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Diffstat (limited to 'mpfr/src/lngamma.c')
| -rw-r--r-- | mpfr/src/lngamma.c | 738 |
1 files changed, 738 insertions, 0 deletions
diff --git a/mpfr/src/lngamma.c b/mpfr/src/lngamma.c new file mode 100644 index 0000000000..c400bd8d3a --- /dev/null +++ b/mpfr/src/lngamma.c @@ -0,0 +1,738 @@ +/* mpfr_lngamma -- lngamma function + +Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 +http://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" + +/* given a precision p, return alpha, such that the argument reduction + will use k = alpha*p*log(2). + + Warning: we should always have alpha >= log(2)/(2Pi) ~ 0.11, + and the smallest value of alpha multiplied by the smallest working + precision should be >= 4. +*/ +static void +mpfr_gamma_alpha (mpfr_t s, mpfr_prec_t p) +{ + if (p <= 100) + mpfr_set_ui_2exp (s, 614, -10, MPFR_RNDN); /* about 0.6 */ + else if (p <= 500) + mpfr_set_ui_2exp (s, 819, -10, MPFR_RNDN); /* about 0.8 */ + else if (p <= 1000) + mpfr_set_ui_2exp (s, 1331, -10, MPFR_RNDN); /* about 1.3 */ + else if (p <= 2000) + mpfr_set_ui_2exp (s, 1741, -10, MPFR_RNDN); /* about 1.7 */ + else if (p <= 5000) + mpfr_set_ui_2exp (s, 2253, -10, MPFR_RNDN); /* about 2.2 */ + else if (p <= 10000) + mpfr_set_ui_2exp (s, 3482, -10, MPFR_RNDN); /* about 3.4 */ + else + mpfr_set_ui_2exp (s, 9, -1, MPFR_RNDN); /* 4.5 */ +} + +#ifdef IS_GAMMA + +/* This function is called in case of intermediate overflow/underflow. + The s1 and s2 arguments are temporary MPFR numbers, having the + working precision. If the result could be determined, then the + flags are updated via pexpo, y is set to the result, and the + (non-zero) ternary value is returned. Otherwise 0 is returned + in order to perform the next Ziv iteration. */ +static int +mpfr_explgamma (mpfr_ptr y, mpfr_srcptr x, mpfr_save_expo_t *pexpo, + mpfr_ptr s1, mpfr_ptr s2, mpfr_rnd_t rnd) +{ + mpfr_t t1, t2; + int inex1, inex2, sign; + MPFR_BLOCK_DECL (flags1); + MPFR_BLOCK_DECL (flags2); + MPFR_GROUP_DECL (group); + + MPFR_BLOCK (flags1, inex1 = mpfr_lgamma (s1, &sign, x, MPFR_RNDD)); + MPFR_ASSERTN (inex1 != 0); + /* s1 = RNDD(lngamma(x)), inexact */ + if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags1))) + { + if (MPFR_SIGN (s1) > 0) + { + MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, MPFR_FLAGS_OVERFLOW); + return mpfr_overflow (y, rnd, sign); + } + else + { + MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, MPFR_FLAGS_UNDERFLOW); + return mpfr_underflow (y, rnd == MPFR_RNDN ? MPFR_RNDZ : rnd, sign); + } + } + + mpfr_set (s2, s1, MPFR_RNDN); /* exact */ + mpfr_nextabove (s2); /* v = RNDU(lngamma(z0)) */ + + if (sign < 0) + rnd = MPFR_INVERT_RND (rnd); /* since the result with be negated */ + MPFR_GROUP_INIT_2 (group, MPFR_PREC (y), t1, t2); + MPFR_BLOCK (flags1, inex1 = mpfr_exp (t1, s1, rnd)); + MPFR_BLOCK (flags2, inex2 = mpfr_exp (t2, s2, rnd)); + /* t1 is the rounding with mode 'rnd' of a lower bound on |Gamma(x)|, + t2 is the rounding with mode 'rnd' of an upper bound, thus if both + are equal, so is the wanted result. If t1 and t2 differ or the flags + differ, at some point of Ziv's loop they should agree. */ + if (mpfr_equal_p (t1, t2) && flags1 == flags2) + { + MPFR_ASSERTN ((inex1 > 0 && inex2 > 0) || (inex1 < 0 && inex2 < 0)); + mpfr_set4 (y, t1, MPFR_RNDN, sign); /* exact */ + if (sign < 0) + inex1 = - inex1; + MPFR_SAVE_EXPO_UPDATE_FLAGS (*pexpo, flags1); + } + else + inex1 = 0; /* couldn't determine the result */ + MPFR_GROUP_CLEAR (group); + + return inex1; +} + +#else + +static int +unit_bit (mpfr_srcptr x) +{ + mpfr_exp_t expo; + mpfr_prec_t prec; + mp_limb_t x0; + + expo = MPFR_GET_EXP (x); + if (expo <= 0) + return 0; /* |x| < 1 */ + + prec = MPFR_PREC (x); + if (expo > prec) + return 0; /* y is a multiple of 2^(expo-prec), thus an even integer */ + + /* Now, the unit bit is represented. */ + + prec = MPFR_PREC2LIMBS (prec) * GMP_NUMB_BITS - expo; + /* number of represented fractional bits (including the trailing 0's) */ + + x0 = *(MPFR_MANT (x) + prec / GMP_NUMB_BITS); + /* limb containing the unit bit */ + + return (x0 >> (prec % GMP_NUMB_BITS)) & 1; +} + +#endif + +/* lngamma(x) = log(gamma(x)). + We use formula [6.1.40] from Abramowitz&Stegun: + lngamma(z) = (z-1/2)*log(z) - z + 1/2*log(2*Pi) + + sum (Bernoulli[2m]/(2m)/(2m-1)/z^(2m-1),m=1..infinity) + According to [6.1.42], if the sum is truncated after m=n, the error + R_n(z) is bounded by |B[2n+2]|*K(z)/(2n+1)/(2n+2)/|z|^(2n+1) + where K(z) = max (z^2/(u^2+z^2)) for u >= 0. + For z real, |K(z)| <= 1 thus R_n(z) is bounded by the first neglected term. + */ +#ifdef IS_GAMMA +#define GAMMA_FUNC mpfr_gamma_aux +#else +#define GAMMA_FUNC mpfr_lngamma_aux +#endif + +static int +GAMMA_FUNC (mpfr_ptr y, mpfr_srcptr z0, mpfr_rnd_t rnd) +{ + mpfr_prec_t precy, w; /* working precision */ + mpfr_t s, t, u, v, z; + unsigned long m, k, maxm; + mpz_t *INITIALIZED(B); /* variable B declared as initialized */ + int compared; + int inexact = 0; /* 0 means: result y not set yet */ + mpfr_exp_t err_s, err_t; + unsigned long Bm = 0; /* number of allocated B[] */ + unsigned long oldBm; + double d; + MPFR_SAVE_EXPO_DECL (expo); + MPFR_ZIV_DECL (loop); + + compared = mpfr_cmp_ui (z0, 1); + + MPFR_SAVE_EXPO_MARK (expo); + +#ifndef IS_GAMMA /* lngamma or lgamma */ + if (compared == 0 || (compared > 0 && mpfr_cmp_ui (z0, 2) == 0)) + { + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_set_ui (y, 0, MPFR_RNDN); /* lngamma(1 or 2) = +0 */ + } + + /* Deal here with tiny inputs. We have for -0.3 <= x <= 0.3: + - log|x| - gamma*x <= log|gamma(x)| <= - log|x| - gamma*x + x^2 */ + if (MPFR_EXP(z0) <= - (mpfr_exp_t) MPFR_PREC(y)) + { + mpfr_t l, h, g; + int ok, inex1, inex2; + mpfr_prec_t prec = MPFR_PREC(y) + 14; + MPFR_ZIV_DECL (loop); + + MPFR_ZIV_INIT (loop, prec); + do + { + mpfr_init2 (l, prec); + if (MPFR_IS_POS(z0)) + { + mpfr_log (l, z0, MPFR_RNDU); /* upper bound for log(z0) */ + mpfr_init2 (h, MPFR_PREC(l)); + } + else + { + mpfr_init2 (h, MPFR_PREC(z0)); + mpfr_neg (h, z0, MPFR_RNDN); /* exact */ + mpfr_log (l, h, MPFR_RNDU); /* upper bound for log(-z0) */ + mpfr_set_prec (h, MPFR_PREC(l)); + } + mpfr_neg (l, l, MPFR_RNDD); /* lower bound for -log(|z0|) */ + mpfr_set (h, l, MPFR_RNDD); /* exact */ + mpfr_nextabove (h); /* upper bound for -log(|z0|), avoids two calls + to mpfr_log */ + mpfr_init2 (g, MPFR_PREC(l)); + /* if z0>0, we need an upper approximation of Euler's constant + for the left bound */ + mpfr_const_euler (g, MPFR_IS_POS(z0) ? MPFR_RNDU : MPFR_RNDD); + mpfr_mul (g, g, z0, MPFR_RNDD); + mpfr_sub (l, l, g, MPFR_RNDD); + mpfr_const_euler (g, MPFR_IS_POS(z0) ? MPFR_RNDD : MPFR_RNDU); /* cached */ + mpfr_mul (g, g, z0, MPFR_RNDU); + mpfr_sub (h, h, g, MPFR_RNDD); + mpfr_mul (g, z0, z0, MPFR_RNDU); + mpfr_add (h, h, g, MPFR_RNDU); + inex1 = mpfr_prec_round (l, MPFR_PREC(y), rnd); + inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd); + /* Caution: we not only need l = h, but both inexact flags should + agree. Indeed, one of the inexact flags might be zero. In that + case if we assume lngamma(z0) cannot be exact, the other flag + should be correct. We are conservative here and request that both + inexact flags agree. */ + ok = SAME_SIGN (inex1, inex2) && mpfr_cmp (l, h) == 0; + if (ok) + mpfr_set (y, h, rnd); /* exact */ + mpfr_clear (l); + mpfr_clear (h); + mpfr_clear (g); + if (ok) + { + MPFR_ZIV_FREE (loop); + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (y, inex1, rnd); + } + /* since we have log|gamma(x)| = - log|x| - gamma*x + O(x^2), + if x ~ 2^(-n), then we have a n-bit approximation, thus + we can try again with a working precision of n bits, + especially when n >> PREC(y). + Otherwise we would use the reflection formula evaluating x-1, + which would need precision n. */ + MPFR_ZIV_NEXT (loop, prec); + } + while (prec <= -MPFR_EXP(z0)); + MPFR_ZIV_FREE (loop); + } +#endif + + precy = MPFR_PREC(y); + + mpfr_init2 (s, MPFR_PREC_MIN); + mpfr_init2 (t, MPFR_PREC_MIN); + mpfr_init2 (u, MPFR_PREC_MIN); + mpfr_init2 (v, MPFR_PREC_MIN); + mpfr_init2 (z, MPFR_PREC_MIN); + + if (compared < 0) + { + mpfr_exp_t err_u; + + /* use reflection formula: + gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x) + thus lngamma(x) = log(Pi*(x-1)/sin(Pi*(2-x))) - lngamma(2-x) */ + + w = precy + MPFR_INT_CEIL_LOG2 (precy); + w += MPFR_INT_CEIL_LOG2 (w) + 14; + MPFR_ZIV_INIT (loop, w); + while (1) + { + MPFR_ASSERTD(w >= 3); + mpfr_set_prec (s, w); + mpfr_set_prec (t, w); + mpfr_set_prec (u, w); + mpfr_set_prec (v, w); + /* In the following, we write r for a real of absolute value + at most 2^(-w). Different instances of r may represent different + values. */ + mpfr_ui_sub (s, 2, z0, MPFR_RNDD); /* s = (2-z0) * (1+2r) >= 1 */ + mpfr_const_pi (t, MPFR_RNDN); /* t = Pi * (1+r) */ + mpfr_lngamma (u, s, MPFR_RNDN); /* lngamma(2-x) */ + /* Let s = (2-z0) + h. By construction, -(2-z0)*2^(1-w) <= h <= 0. + We have lngamma(s) = lngamma(2-z0) + h*Psi(z), z in [2-z0+h,2-z0]. + Since 2-z0+h = s >= 1 and |Psi(x)| <= max(1,log(x)) for x >= 1, + the error on u is bounded by + ulp(u)/2 + (2-z0)*max(1,log(2-z0))*2^(1-w) + = (1/2 + (2-z0)*max(1,log(2-z0))*2^(1-E(u))) ulp(u) */ + d = (double) MPFR_GET_EXP(s) * 0.694; /* upper bound for log(2-z0) */ + err_u = MPFR_GET_EXP(s) + __gmpfr_ceil_log2 (d) + 1 - MPFR_GET_EXP(u); + err_u = (err_u >= 0) ? err_u + 1 : 0; + /* now the error on u is bounded by 2^err_u ulps */ + + mpfr_mul (s, s, t, MPFR_RNDN); /* Pi*(2-x) * (1+r)^4 */ + err_s = MPFR_GET_EXP(s); /* 2-x <= 2^err_s */ + mpfr_sin (s, s, MPFR_RNDN); /* sin(Pi*(2-x)) */ + /* the error on s is bounded by 1/2*ulp(s) + [(1+2^(-w))^4-1]*(2-x) + <= 1/2*ulp(s) + 5*2^(-w)*(2-x) for w >= 3 + <= (1/2 + 5 * 2^(-E(s)) * (2-x)) ulp(s) */ + err_s += 3 - MPFR_GET_EXP(s); + err_s = (err_s >= 0) ? err_s + 1 : 0; + /* the error on s is bounded by 2^err_s ulp(s), thus by + 2^(err_s+1)*2^(-w)*|s| since ulp(s) <= 2^(1-w)*|s|. + Now n*2^(-w) can always be written |(1+r)^n-1| for some + |r|<=2^(-w), thus taking n=2^(err_s+1) we see that + |S - s| <= |(1+r)^(2^(err_s+1))-1| * |s|, where S is the + true value. + In fact if ulp(s) <= ulp(S) the same inequality holds for + |S| instead of |s| in the right hand side, i.e., we can + write s = (1+r)^(2^(err_s+1)) * S. + But if ulp(S) < ulp(s), we need to add one ``bit'' to the error, + to get s = (1+r)^(2^(err_s+2)) * S. This is true since with + E = n*2^(-w) we have |s - S| <= E * |s|, thus + |s - S| <= E/(1-E) * |S|. + Now E/(1-E) is bounded by 2E as long as E<=1/2, + and 2E can be written (1+r)^(2n)-1 as above. + */ + err_s += 2; /* exponent of relative error */ + + mpfr_sub_ui (v, z0, 1, MPFR_RNDN); /* v = (x-1) * (1+r) */ + mpfr_mul (v, v, t, MPFR_RNDN); /* v = Pi*(x-1) * (1+r)^3 */ + mpfr_div (v, v, s, MPFR_RNDN); /* Pi*(x-1)/sin(Pi*(2-x)) */ + mpfr_abs (v, v, MPFR_RNDN); + /* (1+r)^(3+2^err_s+1) */ + err_s = (err_s <= 1) ? 3 : err_s + 1; + /* now (1+r)^M with M <= 2^err_s */ + mpfr_log (v, v, MPFR_RNDN); + /* log(v*(1+e)) = log(v)+log(1+e) where |e| <= 2^(err_s-w). + Since |log(1+e)| <= 2*e for |e| <= 1/4, the error on v is + bounded by ulp(v)/2 + 2^(err_s+1-w). */ + if (err_s + 2 > w) + { + w += err_s + 2; + } + else + { + err_s += 1 - MPFR_GET_EXP(v); + err_s = (err_s >= 0) ? err_s + 1 : 0; + /* the error on v is bounded by 2^err_s ulps */ + err_u += MPFR_GET_EXP(u); /* absolute error on u */ + err_s += MPFR_GET_EXP(v); /* absolute error on v */ + mpfr_sub (s, v, u, MPFR_RNDN); + /* the total error on s is bounded by ulp(s)/2 + 2^(err_u-w) + + 2^(err_s-w) <= ulp(s)/2 + 2^(max(err_u,err_s)+1-w) */ + err_s = (err_s >= err_u) ? err_s : err_u; + err_s += 1 - MPFR_GET_EXP(s); /* error is 2^err_s ulp(s) */ + err_s = (err_s >= 0) ? err_s + 1 : 0; + if (mpfr_can_round (s, w - err_s, MPFR_RNDN, MPFR_RNDZ, precy + + (rnd == MPFR_RNDN))) + goto end; + } + MPFR_ZIV_NEXT (loop, w); + } + MPFR_ZIV_FREE (loop); + } + + /* now z0 > 1 */ + + MPFR_ASSERTD (compared > 0); + + /* since k is O(w), the value of log(z0*...