erf.go

     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  //
   186  //	Erf(+Inf) = 1
   187  //	Erf(-Inf) = -1
   188  //	Erf(NaN) = NaN
   189  func Erf(x float64) float64 {
   190  	if haveArchErf {
   191  		return archErf(x)
   192  	}
   193  	return erf(x)
   194  }
   195  
   196  func erf(x float64) float64 {
   197  	const (
   198  		VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
   199  		Small    = 1.0 / (1 << 28)        // 2**-28
   200  	)
   201  	// special cases
   202  	switch {
   203  	case IsNaN(x):
   204  		return NaN()
   205  	case IsInf(x, 1):
   206  		return 1
   207  	case IsInf(x, -1):
   208  		return -1
   209  	}
   210  	sign := false
   211  	if x < 0 {
   212  		x = -x
   213  		sign = true
   214  	}
   215  	if x < 0.84375 { // |x| < 0.84375
   216  		var temp float64
   217  		if x < Small { // |x| < 2**-28
   218  			if x < VeryTiny {
   219  				temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
   220  			} else {
   221  				temp = x + efx*x
   222  			}
   223  		} else {
   224  			z := x * x
   225  			r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
   226  			s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
   227  			y := r / s
   228  			temp = x + x*y
   229  		}
   230  		if sign {
   231  			return -temp
   232  		}
   233  		return temp
   234  	}
   235  	if x < 1.25 { // 0.84375 <= |x| < 1.25
   236  		s := x - 1
   237  		P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
   238  		Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
   239  		if sign {
   240  			return -erx - P/Q
   241  		}
   242  		return erx + P/Q
   243  	}
   244  	if x >= 6 { // inf > |x| >= 6
   245  		if sign {
   246  			return -1
   247  		}
   248  		return 1
   249  	}
   250  	s := 1 / (x * x)
   251  	var R, S float64
   252  	if x < 1/0.35 { // |x| < 1 / 0.35  ~ 2.857143
   253  		R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
   254  		S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
   255  	} else { // |x| >= 1 / 0.35  ~ 2.857143
   256  		R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
   257  		S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
   258  	}
   259  	z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
   260  	r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
   261  	if sign {
   262  		return r/x - 1
   263  	}
   264  	return 1 - r/x
   265  }
   266  
   267  // Erfc returns the complementary error function of x.
   268  //
   269  // Special cases are:
   270  //
   271  //	Erfc(+Inf) = 0
   272  //	Erfc(-Inf) = 2
   273  //	Erfc(NaN) = NaN
   274  func Erfc(x float64) float64 {
   275  	if haveArchErfc {
   276  		return archErfc(x)
   277  	}
   278  	return erfc(x)
   279  }
   280  
   281  func erfc(x float64) float64 {
   282  	const Tiny = 1.0 / (1 << 56) // 2**-56
   283  	// special cases
   284  	switch {
   285  	case IsNaN(x):
   286  		return NaN()
   287  	case IsInf(x, 1):
   288  		return 0
   289  	case IsInf(x, -1):
   290  		return 2
   291  	}
   292  	sign := false
   293  	if x < 0 {
   294  		x = -x
   295  		sign = true
   296  	}
   297  	if x < 0.84375 { // |x| < 0.84375
   298  		var temp float64
   299  		if x < Tiny { // |x| < 2**-56
   300  			temp = x
   301  		} else {
   302  			z := x * x
   303  			r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
   304  			s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
   305  			y := r / s
   306  			if x < 0.25 { // |x| < 1/4
   307  				temp = x + x*y
   308  			} else {
   309  				temp = 0.5 + (x*y + (x - 0.5))
   310  			}
   311  		}
   312  		if sign {
   313  			return 1 + temp
   314  		}
   315  		return 1 - temp
   316  	}
   317  	if x < 1.25 { // 0.84375 <= |x| < 1.25
   318  		s := x - 1
   319  		P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
   320  		Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
   321  		if sign {
   322  			return 1 + erx + P/Q
   323  		}
   324  		return 1 - erx - P/Q
   325  
   326  	}
   327  	if x < 28 { // |x| < 28
   328  		s := 1 / (x * x)
   329  		var R, S float64
   330  		if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
   331  			R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
   332  			S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
   333  		} else { // |x| >= 1 / 0.35 ~ 2.857143
   334  			if sign && x > 6 {
   335  				return 2 // x < -6
   336  			}
   337  			R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
   338  			S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
   339  		}
   340  		z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
   341  		r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
   342  		if sign {
   343  			return 2 - r/x
   344  		}
   345  		return r / x
   346  	}
   347  	if sign {
   348  		return 2
   349  	}
   350  	return 0
   351  }
   352
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