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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