# Source file src/math/erf.go

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  // ====================================================
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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