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Diffstat (limited to 'deps/jemalloc/test/include/test/math.h')
-rw-r--r-- | deps/jemalloc/test/include/test/math.h | 306 |
1 files changed, 306 insertions, 0 deletions
diff --git a/deps/jemalloc/test/include/test/math.h b/deps/jemalloc/test/include/test/math.h new file mode 100644 index 000000000..efba086dd --- /dev/null +++ b/deps/jemalloc/test/include/test/math.h @@ -0,0 +1,306 @@ +/* + * Compute the natural log of Gamma(x), accurate to 10 decimal places. + * + * This implementation is based on: + * + * Pike, M.C., I.D. Hill (1966) Algorithm 291: Logarithm of Gamma function + * [S14]. Communications of the ACM 9(9):684. + */ +static inline double +ln_gamma(double x) { + double f, z; + + assert(x > 0.0); + + if (x < 7.0) { + f = 1.0; + z = x; + while (z < 7.0) { + f *= z; + z += 1.0; + } + x = z; + f = -log(f); + } else { + f = 0.0; + } + + z = 1.0 / (x * x); + + return f + (x-0.5) * log(x) - x + 0.918938533204673 + + (((-0.000595238095238 * z + 0.000793650793651) * z - + 0.002777777777778) * z + 0.083333333333333) / x; +} + +/* + * Compute the incomplete Gamma ratio for [0..x], where p is the shape + * parameter, and ln_gamma_p is ln_gamma(p). + * + * This implementation is based on: + * + * Bhattacharjee, G.P. (1970) Algorithm AS 32: The incomplete Gamma integral. + * Applied Statistics 19:285-287. + */ +static inline double +i_gamma(double x, double p, double ln_gamma_p) { + double acu, factor, oflo, gin, term, rn, a, b, an, dif; + double pn[6]; + unsigned i; + + assert(p > 0.0); + assert(x >= 0.0); + + if (x == 0.0) { + return 0.0; + } + + acu = 1.0e-10; + oflo = 1.0e30; + gin = 0.0; + factor = exp(p * log(x) - x - ln_gamma_p); + + if (x <= 1.0 || x < p) { + /* Calculation by series expansion. */ + gin = 1.0; + term = 1.0; + rn = p; + + while (true) { + rn += 1.0; + term *= x / rn; + gin += term; + if (term <= acu) { + gin *= factor / p; + return gin; + } + } + } else { + /* Calculation by continued fraction. */ + a = 1.0 - p; + b = a + x + 1.0; + term = 0.0; + pn[0] = 1.0; + pn[1] = x; + pn[2] = x + 1.0; + pn[3] = x * b; + gin = pn[2] / pn[3]; + + while (true) { + a += 1.0; + b += 2.0; + term += 1.0; + an = a * term; + for (i = 0; i < 2; i++) { + pn[i+4] = b * pn[i+2] - an * pn[i]; + } + if (pn[5] != 0.0) { + rn = pn[4] / pn[5]; + dif = fabs(gin - rn); + if (dif <= acu && dif <= acu * rn) { + gin = 1.0 - factor * gin; + return gin; + } + gin = rn; + } + for (i = 0; i < 4; i++) { + pn[i] = pn[i+2]; + } + + if (fabs(pn[4]) >= oflo) { + for (i = 0; i < 4; i++) { + pn[i] /= oflo; + } + } + } + } +} + +/* + * Given a value p in [0..1] of the lower tail area of the normal distribution, + * compute the limit on the definite integral from [-inf..z] that satisfies p, + * accurate to 16 decimal places. + * + * This implementation is based on: + * + * Wichura, M.J. (1988) Algorithm AS 241: The percentage points of the normal + * distribution. Applied Statistics 37(3):477-484. + */ +static inline double +pt_norm(double p) { + double q, r, ret; + + assert(p > 0.0 && p < 1.0); + + q = p - 0.5; + if (fabs(q) <= 0.425) { + /* p close to 1/2. */ + r = 0.180625 - q * q; + return q * (((((((2.5090809287301226727e3 * r + + 3.3430575583588128105e4) * r + 6.7265770927008700853e4) * r + + 4.5921953931549871457e4) * r + 1.3731693765509461125e4) * + r + 1.9715909503065514427e3) * r + 1.3314166789178437745e2) + * r + 3.3871328727963666080e0) / + (((((((5.2264952788528545610e3 * r + + 2.8729085735721942674e4) * r + 3.9307895800092710610e4) * r + + 2.1213794301586595867e4) * r + 5.3941960214247511077e3) * + r + 6.8718700749205790830e2) * r + 4.2313330701600911252e1) + * r + 1.0); + } else { + if (q < 0.0) { + r = p; + } else { + r = 1.0 - p; + } + assert(r > 0.0); + + r = sqrt(-log(r)); + if (r <= 5.0) { + /* p neither close to 1/2 nor 0 or 1. */ + r -= 1.6; + ret = ((((((((7.74545014278341407640e-4 * r + + 2.27238449892691845833e-2) * r + + 2.41780725177450611770e-1) * r + + 1.27045825245236838258e0) * r + + 3.64784832476320460504e0) * r + + 5.76949722146069140550e0) * r + + 4.63033784615654529590e0) * r + + 1.42343711074968357734e0) / + (((((((1.05075007164441684324e-9 * r + + 5.47593808499534494600e-4) * r + + 1.51986665636164571966e-2) + * r + 1.48103976427480074590e-1) * r + + 6.89767334985100004550e-1) * r + + 1.67638483018380384940e0) * r + + 2.05319162663775882187e0) * r + 1.0)); + } else { + /* p near 0 or 1. */ + r -= 5.0; + ret = ((((((((2.01033439929228813265e-7 * r + + 2.71155556874348757815e-5) * r + + 1.24266094738807843860e-3) * r + + 2.65321895265761230930e-2) * r + + 2.96560571828504891230e-1) * r + + 1.78482653991729133580e0) * r + + 5.46378491116411436990e0) * r + + 6.65790464350110377720e0) / + (((((((2.04426310338993978564e-15 * r + + 1.42151175831644588870e-7) * r + + 1.84631831751005468180e-5) * r + + 7.86869131145613259100e-4) * r + + 1.48753612908506148525e-2) * r + + 1.36929880922735805310e-1) * r + + 5.99832206555887937690e-1) + * r + 1.0)); + } + if (q < 0.0) { + ret = -ret; + } + return ret; + } +} + +/* + * Given a value p in [0..1] of the lower tail area of the Chi^2 distribution + * with df degrees of freedom, where ln_gamma_df_2 is ln_gamma(df/2.0), compute + * the upper limit on the definite integral from [0..z] that satisfies p, + * accurate to 12 decimal places. + * + * This implementation is based on: + * + * Best, D.J., D.E. Roberts (1975) Algorithm AS 91: The percentage points of + * the Chi^2 distribution. Applied Statistics 24(3):385-388. + * + * Shea, B.L. (1991) Algorithm AS R85: A remark on AS 91: The percentage + * points of the Chi^2 distribution. Applied Statistics 40(1):233-235. + */ +static inline double +pt_chi2(double p, double df, double ln_gamma_df_2) { + double e, aa, xx, c, ch, a, q, p1, p2, t, x, b, s1, s2, s3, s4, s5, s6; + unsigned i; + + assert(p >= 0.0 && p < 1.0); + assert(df > 0.0); + + e = 5.0e-7; + aa = 0.6931471805; + + xx = 0.5 * df; + c = xx - 1.0; + + if (df < -1.24 * log(p)) { + /* Starting approximation for small Chi^2. */ + ch = pow(p * xx * exp(ln_gamma_df_2 + xx * aa), 1.0 / xx); + if (ch - e < 0.0) { + return ch; + } + } else { + if (df > 0.32) { + x = pt_norm(p); + /* + * Starting approximation using Wilson and Hilferty + * estimate. + */ + p1 = 0.222222 / df; + ch = df * pow(x * sqrt(p1) + 1.0 - p1, 3.0); + /* Starting approximation for p tending to 1. */ + if (ch > 2.2 * df + 6.0) { + ch = -2.0 * (log(1.0 - p) - c * log(0.5 * ch) + + ln_gamma_df_2); + } + } else { + ch = 0.4; + a = log(1.0 - p); + while (true) { + q = ch; + p1 = 1.0 + ch * (4.67 + ch); + p2 = ch * (6.73 + ch * (6.66 + ch)); + t = -0.5 + (4.67 + 2.0 * ch) / p1 - (6.73 + ch + * (13.32 + 3.0 * ch)) / p2; + ch -= (1.0 - exp(a + ln_gamma_df_2 + 0.5 * ch + + c * aa) * p2 / p1) / t; + if (fabs(q / ch - 1.0) - 0.01 <= 0.0) { + break; + } + } + } + } + + for (i = 0; i < 20; i++) { + /* Calculation of seven-term Taylor series. */ + q = ch; + p1 = 0.5 * ch; + if (p1 < 0.0) { + return -1.0; + } + p2 = p - i_gamma(p1, xx, ln_gamma_df_2); + t = p2 * exp(xx * aa + ln_gamma_df_2 + p1 - c * log(ch)); + b = t / ch; + a = 0.5 * t - b * c; + s1 = (210.0 + a * (140.0 + a * (105.0 + a * (84.0 + a * (70.0 + + 60.0 * a))))) / 420.0; + s2 = (420.0 + a * (735.0 + a * (966.0 + a * (1141.0 + 1278.0 * + a)))) / 2520.0; + s3 = (210.0 + a * (462.0 + a * (707.0 + 932.0 * a))) / 2520.0; + s4 = (252.0 + a * (672.0 + 1182.0 * a) + c * (294.0 + a * + (889.0 + 1740.0 * a))) / 5040.0; + s5 = (84.0 + 264.0 * a + c * (175.0 + 606.0 * a)) / 2520.0; + s6 = (120.0 + c * (346.0 + 127.0 * c)) / 5040.0; + ch += t * (1.0 + 0.5 * t * s1 - b * c * (s1 - b * (s2 - b * (s3 + - b * (s4 - b * (s5 - b * s6)))))); + if (fabs(q / ch - 1.0) <= e) { + break; + } + } + + return ch; +} + +/* + * Given a value p in [0..1] and Gamma distribution shape and scale parameters, + * compute the upper limit on the definite integral from [0..z] that satisfies + * p. + */ +static inline double +pt_gamma(double p, double shape, double scale, double ln_gamma_shape) { + return pt_chi2(p, shape * 2.0, ln_gamma_shape) * 0.5 * scale; +} |