1 #ifndef JEMALLOC_ENABLE_INLINE 2 double ln_gamma(double x); 3 double i_gamma(double x, double p, double ln_gamma_p); 4 double pt_norm(double p); 5 double pt_chi2(double p, double df, double ln_gamma_df_2); 6 double pt_gamma(double p, double shape, double scale, double ln_gamma_shape); 7 #endif 8 9 #if (defined(JEMALLOC_ENABLE_INLINE) || defined(MATH_C_)) 10 /* 11 * Compute the natural log of Gamma(x), accurate to 10 decimal places. 12 * 13 * This implementation is based on: 14 * 15 * Pike, M.C., I.D. Hill (1966) Algorithm 291: Logarithm of Gamma function 16 * [S14]. Communications of the ACM 9(9):684. 17 */ 18 JEMALLOC_INLINE double 19 ln_gamma(double x) 20 { 21 double f, z; 22 23 assert(x > 0.0); 24 25 if (x < 7.0) { 26 f = 1.0; 27 z = x; 28 while (z < 7.0) { 29 f *= z; 30 z += 1.0; 31 } 32 x = z; 33 f = -log(f); 34 } else 35 f = 0.0; 36 37 z = 1.0 / (x * x); 38 39 return (f + (x-0.5) * log(x) - x + 0.918938533204673 + 40 (((-0.000595238095238 * z + 0.000793650793651) * z - 41 0.002777777777778) * z + 0.083333333333333) / x); 42 } 43 44 /* 45 * Compute the incomplete Gamma ratio for [0..x], where p is the shape 46 * parameter, and ln_gamma_p is ln_gamma(p). 47 * 48 * This implementation is based on: 49 * 50 * Bhattacharjee, G.P. (1970) Algorithm AS 32: The incomplete Gamma integral. 51 * Applied Statistics 19:285-287. 52 */ 53 JEMALLOC_INLINE double 54 i_gamma(double x, double p, double ln_gamma_p) 55 { 56 double acu, factor, oflo, gin, term, rn, a, b, an, dif; 57 double pn[6]; 58 unsigned i; 59 60 assert(p > 0.0); 61 assert(x >= 0.0); 62 63 if (x == 0.0) 64 return (0.0); 65 66 acu = 1.0e-10; 67 oflo = 1.0e30; 68 gin = 0.0; 69 factor = exp(p * log(x) - x - ln_gamma_p); 70 71 if (x <= 1.0 || x < p) { 72 /* Calculation by series expansion. */ 73 gin = 1.0; 74 term = 1.0; 75 rn = p; 76 77 while (true) { 78 rn += 1.0; 79 term *= x / rn; 80 gin += term; 81 if (term <= acu) { 82 gin *= factor / p; 83 return (gin); 84 } 85 } 86 } else { 87 /* Calculation by continued fraction. */ 88 a = 1.0 - p; 89 b = a + x + 1.0; 90 term = 0.0; 91 pn[0] = 1.0; 92 pn[1] = x; 93 pn[2] = x + 1.0; 94 pn[3] = x * b; 95 gin = pn[2] / pn[3]; 96 97 while (true) { 98 a += 1.0; 99 b += 2.0; 100 term += 1.0; 101 an = a * term; 102 for (i = 0; i < 2; i++) 103 pn[i+4] = b * pn[i+2] - an * pn[i]; 104 if (pn[5] != 0.0) { 105 rn = pn[4] / pn[5]; 106 dif = fabs(gin - rn); 107 if (dif <= acu && dif <= acu * rn) { 108 gin = 1.0 - factor * gin; 109 return (gin); 110 } 111 gin = rn; 112 } 113 for (i = 0; i < 4; i++) 114 pn[i] = pn[i+2]; 115 116 if (fabs(pn[4]) >= oflo) { 117 for (i = 0; i < 4; i++) 118 pn[i] /= oflo; 119 } 120 } 121 } 122 } 123 124 /* 125 * Given a value p in [0..1] of the lower tail area of the normal distribution, 126 * compute the limit on the definite integral from [-inf..z] that satisfies p, 127 * accurate to 16 decimal places. 