| 1 | module math |
| 2 | |
| 3 | /* |
| 4 | * x |
| 5 | * 2 |\ |
| 6 | * erf(x) = --------- | exp(-t*t)dt |
| 7 | * sqrt(pi) \| |
| 8 | * 0 |
| 9 | * |
| 10 | * erfc(x) = 1-erf(x) |
| 11 | * Note that |
| 12 | * erf(-x) = -erf(x) |
| 13 | * erfc(-x) = 2 - erfc(x) |
| 14 | * |
| 15 | * Method: |
| 16 | * 1. For |x| in [0, 0.84375] |
| 17 | * erf(x) = x + x*R(x**2) |
| 18 | * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] |
| 19 | * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] |
| 20 | * where R = P/Q where P is an odd poly of degree 8 and |
| 21 | * Q is an odd poly of degree 10. |
| 22 | * -57.90 |
| 23 | * | R - (erf(x)-x)/x | <= 2 |
| 24 | * |
| 25 | * |
| 26 | * Remark. The formula is derived by noting |
| 27 | * erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....) |
| 28 | * and that |
| 29 | * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 |
| 30 | * is close to one. The interval is chosen because the fix |
| 31 | * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is |
| 32 | * near 0.6174), and by some experiment, 0.84375 is chosen to |
| 33 | * guarantee the error is less than one ulp for erf. |
| 34 | * |
| 35 | * 2. For |x| in [0.84375,1.25], let s_ = |x| - 1, and |
| 36 | * c = 0.84506291151 rounded to single (24 bits) |
| 37 | * erf(x) = sign(x) * (c + P1(s_)/Q1(s_)) |
| 38 | * erfc(x) = (1-c) - P1(s_)/Q1(s_) if x > 0 |
| 39 | * 1+(c+P1(s_)/Q1(s_)) if x < 0 |
| 40 | * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 |
| 41 | * Remark: here we use the taylor series expansion at x=1. |
| 42 | * erf(1+s_) = erf(1) + s_*Poly(s_) |
| 43 | * = 0.845.. + P1(s_)/Q1(s_) |
| 44 | * That is, we use rational approximation to approximate |
| 45 | * erf(1+s_) - (c = (single)0.84506291151) |
| 46 | * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] |
| 47 | * where |
| 48 | * P1(s_) = degree 6 poly in s_ |
| 49 | * Q1(s_) = degree 6 poly in s_ |
| 50 | * |
| 51 | * 3. For x in [1.25,1/0.35(~2.857143)], |
| 52 | * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/s1) |
| 53 | * erf(x) = 1 - erfc(x) |
| 54 | * where |
| 55 | * R1(z) = degree 7 poly in z, (z=1/x**2) |
| 56 | * s1(z) = degree 8 poly in z |
| 57 | * |
| 58 | * 4. For x in [1/0.35,28] |
| 59 | * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/s2) if x > 0 |
| 60 | * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/s2) if -6<x<0 |
| 61 | * = 2.0 - tiny (if x <= -6) |
| 62 | * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else |
| 63 | * erf(x) = sign(x)*(1.0 - tiny) |
| 64 | * where |
| 65 | * R2(z) = degree 6 poly in z, (z=1/x**2) |
| 66 | * s2(z) = degree 7 poly in z |
| 67 | * |
| 68 | * Note1: |
| 69 | * To compute exp(-x*x-0.5625+R/s), let s_ be a single |
| 70 | * precision number and s_ := x; then |
| 71 | * -x*x = -s_*s_ + (s_-x)*(s_+x) |
| 72 | * exp(-x*x-0.5626+R/s) = |
| 73 | * exp(-s_*s_-0.5625)*exp((s_-x)*(s_+x)+R/s); |
| 74 | * Note2: |
| 75 | * Here 4 and 5 make use of the asymptotic series |
| 76 | * exp(-x*x) |
| 77 | * erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) ) |
| 78 | * x*sqrt(pi) |
| 79 | * We use rational approximation to approximate |
| 80 | * g(s_)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625 |
| 81 | * Here is the error bound for R1/s1 and R2/s2 |
| 82 | * |R1/s1 - f(x)| < 2**(-62.57) |
| 83 | * |R2/s2 - f(x)| < 2**(-61.52) |
| 84 | * |
| 85 | * 5. For inf > x >= 28 |
| 86 | * erf(x) = sign(x) *(1 - tiny) (raise inexact) |
| 87 | * erfc(x) = tiny*tiny (raise underflow) if x > 0 |
| 88 | * = 2 - tiny if x<0 |
| 89 | * |
| 90 | * 7. special case: |
| 91 | * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, |
| 92 | * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, |
| 93 | * erfc/erf(nan) is nan |
| 94 | */ |
| 95 | const ( |
| 96 | erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000 |
| 97 | // Coefficients for approximation to erf in [0, 0.84375] |
| 98 | efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69 |
| 99 | efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69 |
| 100 | pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68 |
| 101 | pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913 |
| 102 | pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F |
| 103 | pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4 |
| 104 | pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC |
| 105 | qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09 |
