/*

 * 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;

}