/* mpfr_gamma_inc -- incomplete gamma function Copyright 2016-2021 Free Software Foundation, Inc. Contributed by the AriC and Caramba projects, INRIA. This file is part of the GNU MPFR Library. The GNU MPFR Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. The GNU MPFR Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ #define MPFR_NEED_LONGLONG_H #include "mpfr-impl.h" /* The incomplete gamma function is defined for x >= 0 and a not a negative integer by: gamma_inc(a,x) := Gamma(a,x) = int(t^(a-1) * exp(-t), t=x..infinity) = Gamma(a) - gamma(a,x) with: gamma(a,x) = int(t^(a-1) * exp(-t), t=0..x). The function gamma(a,x) satisfies the Taylor expansions (we use the second one in the code below): gamma(a,x) = x^a * sum((-x)^k/k!/(a+k), k=0..infinity) gamma(a,x) = x^a * exp(-x) * sum(x^k/(a*(a+1)*...*(a+k)), k=0..infinity) */ static int mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t r); int mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) { mpfr_prec_t w; mpfr_t s, t, u; int inex; unsigned long k; mpfr_exp_t e0, e1, e2, err; MPFR_GROUP_DECL(group); MPFR_ZIV_DECL(loop); MPFR_SAVE_EXPO_DECL (expo); if (MPFR_ARE_SINGULAR (a, x)) { /* if a or x is NaN, return NaN */ if (MPFR_IS_NAN (a) || MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } /* Note: for x < 0, gamma_inc (a, x) is a complex number */ if (MPFR_IS_INF (a) || MPFR_IS_INF (x)) { if (MPFR_IS_INF (a) && MPFR_IS_INF (x)) { if ((MPFR_IS_POS (a) && MPFR_IS_POS (x)) || MPFR_IS_NEG (x)) { /* (a) gamma_inc(+Inf,+Inf) = NaN because gamma_inc(x,x) tends to +Inf but gamma_inc(x,x^2) tends to +0. (b) gamma_inc(+/-Inf,-Inf) = NaN, for example gamma_inc (a, -a) is a complex number for a not an integer */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else { /* gamma_inc(-Inf,+Inf) = +0 */ MPFR_SET_ZERO (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact */ } } else /* only one of a, x is infinite */ { if (MPFR_IS_INF (a)) { MPFR_ASSERTD (MPFR_IS_INF (a) && MPFR_IS_FP (x)); if (MPFR_IS_POS (a)) { /* gamma_inc(+Inf, x) = +Inf */ MPFR_SET_INF (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact */ } else /* a = -Inf */ { /* gamma_inc(-Inf, x) = NaN for x <= 0 +Inf for 0 < x < 1 +0 for 1 <= x */ if (mpfr_cmp_ui (x, 0) <= 0) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (mpfr_cmp_ui (x, 1) < 0) { MPFR_SET_INF (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact */ } else { MPFR_SET_ZERO (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact */ } } } else { MPFR_ASSERTD (MPFR_IS_FP (a) && MPFR_IS_INF (x)); if (MPFR_IS_POS (x)) { /* x is +Inf: integral tends to zero */ MPFR_SET_ZERO (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact */ } else /* NaN for x < 0 */ { MPFR_SET_NAN (y); MPFR_RET_NAN; } } } } if (MPFR_IS_ZERO (a) || MPFR_IS_ZERO (x)) { if (MPFR_IS_ZERO (a)) { if (mpfr_cmp_ui (x, 0) < 0) { /* gamma_inc(a,x) = NaN for x < 0 */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_ZERO (x)) /* gamma_inc(a,0) = gamma(a) */ return mpfr_gamma (y, a, rnd); /* a=+0->+Inf, a=-0->-Inf */ else { /* gamma_inc (0, x) = int (exp(-t), t=x..infinity) = E1(x) */ mpfr_t minus_x; MPFR_TMP_INIT_NEG(minus_x, x); /* mpfr_eint(x) for x < 0 returns -E1(-x) */ inex = mpfr_eint (y, minus_x, MPFR_INVERT_RND(rnd)); MPFR_CHANGE_SIGN(y); return -inex; } } else /* x = 0: gamma_inc(a,0) = gamma(a) */ return mpfr_gamma (y, a, rnd); } } /* for x < 0 return NaN */ if (MPFR_SIGN(x) < 0) { MPFR_SET_NAN (y); MPFR_RET_NAN; } if (mpfr_integer_p (a) && MPFR_SIGN(a) < 0) return mpfr_gamma_inc_negint (y, a, x, rnd); MPFR_SAVE_EXPO_MARK (expo); w = MPFR_PREC(y) + 13; /* working precision */ MPFR_GROUP_INIT_2(group, w, s, t); mpfr_init2 (u, 2); /* u is special (see below) */ MPFR_ZIV_INIT (loop, w); for (;;) { mpfr_exp_t expu, precu, exps; mpfr_t s_abs; mpfr_exp_t decay = 0; MPFR_BLOCK_DECL (flags); /* Note: in the error analysis below, theta represents any value of absolute value less than 2^(-w) where w is the working precision (two instances of theta may represent different values), cf Higham's book. */ /* to ensure that u = a + k is exact, we have three cases: (1) EXP(a) <= 0, then we need PREC(u) >= 1 - EXP(a) + PREC(a) (2) EXP(a) - PREC(a) <= 0 < E(a), then PREC(u) >= PREC(a) (3) 0 < EXP(a) - PREC(a), then PREC(u) >= EXP(a) */ precu = MPFR_GET_EXP(a) <= 0 ? MPFR_ADD_PREC (MPFR_PREC(a), 1 - MPFR_EXP(a)) : (MPFR_EXP(a) <= MPFR_PREC(a)) ? MPFR_PREC(a) : MPFR_EXP(a); MPFR_ASSERTD (precu + 1 <= MPFR_PREC_MAX); mpfr_set_prec (u, precu + 1); expu = (MPFR_EXP(a) > 0) ? MPFR_EXP(a) : 1; /* estimate Taylor series */ mpfr_ui_div (t, 1, a, MPFR_RNDA); /* t = 1/a * (1 + theta) */ mpfr_set (s, t, MPFR_RNDA); /* s = 1/a * (1 + theta) */ if (MPFR_IS_NEG(a)) { mpfr_init2 (s_abs, 32); mpfr_abs (s_abs, s, MPFR_RNDU); } for (k = 1;; k++) { mpfr_mul (t, t, x, MPFR_RNDU); /* t = x^k/(a * ... * (a+k-1)) * (1 + theta)^(2k) */ inex = mpfr_add_ui (u, a, k, MPFR_RNDZ); /* u = a+k exactly */ MPFR_ASSERTD(inex == 0); mpfr_div (t, t, u, MPFR_RNDA); /* t = x^k/(a * ... * (a+k)) * (1 + theta)^(2k+1) */ mpfr_add (s, s, t, MPFR_RNDZ); /* when s is zero, we consider ulp(s) = ulp(t) */ exps = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s); if (MPFR_IS_NEG(a)) { if (MPFR_IS_POS(t)) mpfr_add (s_abs, s_abs, t, MPFR_RNDU); else mpfr_sub (s_abs, s_abs, t, MPFR_RNDU); } /* we stop when |t| < ulp(s), u > 0 and |x/u| < 1/2, which ensures that the tail is at most 2*ulp(s) */ MPFR_ASSERTD (MPFR_NOTZERO(t)); if (MPFR_GET_EXP(t) + w <= exps && MPFR_IS_POS(u) && MPFR_GET_EXP(x) + 