| 106 | qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA |
| 107 | qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F |
| 108 | qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10 |
| 109 | qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120 |
| 110 | // Coefficients for approximation to erf in [0.84375, 1.25] |
| 111 | pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538 |
| 112 | pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D |
| 113 | pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1 |
| 114 | pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4 |
| 115 | pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC |
| 116 | pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB |
| 117 | pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F |
| 118 | qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323 |
| 119 | qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33 |
| 120 | qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7 |
| 121 | qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F |
| 122 | qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C |
| 123 | qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D |
| 124 | // Coefficients for approximation to erfc in [1.25, 1/0.35] |
| 125 | ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435 |
| 126 | ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360 |
| 127 | ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726 |
| 128 | ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D |
| 129 | ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266 |
| 130 | ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2 |
| 131 | ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2 |
| 132 | ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C |
| 133 | sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687 |
| 134 | sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721 |
| 135 | sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71 |
| 136 | sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868 |
| 137 | sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314 |
| 138 | sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C |
| 139 | sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93 |
| 140 | sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62 |
| 141 | // Coefficients for approximation to erfc in [1/.35, 28] |
| 142 | rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A |
| 143 | rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE |
| 144 | rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A |
| 145 | rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98 |
| 146 | rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228 |
| 147 | rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992 |
| 148 | rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F |
| 149 | sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190 |
| 150 | sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A |
| 151 | sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118 |
| 152 | sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A |
| 153 | sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6 |
| 154 | sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763 |
| 155 | sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62 |
| 156 | ) |
| 157 | |
| 158 | // erf returns the error function of x. |
| 159 | // |
| 160 | // special cases are: |
| 161 | // erf(+inf) = 1 |
| 162 | // erf(-inf) = -1 |
| 163 | // erf(nan) = nan |
| 164 | pub fn erf(a f64) f64 { |
| 165 | mut x := a |
| 166 | very_tiny := 2.848094538889218e-306 // 0x0080000000000000 |
| 167 | small_ := 1.0 / f64(u64(1) << 28) // 2**-28 |
| 168 | if is_nan(x) { |
| 169 | return nan() |
| 170 | } |
| 171 | if is_inf(x, 1) { |
| 172 | return 1.0 |
| 173 | } |
| 174 | if is_inf(x, -1) { |
| 175 | return f64(-1) |
| 176 | } |
| 177 | mut sign := false |
| 178 | if x < 0 { |
| 179 | x = -x |
| 180 | sign = true |
| 181 | } |
| 182 | if x < 0.84375 { // |x| < 0.84375 |
| 183 | mut temp := 0.0 |
| 184 | if x < small_ { // |x| < 2**-28 |
| 185 | if x < very_tiny { |
| 186 | temp = 0.125 * (8.0 * x + math.efx8 * x) // avoid underflow |
| 187 | } else { |
| 188 | temp = x + math.efx * x |
| 189 | } |
| 190 | } else { |
| 191 | z := x * x |
| 192 | r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4))) |
| 193 | s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 + |
| 194 | z * math.qq5)))) |
