x
	 212  		sign = true
	 213  	}
	 214  	if x < 0.84375 { // |x| < 0.84375
	 215  		var temp float64
	 216  		if x < Small { // |x| < 2**-28
	 217  			if x < VeryTiny {
	 218  				temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
	 219  			} else {
	 220  				temp = x + efx*x
	 221  			}
	 222  		} else {
	 223  			z := x * x
	 224  			r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
	 225  			s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
	 226  			y := r / s
	 227  			temp = x + x*y
	 228  		}
	 229  		if sign {
	 230  			return -temp
	 231  		}
	 232  		return temp
	 233  	}
	 234  	if x < 1.25 { // 0.84375 <= |x| < 1.25
	 235  		s := x - 1
	 236  		P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
	 237  		Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
	 238  		if sign {
	 239  			return -erx - P/Q
	 240  		}
	 241  		return erx + P/Q
	 242  	}
	 243  	if x >= 6 { // inf > |x| >= 6
	 244  		if sign {
	 245  			return -1
	 246  		}
	 247  		return 1
	 248  	}
	 249  	s := 1 / (x * x)
	 250  	var R, S float64
	 251  	if x < 1/0.35 { // |x| < 1 / 0.35	~ 2.857143
	 252  		R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
	 253  		S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
	 254  	} else { // |x| >= 1 / 0.35	~ 2.857143
	 255  		R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
	 256  		S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
	 257  	}
	 258  	z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
	 259  	r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
	 260  	if sign {
	 261  		return r/x - 1
	 262  	}
	 263  	return 1 - r/x
	 264  }
	 265  
	 266  // Erfc returns the complementary error function of x.
	 267  //
	 268  // Special cases are:
	 269  //	Erfc(+Inf) = 0
	 270  //	Erfc(-Inf) = 2
	 271  //	Erfc(NaN) = NaN
	 272  func Erfc(x float64) float64 {
	 273  	if haveArchErfc {
	 274  		return archErfc(x)
	 275  	}
	 276  	return erfc(x)
	 277  }
	 278  
	 279  func erfc(x float64) float64 {
	 280  	const Tiny = 1.0 / (1 << 56) // 2**-56
	 281  	// special cases
	 282  	switch {
	 283  	case IsNaN(x):
	 284  		return NaN()
	 285  	case IsInf(x, 1):
	 286  		return 0
	 287  	case IsInf(x, -1):
	 288  		return 2
	 289  	}
	 290  	sign := false
	 291  	if x < 0 {
	 292  		x = -x
	 293  		sign = true
	 294  	}
	 295  	if x < 0.84375 { // |x| < 0.84375
	 296  		var temp float64
	 297  		if x < Tiny { // |x| < 2**-56
	 298  			temp = x
	 299  		} else {
	 300  			z := x * x
	 301  			r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
	 302  			s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
	 303  			y := r / s
	 304  			if x < 0.25 { // |x| < 1/4
	 305  				temp = x + x*y
	 306  			} else {
	 307  				temp = 0.5 + (x*y + (x - 0.5))
	 308  			}
	 309  		}
	 310  		if sign {
	 311  			return 1 + temp
	 312  		}
	 313  		return 1 - temp
	 314  	}
	 315  	if x < 1.25 { // 0.84375 <= |x| < 1.25
	 316  		s := x - 1
	 317  		P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
	 318  		Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
	 319  		if sign {
	 320  			return 1 + erx + P/Q
	 321  		}
	 322  		return 1 - erx - P/Q
	 323  
	 324  	}
	 325  	if x < 28 { // |x| < 28
	 326  		s := 1 / (x * x)
	 327  		var R, S float64
	 328  		if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
	 329  			R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
	 330  			S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
	 331  		} else { // |x| >= 1 / 0.35 ~ 2.857143
	 332  			if sign && x > 6 {
	 333  				return 2 // x < -6
	 334  			}
	 335  			R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
	 336  			S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
	 337  		}
	 338  		z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
	 339  		r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
	 340  		if sign {
	 341  			return 2 - r/x
	 342  		}
	 343  		return r / x
	 344  	}
	 345  	if sign {
	 346  		return 2
	 347  	}
	 348  	return 0
	 349  }
	 350
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