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

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