128 * 129 * This implementation is based on: 130 * 131 * Wichura, M.J. (1988) Algorithm AS 241: The percentage points of the normal 132 * distribution. Applied Statistics 37(3):477-484. 133 */ 134 JEMALLOC_INLINE double 135 pt_norm(double p) 136 { 137 double q, r, ret; 138 139 assert(p > 0.0 && p < 1.0); 140 141 q = p - 0.5; 142 if (fabs(q) <= 0.425) { 143 /* p close to 1/2. */ 144 r = 0.180625 - q * q; 145 return (q * (((((((2.5090809287301226727e3 * r + 146 3.3430575583588128105e4) * r + 6.7265770927008700853e4) * r 147 + 4.5921953931549871457e4) * r + 1.3731693765509461125e4) * 148 r + 1.9715909503065514427e3) * r + 1.3314166789178437745e2) 149 * r + 3.3871328727963666080e0) / 150 (((((((5.2264952788528545610e3 * r + 151 2.8729085735721942674e4) * r + 3.9307895800092710610e4) * r 152 + 2.1213794301586595867e4) * r + 5.3941960214247511077e3) * 153 r + 6.8718700749205790830e2) * r + 4.2313330701600911252e1) 154 * r + 1.0)); 155 } else { 156 if (q < 0.0) 157 r = p; 158 else 159 r = 1.0 - p; 160 assert(r > 0.0); 161 162 r = sqrt(-log(r)); 163 if (r <= 5.0) { 164 /* p neither close to 1/2 nor 0 or 1. */ 165 r -= 1.6; 166 ret = ((((((((7.74545014278341407640e-4 * r + 167 2.27238449892691845833e-2) * r + 168 2.41780725177450611770e-1) * r + 169 1.27045825245236838258e0) * r + 170 3.64784832476320460504e0) * r + 171 5.76949722146069140550e0) * r + 172 4.63033784615654529590e0) * r + 173 1.42343711074968357734e0) / 174 (((((((1.05075007164441684324e-9 * r + 175 5.47593808499534494600e-4) * r + 176 1.51986665636164571966e-2) 177 * r + 1.48103976427480074590e-1) * r + 178 6.89767334985100004550e-1) * r + 179 1.67638483018380384940e0) * r + 180 2.05319162663775882187e0) * r + 1.0)); 181 } else { 182 /* p near 0 or 1. */ 183 r -= 5.0; 184 ret = ((((((((2.01033439929228813265e-7 * r + 185 2.71155556874348757815e-5) * r + 186 1.24266094738807843860e-3) * r + 187 2.65321895265761230930e-2) * r + 188 2.96560571828504891230e-1) * r + 189 1.78482653991729133580e0) * r + 190 5.46378491116411436990e0) * r + 191 6.65790464350110377720e0) / 192 (((((((2.04426310338993978564e-15 * r + 193 1.42151175831644588870e-7) * r + 194 1.84631831751005468180e-5) * r + 195 7.86869131145613259100e-4) * r + 196 1.48753612908506148525e-2) * r + 197 1.36929880922735805310e-1) * r + 198 5.99832206555887937690e-1) 199 * r + 1.0)); 200 } 201 if (q < 0.0) 202 ret = -ret; 203 return (ret); 204 } 205 } 206 207 /* 208 * Given a value p in [0..1] of the lower tail area of the Chi^2 distribution 209 * with df degrees of freedom, where ln_gamma_df_2 is ln_gamma(df/2.0), compute 210 * the upper limit on the definite integral from [0..z] that satisfies p, 211 * accurate to 12 decimal places. 212 * 213 * This implementation is based on: 214 * 215 * Best, D.J., D.E. Roberts (1975) Algorithm AS 91: The percentage points of 216 * the Chi^2 distribution. Applied Statistics 24(3):385-388. 