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Diffstat (limited to 'v_windows/v/vlib/math/erf.v')
| -rw-r--r-- | v_windows/v/vlib/math/erf.v | 327 |
1 files changed, 327 insertions, 0 deletions
diff --git a/v_windows/v/vlib/math/erf.v b/v_windows/v/vlib/math/erf.v new file mode 100644 index 0000000..2375789 --- /dev/null +++ b/v_windows/v/vlib/math/erf.v @@ -0,0 +1,327 @@ +module math + +/* +* x + * 2 |\ + * erf(x) = --------- | exp(-t*t)dt + * sqrt(pi) \| + * 0 + * + * erfc(x) = 1-erf(x) + * Note that + * erf(-x) = -erf(x) + * erfc(-x) = 2 - erfc(x) + * + * Method: + * 1. For |x| in [0, 0.84375] + * erf(x) = x + x*R(x**2) + * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] + * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] + * where R = P/Q where P is an odd poly of degree 8 and + * Q is an odd poly of degree 10. + * -57.90 + * | R - (erf(x)-x)/x | <= 2 + * + * + * Remark. The formula is derived by noting + * erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....) + * and that + * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 + * is close to one. The interval is chosen because the fix + * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is + * near 0.6174), and by some experiment, 0.84375 is chosen to + * guarantee the error is less than one ulp for erf. + * + * 2. For |x| in [0.84375,1.25], let s_ = |x| - 1, and + * c = 0.84506291151 rounded to single (24 bits) + * erf(x) = sign(x) * (c + P1(s_)/Q1(s_)) + * erfc(x) = (1-c) - P1(s_)/Q1(s_) if x > 0 + * 1+(c+P1(s_)/Q1(s_)) if x < 0 + * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 + * Remark: here we use the taylor series expansion at x=1. + * erf(1+s_) = erf(1) + s_*Poly(s_) + * = 0.845.. + P1(s_)/Q1(s_) + * That is, we use rational approximation to approximate + * erf(1+s_) - (c = (single)0.84506291151) + * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] + * where + * P1(s_) = degree 6 poly in s_ + * Q1(s_) = degree 6 poly in s_ + * + * 3. For x in [1.25,1/0.35(~2.857143)], + * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/s1) + * erf(x) = 1 - erfc(x) + * where + * R1(z) = degree 7 poly in z, (z=1/x**2) + * s1(z) = degree 8 poly in z + * + * 4. For x in [1/0.35,28] + * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/s2) if x > 0 + * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/s2) if -6<x<0 + * = 2.0 - tiny (if x <= -6) + * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else + * erf(x) = sign(x)*(1.0 - tiny) + * where + * R2(z) = degree 6 poly in z, (z=1/x**2) + * s2(z) = degree 7 poly in z + * + * Note1: + * To compute exp(-x*x-0.5625+R/s), let s_ be a single + * precision number and s_ := x; then + * -x*x = -s_*s_ + (s_-x)*(s_+x) + * exp(-x*x-0.5626+R/s) = + * exp(-s_*s_-0.5625)*exp((s_-x)*(s_+x)+R/s); + * Note2: + * Here 4 and 5 make use of the asymptotic series + * exp(-x*x) + * erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) ) + * x*sqrt(pi) + * We use rational approximation to approximate + * g(s_)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625 + * Here is the error bound for R1/s1 and R2/s2 + * |R1/s1 - f(x)| < 2**(-62.57) + * |R2/s2 - f(x)| < 2**(-61.52) + * + * 5. For inf > x >= 28 + * erf(x) = sign(x) *(1 - tiny) (raise inexact) + * erfc(x) = tiny*tiny (raise underflow) if x > 0 + * = 2 - tiny if x<0 + * + * 7. special case: + * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, + * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, + * erfc/erf(nan) is nan +*/ +const ( + erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000 + // Coefficients for approximation to erf in [0, 0.84375] + efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69 + efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69 + pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68 + pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913 + pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F + pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4 + pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC + qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09 + qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA + qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F + qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10 + qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120 + // Coefficients for approximation to erf in [0.84375, 1.25] + pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538 + pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D + pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1 + pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4 + pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC + pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB + pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F + qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323 + qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33 + qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7 + qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F + qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C + qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D + // Coefficients for approximation to erfc in [1.25, 1/0.35] + ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435 + ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360 + ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726 + ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D + ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266 + ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2 + ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2 + ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C + sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687 + sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721 + sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71 + sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868 + sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314 + sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C + sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93 + sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62 + // Coefficients for approximation to erfc in [1/.35, 28] + rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A + rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE + rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A + rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98 + rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228 + rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992 + rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F + sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190 + sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A + sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118 + sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A + sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6 + sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763 + sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62 +) + +// erf returns the error function of x. +// +// special cases are: +// erf(+inf) = 1 +// erf(-inf) = -1 +// erf(nan) = nan +pub fn erf(a f64) f64 { + mut x := a + very_tiny := 2.848094538889218e-306 // 0x0080000000000000 + small := 1.0 / f64(u64(1) << 28) // 2**-28 + if is_nan(x) { + return nan() + } + if is_inf(x, 1) { + return 1.0 + } + if is_inf(x, -1) { + return f64(-1) + } + mut sign := false + if x < 0 { + x = -x + sign = true + } + if x < 0.84375 { // |x| < 0.84375 + mut temp := 0.0 + if x < small { // |x| < 2**-28 + if x < very_tiny { + temp = 0.125 * (8.0 * x + math.efx8 * x) // avoid underflow + } else { + temp = x + math.efx * x + } + } else { + z := x * x + r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4))) + s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 + + z * math.qq5)))) + y := r / s_ + temp = x + x * y + } + if sign { + return -temp + } + return temp + } + if x < 1.25 { // 0.84375 <= |x| < 1.25 + s_ := x - 1 + p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 + + s_ * (math.pa5 + s_ * math.pa6))))) + q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 + + s_ * (math.qa5 + s_ * math.qa6))))) + if sign { + return -math.erx - p / q + } + return math.erx + p / q + } + if x >= 6 { // inf > |x| >= 6 + if sign { + return -1 + } + return 1.0 + } + s_ := 1.0 / (x * x) + mut r := 0.0 + mut s := 0.0 + if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143 + r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 + + s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7)))))) + s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 + + s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8))))))) + } else { // |x| >= 1 / 0.35 ~ 2.857143 + r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 + + s_ * (math.rb5 + s_ * math.rb6))))) + s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 + + s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7)))))) + } + z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x + r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s) + if sign { + return r_ / x - 1.0 + } + return 1.0 - r_ / x +} + +// erfc returns the complementary error function of x. +// +// special cases are: +// erfc(+inf) = 0 +// erfc(-inf) = 2 +// erfc(nan) = nan +pub fn erfc(a f64) f64 { + mut x := a + tiny := 1.0 / f64(u64(1) << 56) // 2**-56 + // special cases + if is_nan(x) { + return nan() + } + if is_inf(x, 1) { + return 0.0 + } + if is_inf(x, -1) { + return 2.0 + } + mut sign := false + if x < 0 { + x = -x + sign = true + } + if x < 0.84375 { // |x| < 0.84375 + mut temp := 0.0 + if x < tiny { // |x| < 2**-56 + temp = x + } else { + z := x * x + r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4))) + s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 + + z * math.qq5)))) + y := r / s_ + if x < 0.25 { // |x| < 1.0/4 + temp = x + x * y + } else { + temp = 0.5 + (x * y + (x - 0.5)) + } + } + if sign { + return 1.0 + temp + } + return 1.0 - temp + } + if x < 1.25 { // 0.84375 <= |x| < 1.25 + s_ := x - 1 + p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 + + s_ * (math.pa5 + s_ * math.pa6))))) + q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 + + s_ * (math.qa5 + s_ * math.qa6))))) + if sign { + return 1.0 + math.erx + p / q + } + return 1.0 - math.erx - p / q + } + if x < 28 { // |x| < 28 + s_ := 1.0 / (x * x) + mut r := 0.0 + mut s := 0.0 + if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143 + r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 + + s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7)))))) + s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 + + s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8))))))) + } else { // |x| >= 1 / 0.35 ~ 2.857143 + if sign && x > 6 { + return 2.0 // x < -6 + } + r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 + + s_ * (math.rb5 + s_ * math.rb6))))) + s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 + + s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7)))))) + } + z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x + r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s) + if sign { + return 2.0 - r_ / x + } + return r_ / x + } + if sign { + return 2.0 + } + return 0.0 +} |