| 195 | y := r / s_ |
| 196 | temp = x + x * y |
| 197 | } |
| 198 | if sign { |
| 199 | return -temp |
| 200 | } |
| 201 | return temp |
| 202 | } |
| 203 | if x < 1.25 { // 0.84375 <= |x| < 1.25 |
| 204 | s_ := x - 1 |
| 205 | p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 + |
| 206 | s_ * (math.pa5 + s_ * math.pa6))))) |
| 207 | q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 + |
| 208 | s_ * (math.qa5 + s_ * math.qa6))))) |
| 209 | if sign { |
| 210 | return -math.erx - p / q |
| 211 | } |
| 212 | return math.erx + p / q |
| 213 | } |
| 214 | if x >= 6 { // inf > |x| >= 6 |
| 215 | if sign { |
| 216 | return -1 |
| 217 | } |
| 218 | return 1.0 |
| 219 | } |
| 220 | s_ := 1.0 / (x * x) |
| 221 | mut r := 0.0 |
| 222 | mut s := 0.0 |
| 223 | if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143 |
| 224 | r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 + |
| 225 | s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7)))))) |
| 226 | s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 + |
| 227 | s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8))))))) |
| 228 | } else { // |x| >= 1 / 0.35 ~ 2.857143 |
| 229 | r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 + |
| 230 | s_ * (math.rb5 + s_ * math.rb6))))) |
| 231 | s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 + |
| 232 | s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7)))))) |
| 233 | } |
| 234 | z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x |
| 235 | r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s) |
| 236 | if sign { |
| 237 | return r_ / x - 1.0 |
| 238 | } |
| 239 | return 1.0 - r_ / x |
| 240 | } |
| 241 | |
| 242 | // erfc returns the complementary error function of x. |
| 243 | // |
| 244 | // special cases are: |
| 245 | // erfc(+inf) = 0 |
| 246 | // erfc(-inf) = 2 |
| 247 | // erfc(nan) = nan |
| 248 | pub fn erfc(a f64) f64 { |
| 249 | mut x := a |
| 250 | tiny := 1.0 / f64(u64(1) << 56) // 2**-56 |
| 251 | // special cases |
| 252 | if is_nan(x) { |
| 253 | return nan() |
| 254 | } |
| 255 | if is_inf(x, 1) { |
| 256 | return 0.0 |
| 257 | } |
| 258 | if is_inf(x, -1) { |
| 259 | return 2.0 |
| 260 | } |
| 261 | mut sign := false |
| 262 | if x < 0 { |
| 263 | x = -x |
| 264 | sign = true |
| 265 | } |
| 266 | if x < 0.84375 { // |x| < 0.84375 |
| 267 | mut temp := 0.0 |
| 268 | if x < tiny { // |x| < 2**-56 |
| 269 | temp = x |
| 270 | } else { |
| 271 | z := x * x |
| 272 | r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4))) |
| 273 | s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 + |
| 274 | z * math.qq5)))) |
| 275 | y := r / s_ |
| 276 | if x < 0.25 { // |x| < 1.0/4 |
| 277 | temp = x + x * y |
| 278 | } else { |
| 279 | temp = 0.5 + (x * y + (x - 0.5)) |
| 280 | } |
| 281 | } |
| 282 | if sign { |
| 283 | return 1.0 + temp |
| 284 | } |
| 285 | return 1.0 - temp |
| 286 | } |
| 287 | if x < 1.25 { // 0.84375 <= |x| < 1.25 |
| 288 | s_ := x - 1 |
| 289 | p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 + |
| 290 | s_ * (math.pa5 + s_ * math.pa6))))) |
| 291 | q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 + |
| 292 | s_ * (math.qa5 + s_ * math.qa6))))) |
| 293 | if sign { |
| 294 | return 1.0 + math.erx + p / q |
| 295 | } |
| 296 | return 1.0 - math.erx - p / q |
| 297 | } |
| 298 | if x < 28 { // |x| < 28 |
| 299 | s_ := 1.0 / (x * x) |
| 300 | mut r := 0.0 |
| 301 | mut s := 0.0 |
| 302 | if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143 |
| 303 | r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 + |
| 304 | s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7)))))) |
| 305 | s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 + |
| 306 | s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8))))))) |
| 307 | } else { // |x| >= 1 / 0.35 ~ 2.857143 |
| 308 | if sign && x > 6 { |
| 309 | return 2.0 // x < -6 |
| 310 | } |
| 311 | r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 + |
| 312 | s_ * (math.rb5 + s_ * math.rb6))))) |
| 313 | s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 + |
| 314 | s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7)))))) |
| 315 | } |
| 316 | z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x |
| 317 | r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s) |
| 318 | if sign { |
| 319 | return 2.0 - r_ / x |
| 320 | } |
| 321 | return r_ / x |
| 322 | } |
| 323 | if sign { |
| 324 | return 2.0 |
| 325 | } |
| 326 | return 0.0 |
| 327 | } |