217 * 218 * Shea, B.L. (1991) Algorithm AS R85: A remark on AS 91: The percentage 219 * points of the Chi^2 distribution. Applied Statistics 40(1):233-235. 220 */ 221 JEMALLOC_INLINE double 222 pt_chi2(double p, double df, double ln_gamma_df_2) 223 { 224 double e, aa, xx, c, ch, a, q, p1, p2, t, x, b, s1, s2, s3, s4, s5, s6; 225 unsigned i; 226 227 assert(p >= 0.0 && p < 1.0); 228 assert(df > 0.0); 229 230 e = 5.0e-7; 231 aa = 0.6931471805; 232 233 xx = 0.5 * df; 234 c = xx - 1.0; 235 236 if (df < -1.24 * log(p)) { 237 /* Starting approximation for small Chi^2. */ 238 ch = pow(p * xx * exp(ln_gamma_df_2 + xx * aa), 1.0 / xx); 239 if (ch - e < 0.0) 240 return (ch); 241 } else { 242 if (df > 0.32) { 243 x = pt_norm(p); 244 /* 245 * Starting approximation using Wilson and Hilferty 246 * estimate. 247 */ 248 p1 = 0.222222 / df; 249 ch = df * pow(x * sqrt(p1) + 1.0 - p1, 3.0); 250 /* Starting approximation for p tending to 1. */ 251 if (ch > 2.2 * df + 6.0) { 252 ch = -2.0 * (log(1.0 - p) - c * log(0.5 * ch) + 253 ln_gamma_df_2); 254 } 255 } else { 256 ch = 0.4; 257 a = log(1.0 - p); 258 while (true) { 259 q = ch; 260 p1 = 1.0 + ch * (4.67 + ch); 261 p2 = ch * (6.73 + ch * (6.66 + ch)); 262 t = -0.5 + (4.67 + 2.0 * ch) / p1 - (6.73 + ch 263 * (13.32 + 3.0 * ch)) / p2; 264 ch -= (1.0 - exp(a + ln_gamma_df_2 + 0.5 * ch + 265 c * aa) * p2 / p1) / t; 266 if (fabs(q / ch - 1.0) - 0.01 <= 0.0) 267 break; 268 } 269 } 270 } 271 272 for (i = 0; i < 20; i++) { 273 /* Calculation of seven-term Taylor series. */ 274 q = ch; 275 p1 = 0.5 * ch; 276 if (p1 < 0.0) 277 return (-1.0); 278 p2 = p - i_gamma(p1, xx, ln_gamma_df_2); 279 t = p2 * exp(xx * aa + ln_gamma_df_2 + p1 - c * log(ch)); 280 b = t / ch; 281 a = 0.5 * t - b * c; 282 s1 = (210.0 + a * (140.0 + a * (105.0 + a * (84.0 + a * (70.0 + 283 60.0 * a))))) / 420.0; 284 s2 = (420.0 + a * (735.0 + a * (966.0 + a * (1141.0 + 1278.0 * 285 a)))) / 2520.0; 286 s3 = (210.0 + a * (462.0 + a * (707.0 + 932.0 * a))) / 2520.0; 287 s4 = (252.0 + a * (672.0 + 1182.0 * a) + c * (294.0 + a * 288 (889.0 + 1740.0 * a))) / 5040.0; 289 s5 = (84.0 + 264.0 * a + c * (175.0 + 606.0 * a)) / 2520.0; 290 s6 = (120.0 + c * (346.0 + 127.0 * c)) / 5040.0; 291 ch += t * (1.0 + 0.5 * t * s1 - b * c * (s1 - b * (s2 - b * (s3 292 - b * (s4 - b * (s5 - b * s6)))))); 293 if (fabs(q / ch - 1.0) <= e) 294 break; 295 } 296 297 return (ch); 298 } 299 300 /* 301 * Given a value p in [0..1] and Gamma distribution shape and scale parameters, 302 * compute the upper limit on the definite integral from [0..z] that satisfies 303 * p. 304 */ 305 JEMALLOC_INLINE double 306 pt_gamma(double p, double shape, double scale, double ln_gamma_shape) 307 { 308 309 return (pt_chi2(p, shape * 2.0, ln_gamma_shape) * 0.5 * scale); 310 } 311 #endif 312
