328 lines
12 KiB
V
328 lines
12 KiB
V
module math
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/*
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x**2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s_ = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s_)/Q1(s_))
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* erfc(x) = (1-c) - P1(s_)/Q1(s_) if x > 0
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* 1+(c+P1(s_)/Q1(s_)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s_) = erf(1) + s_*Poly(s_)
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* = 0.845.. + P1(s_)/Q1(s_)
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* That is, we use rational approximation to approximate
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* erf(1+s_) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s_) = degree 6 poly in s_
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* Q1(s_) = degree 6 poly in s_
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*
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* 3. For x in [1.25,1/0.35(~2.857143)],
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/s1)
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* erf(x) = 1 - erfc(x)
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* where
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* R1(z) = degree 7 poly in z, (z=1/x**2)
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* s1(z) = degree 8 poly in z
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*
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* 4. For x in [1/0.35,28]
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/s2) if x > 0
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* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/s2) if -6<x<0
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* = 2.0 - tiny (if x <= -6)
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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* erf(x) = sign(x)*(1.0 - tiny)
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* where
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* R2(z) = degree 6 poly in z, (z=1/x**2)
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* s2(z) = degree 7 poly in z
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*
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* Note1:
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* To compute exp(-x*x-0.5625+R/s), let s_ be a single
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* precision number and s_ := x; then
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* -x*x = -s_*s_ + (s_-x)*(s_+x)
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* exp(-x*x-0.5626+R/s) =
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* exp(-s_*s_-0.5625)*exp((s_-x)*(s_+x)+R/s);
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* Note2:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
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* x*sqrt(pi)
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* We use rational approximation to approximate
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* g(s_)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
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* Here is the error bound for R1/s1 and R2/s2
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* |R1/s1 - f(x)| < 2**(-62.57)
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* |R2/s2 - f(x)| < 2**(-61.52)
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*
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* 5. For inf > x >= 28
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* erf(x) = sign(x) *(1 - tiny) (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow) if x > 0
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* = 2 - tiny if x<0
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*
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* 7. special case:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(nan) is nan
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*/
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const (
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erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
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// Coefficients for approximation to erf in [0, 0.84375]
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efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69
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efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69
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pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68
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pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
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pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
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pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4
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pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
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qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09
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qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA
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qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F
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qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10
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qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120
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// Coefficients for approximation to erf in [0.84375, 1.25]
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pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
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pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D
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pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
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pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4
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pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
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pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB
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pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
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qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323
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qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33
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qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7
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qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F
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qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C
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qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D
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// Coefficients for approximation to erfc in [1.25, 1/0.35]
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ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
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ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
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ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
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ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
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ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
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ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
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ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
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ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
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sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687
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sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721
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sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71
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sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868
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sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314
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sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C
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sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93
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sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
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// Coefficients for approximation to erfc in [1/.35, 28]
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rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
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rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
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rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
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rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
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rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
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rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
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rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
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sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190
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sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A
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sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118
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sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A
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sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6
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sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763
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sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
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)
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// erf returns the error function of x.
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//
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// special cases are:
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// erf(+inf) = 1
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// erf(-inf) = -1
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// erf(nan) = nan
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pub fn erf(a f64) f64 {
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mut x := a
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very_tiny := 2.848094538889218e-306 // 0x0080000000000000
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small_ := 1.0 / f64(u64(1) << 28) // 2**-28
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if is_nan(x) {
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return nan()
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}
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if is_inf(x, 1) {
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return 1.0
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}
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if is_inf(x, -1) {
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return f64(-1)
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}
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mut sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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if x < 0.84375 { // |x| < 0.84375
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mut temp := 0.0
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if x < small_ { // |x| < 2**-28
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if x < very_tiny {
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temp = 0.125 * (8.0 * x + math.efx8 * x) // avoid underflow
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} else {
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temp = x + math.efx * x
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}
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} else {
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z := x * x
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r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4)))
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s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 +
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z * math.qq5))))
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y := r / s_
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temp = x + x * y
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}
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if sign {
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return -temp
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}
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return temp
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}
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if x < 1.25 { // 0.84375 <= |x| < 1.25
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s_ := x - 1
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p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 +
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s_ * (math.pa5 + s_ * math.pa6)))))
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q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 +
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s_ * (math.qa5 + s_ * math.qa6)))))
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if sign {
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return -math.erx - p / q
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}
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return math.erx + p / q
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}
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if x >= 6 { // inf > |x| >= 6
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if sign {
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return -1
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}
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return 1.0
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}
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s_ := 1.0 / (x * x)
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mut r := 0.0
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mut s := 0.0
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if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143
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r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 +
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s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7))))))
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s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 +
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s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8)))))))
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} else { // |x| >= 1 / 0.35 ~ 2.857143
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r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 +
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s_ * (math.rb5 + s_ * math.rb6)))))
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s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 +
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s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7))))))
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}
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z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
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r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s)
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if sign {
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return r_ / x - 1.0
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}
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return 1.0 - r_ / x
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}
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// erfc returns the complementary error function of x.
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//
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// special cases are:
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// erfc(+inf) = 0
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// erfc(-inf) = 2
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// erfc(nan) = nan
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pub fn erfc(a f64) f64 {
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mut x := a
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tiny := 1.0 / f64(u64(1) << 56) // 2**-56
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// special cases
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if is_nan(x) {
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return nan()
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}
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if is_inf(x, 1) {
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return 0.0
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}
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if is_inf(x, -1) {
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return 2.0
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}
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mut sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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if x < 0.84375 { // |x| < 0.84375
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mut temp := 0.0
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if x < tiny { // |x| < 2**-56
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temp = x
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} else {
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z := x * x
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r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4)))
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s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 +
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z * math.qq5))))
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y := r / s_
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if x < 0.25 { // |x| < 1.0/4
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temp = x + x * y
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} else {
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temp = 0.5 + (x * y + (x - 0.5))
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}
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}
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if sign {
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return 1.0 + temp
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}
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return 1.0 - temp
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}
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if x < 1.25 { // 0.84375 <= |x| < 1.25
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s_ := x - 1
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p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 +
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s_ * (math.pa5 + s_ * math.pa6)))))
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q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 +
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s_ * (math.qa5 + s_ * math.qa6)))))
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if sign {
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return 1.0 + math.erx + p / q
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}
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return 1.0 - math.erx - p / q
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}
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if x < 28 { // |x| < 28
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s_ := 1.0 / (x * x)
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mut r := 0.0
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mut s := 0.0
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if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143
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r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 +
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s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7))))))
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s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 +
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s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8)))))))
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} else { // |x| >= 1 / 0.35 ~ 2.857143
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if sign && x > 6 {
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return 2.0 // x < -6
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}
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r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 +
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s_ * (math.rb5 + s_ * math.rb6)))))
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s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 +
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s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7))))))
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}
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z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
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r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s)
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if sign {
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return 2.0 - r_ / x
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}
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return r_ / x
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}
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if sign {
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return 2.0
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}
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return 0.0
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}
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