1 < MPFR_GET_EXP(u)) break; /* if there was an exponent shift in u, increase the precision of u so that mpfr_add_ui (u, a, k) remains exact */ if (MPFR_EXP(u) > expu) /* exponent shift in u */ { MPFR_ASSERTD(MPFR_EXP(u) == expu + 1); expu = MPFR_EXP(u); mpfr_set_prec (u, mpfr_get_prec (u) + 1); } } if (MPFR_IS_NEG(a)) { decay = MPFR_GET_EXP(s_abs) - MPFR_GET_EXP(s); mpfr_clear (s_abs); } /* For a > 0, since all terms are positive, we have s = S * (1 + theta)^(2k+3) with S being the infinite Taylor series. For a < 0, the error is bounded by that on the sum s_abs of absolute values of the terms, i.e., S_abs * [(1 + theta)^(2k+3) - 1]. Thus we can simply use the same error analysis as for a > 0, adding an error corresponding to the decay of exponent between s_abs and s. */ /* multiply by exp(-x) */ mpfr_exp (t, x, MPFR_RNDZ); /* t = exp(x) * (1+theta) */ mpfr_div (s, s, t, MPFR_RNDZ); /* s = * (1+theta)^(2k+5) */ /* multiply by x^a */ mpfr_pow (t, x, a, MPFR_RNDZ); /* t = x^a * (1+theta) */ mpfr_mul (s, s, t, MPFR_RNDZ); /* s = Gamma(a,x) * (1+theta)^(2k+7) */ /* Since |theta| < 2^(-w) using the Taylor expansion of log(1+x) we have log(1+theta) = theta1 with |theta1| < 1.16*2^(-w) for w >= 2, thus (1+theta)^(2k+7) = exp((2k+7)*theta1). Assuming 2k+7 = t*2^w for |t| < 0.5, we have |(2k+7)*theta1| = |t*2^w*theta1| < 0.58. For |u| < 0.58 we have |exp(u)-1| < 1.36*|u| thus |(1+theta)^(2k+7) - 1| < 1.36*0.58*(2k+7)/2^w < 0.79*(2k+7)/2^w. Since one ulp is at worst a relative error of 2^(1-w), the error on s is at most 2^(decay+1)*(2k+7) ulps. */ /* subtract from gamma(a) */ MPFR_BLOCK (flags, mpfr_gamma (t, a, MPFR_RNDZ)); MPFR_ASSERTN (!MPFR_OVERFLOW (flags)); /* FIXME: support overflow */ /* t = gamma(a) * (1+theta) */ e0 = MPFR_GET_EXP (t); e1 = (MPFR_IS_ZERO(s)) ? __gmpfr_emin : MPFR_GET_EXP (s); mpfr_sub (s, t, s, MPFR_RNDZ); /* if s is zero, we can assume ulp(s) = ulp(t), but anyway we won't be able to round */ e2 = (MPFR_IS_ZERO(s)) ? e0 : MPFR_GET_EXP (s); /* the final error is at most 1 ulp (for the final subtraction) + 2^(e0-e2) ulps # for the error in t + 2^(decay+1)*(2k+7) ulps * 2^(e1-e2) # for the error in gamma(a,x) */ e1 += decay + 1 + MPFR_INT_CEIL_LOG2 (2*k+7); /* Now the error is <= 1 + 2^(e0-e2) + 2^(e1-e2). Since the formula is symmetric in e0 and e1, we can assume without loss of generality e0 >= e1, then: if e0 = e1: err <= 1 + 2*2^(e0-e2) <= 2^(e0-e2+2) if e0 > e1: err <= 1 + 1.5*2^(e0-e2) <= 2^(e0-e2+1) if e0 > e2 <= 2^2 otherwise */ if (e0 == e1) { /* Check that e0 - e2 + 2 <= MPFR_EXP_MAX */ MPFR_ASSERTD (e2 >= 2 || e0 <= (MPFR_EXP_MAX - 2) + e2); /* Check that e0 - e2 + 2 >= MPFR_EXP_MIN */ MPFR_ASSERTD (e2 <= 2 || e0 >= MPFR_EXP_MIN + (e2 - 2)); err = e0 - e2 + 2; } else { e0 = (e0 > e1) ? e0 : e1; /* max(e0,e1) */ MPFR_ASSERTD (e0 <= e2 || e2 >= 1 || e0 <= (MPFR_EXP_MAX - 1) + e2); err = (e0 > e2) ? e0 - e2 + 1 : 2; } if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err, MPFR_PREC(y), rnd))) break; MPFR_ZIV_NEXT (loop, w); MPFR_GROUP_REPREC_2(group, w, s, t); } MPFR_ZIV_FREE (loop); mpfr_clear (u); inex = mpfr_set (y, s, rnd); MPFR_GROUP_CLEAR(group); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inex, rnd); } /* For a negative integer, we have (formula 6.5.19): gamma(-n,x) = (-1)^n/n! [E_1(x) - exp(-x) sum((-1)^j*j!/x^(j+1), j=0..n-1)] See also https://arxiv.org/pdf/1407.0349v1.pdf. Assumes 'a' is a negative integer. */ static int mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) { mpfr_t s, t, abs_a, neg_x; unsigned long j; mpfr_prec_t w; int inex; mpfr_exp_t exp_s, new_exp_s, exp_t, err_s, logj; MPFR_GROUP_DECL(group); MPFR_ZIV_DECL(loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_ASSERTD(mpfr_integer_p (a)); MPFR_ASSERTD(mpfr_cmp_ui (a, 0) < 0); MPFR_TMP_INIT_ABS(abs_a, a); /* below, theta represents any value such that |theta| <= 2^(-w) */ w = MPFR_PREC(y) + 10; /* initial working precision */ MPFR_SAVE_EXPO_MARK (expo); MPFR_GROUP_INIT_2(group, w, s, t); MPFR_ZIV_INIT (loop, w); for (;;) { /* we require |a| <= 2^(w-3) for the error analysis below */ if (MPFR_GET_EXP(a) + 3 > w) w = MPFR_GET_EXP(a) + 3; mpfr_ui_div (t, 1, x, MPFR_RNDN); /* t = 1/x * (1 + theta) */ mpfr_set (s, t, MPFR_RNDN); MPFR_ASSERTD (MPFR_NOTZERO(s)); exp_t = exp_s = MPFR_GET_EXP(s); /* max. exponent of s/t during loop */ new_exp_s = exp_s; for (j = 1; mpfr_cmp_ui (abs_a, j) > 0; j++) { /* invariant: t = (-1)^(j-1)*(j-1)!/x^j * (1 + theta)^(2j-1) */ mpfr_mul_ui (t, t, j, MPFR_RNDN); mpfr_neg (t, t, MPFR_RNDN); /* exact */ mpfr_div (t, t, x, MPFR_RNDN); /* now t = (-1)^j*j!/x^(j+1) * (1 + theta)^(2j+1). We have (1 + theta)^(2j+1) = exp((2j+1)*log(1+theta)). For |u| <= 1/2, we have |log(1+u)| <= 1.4 |u| thus: |(1+theta)^(2j+1)-1| <= max |exp(1.4*(2j+1)*u)-1| for |u|<=2^(-w). Now for |v| <= 1/2 we have |exp(v)-1| <= 0.7*|v| thus: |(1+theta)^(2j+1) - 1| <= 2*(2j+1)*2^(-w) as long as 1.4*(2j+1)*2^(-w) <= 1/2, which is true when j<2^(w-3). Since j < |a| it suffices that |a| <= 2^(w-3). In that case the rel. error on t is bounded by 2*(2j+1)*2^(-w), thus the error in ulps is bounded by 2*(2j+1) ulp(t). */ if (MPFR_IS_ZERO(t)) /* underflow on t */ break; if (MPFR_GET_EXP(t) > exp_t) exp_t = MPFR_GET_EXP(t); mpfr_add (s, s, t, MPFR_RNDN); /* if s is zero, we can assume its ulp is that of t */ new_exp_s = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s); if (new_exp_s > exp_s) exp_s = new_exp_s; } /* the error on s is bounded by (j-1) * 2^(exp_s - EXP(s)) * 1/2 for the