math.h revision b1941c615023cab9baf0a78a28df1e3b4972434f 
1#ifndef JEMALLOC_ENABLE_INLINE
2double	ln_gamma(double x);
3double	i_gamma(double x, double p, double ln_gamma_p);
4double	pt_norm(double p);
5double	pt_chi2(double p, double df, double ln_gamma_df_2);
6double	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 */
18JEMALLOC_INLINE double
19ln_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 */
53JEMALLOC_INLINE double
54i_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 */
134JEMALLOC_INLINE double
135pt_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 */
221JEMALLOC_INLINE double
222pt_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 integeral from [0..z] that satisfies
303 * p.
304 */
305JEMALLOC_INLINE double
306pt_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