mpfr_add roundings, plus sum(2*(2i+1), i=1..j-1) * 2^(exp_t - EXP(s)) for the error on t. The latter sum is (2*j^2-2) * 2^(exp_t - EXP(s)). */ logj = MPFR_INT_CEIL_LOG2(j); exp_s += logj - 1; exp_t += 1 + 2 * logj; /* now the error on s is bounded by 2^(exp_s-EXP(s))+2^(exp_t-EXP(s)) */ exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1; err_s = exp_s - new_exp_s; /* now the error on the sum S := sum((-1)^j*j!/x^(j+1), j=0..n-1) is bounded by 2^err_s ulp(s) */ MPFR_TMP_INIT_NEG(neg_x, x); mpfr_exp (t, neg_x, MPFR_RNDN); /* t = exp(-x) * (1 + theta) */ mpfr_mul (s, s, t, MPFR_RNDN); if (MPFR_IS_ZERO(s)) { MPFR_ASSERTD (MPFR_NOTZERO(t)); new_exp_s += MPFR_GET_EXP(t); } /* s = exp(-x) * (S +/- 2^err_s ulp(S)) * (1 + theta)^2. = exp(-x) * (S +/- 2^err_s ulp(S)) * (1 +/- 3 ulp(1)) The error on s is bounded by: exp(-x) * [2^err_s*ulp(S) + S*3*ulp(1) + 2^err_s*ulp(S)*3*ulp(1)] <= ulp(s) * [2^(err_s+1) + 6 + 1] <= ulp(s) * 2^(err_s+2) as long as err_s >= 2. */ err_s = (err_s >= 2) ? err_s + 2 : 4; /* now the error on s is bounded by 2^err_s ulp(s) */ mpfr_eint (t, neg_x, MPFR_RNDN); /* t = -E1(-x) * (1 + theta) */ mpfr_neg (t, t, MPFR_RNDN); /* exact */ exp_s = (MPFR_IS_ZERO(s)) ? new_exp_s : MPFR_GET_EXP(s); MPFR_ASSERTD (MPFR_NOTZERO(t)); exp_t = MPFR_GET_EXP(t); mpfr_sub (s, t, s, MPFR_RNDN); /* E_1(x) - exp(-x) * S */ if (MPFR_IS_ZERO(s)) /* cancellation: increase working precision */ goto next_w; /* err(s) <= 1/2 * ulp(s) [mpfr_sub] + 2^err_s * 2^(exp_s-EXP(s)) * ulp(s) [previous error on s] + 1/2 * 2^(exp_t-EXP(s)) * ulp(s) [error on t] */ exp_s += err_s; exp_t -= 1; exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1; MPFR_ASSERTD (MPFR_NOTZERO(s)); err_s = exp_s - MPFR_GET_EXP(s); /* err(s) <= 1/2 * ulp(s) + 2^err_s * ulp(s) */ /* divide by n! */ mpfr_gamma (t, abs_a, MPFR_RNDN); /* t = (n-1)! * (1 + theta) */ mpfr_mul (t, t, abs_a, MPFR_RNDN); /* t = n! * (1 + theta)^2 */ mpfr_div (s, s, t, MPFR_RNDN); /* since (1 + theta)^2 converts to an error of at most 3 ulps for w >= 2, the final error is at most: 2 * (1/2 + 2^err_s) * ulp(s) [error on previous s] + 2 * 3 * ulp(s) [error on t] + 1 * ulp(s) [product of errors] = ulp(s) * (2^(err_s+1) + 8) */ err_s = (err_s >= 2) ? err_s + 1 : 4; /* the final error is bounded by 2^err_s * ulp(s) */ /* Is there a better way to compute (-1)^n? */ mpfr_set_si (t, -1, MPFR_RNDN); mpfr_pow (t, t, abs_a, MPFR_RNDN); if (MPFR_IS_NEG(t)) mpfr_neg (s, s, MPFR_RNDN); if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err_s, MPFR_PREC(y), rnd))) break; next_w: MPFR_ZIV_NEXT (loop, w); MPFR_GROUP_REPREC_2(group, w, s, t); } MPFR_ZIV_FREE (loop); inex = mpfr_set (y, s, rnd); MPFR_GROUP_CLEAR(group); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inex, rnd); }