*(z0+k-1)) is about w*log(w), + so there is a cancellation of ~log(w) in the argument reconstruction */ + w = precy + MPFR_INT_CEIL_LOG2 (precy); + w += MPFR_INT_CEIL_LOG2 (w) + 13; + MPFR_ZIV_INIT (loop, w); + while (1) + { + MPFR_ASSERTD (w >= 3); + + /* argument reduction: we compute gamma(z0 + k), where the series + has error term B_{2n}/(z0+k)^(2n) ~ (n/(Pi*e*(z0+k)))^(2n) + and we need k steps of argument reconstruction. Assuming k is large + with respect to z0, and k = n, we get 1/(Pi*e)^(2n) ~ 2^(-w), i.e., + k ~ w*log(2)/2/log(Pi*e) ~ 0.1616 * w. + However, since the series is more expensive to compute, the optimal + value seems to be k ~ 4.5 * w experimentally. */ + mpfr_set_prec (s, 53); + mpfr_gamma_alpha (s, w); + mpfr_set_ui_2exp (s, 9, -1, MPFR_RNDU); + mpfr_mul_ui (s, s, w, MPFR_RNDU); + if (mpfr_cmp (z0, s) < 0) + { + mpfr_sub (s, s, z0, MPFR_RNDU); + k = mpfr_get_ui (s, MPFR_RNDU); + if (k < 3) + k = 3; + } + else + k = 3; + + mpfr_set_prec (s, w); + mpfr_set_prec (t, w); + mpfr_set_prec (u, w); + mpfr_set_prec (v, w); + mpfr_set_prec (z, w); + + mpfr_add_ui (z, z0, k, MPFR_RNDN); + /* z = (z0+k)*(1+t1) with |t1| <= 2^(-w) */ + + /* z >= 4 ensures the relative error on log(z) is small, + and also (z-1/2)*log(z)-z >= 0 */ + MPFR_ASSERTD (mpfr_cmp_ui (z, 4) >= 0); + + mpfr_log (s, z, MPFR_RNDN); /* log(z) */ + /* we have s = log((z0+k)*(1+t1))*(1+t2) with |t1|, |t2| <= 2^(-w). + Since w >= 2 and z0+k >= 4, we can write log((z0+k)*(1+t1)) + = log(z0+k) * (1+t3) with |t3| <= 2^(-w), thus we have + s = log(z0+k) * (1+t4)^2 with |t4| <= 2^(-w) */ + mpfr_mul_2ui (t, z, 1, MPFR_RNDN); /* t = 2z * (1+t5) */ + mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* t = 2z-1 * (1+t6)^3 */ + /* since we can write 2z*(1+t5) = (2z-1)*(1+t5') with + t5' = 2z/(2z-1) * t5, thus |t5'| <= 8/7 * t5 */ + mpfr_mul (s, s, t, MPFR_RNDN); /* (2z-1)*log(z) * (1+t7)^6 */ + mpfr_div_2ui (s, s, 1, MPFR_RNDN); /* (z-1/2)*log(z) * (1+t7)^6 */ + mpfr_sub (s, s, z, MPFR_RNDN); /* (z-1/2)*log(z)-z */ + /* s = [(z-1/2)*log(z)-z]*(1+u)^14, s >= 1/2 */ + + mpfr_ui_div (u, 1, z, MPFR_RNDN); /* 1/z * (1+u), u <= 1/4 since z >= 4 */ + + /* the first term is B[2]/2/z = 1/12/z: t=1/12/z, C[2]=1 */ + mpfr_div_ui (t, u, 12, MPFR_RNDN); /* 1/(12z) * (1+u)^2, t <= 3/128 */ + mpfr_set (v, t, MPFR_RNDN); /* (1+u)^2, v < 2^(-5) */ + mpfr_add (s, s, v, MPFR_RNDN); /* (1+u)^15 */ + + mpfr_mul (u, u, u, MPFR_RNDN); /* 1/z^2 * (1+u)^3 */ + + if (Bm == 0) + { + B = mpfr_bernoulli_internal ((mpz_t *) 0, 0); + B = mpfr_bernoulli_internal (B, 1); + Bm = 2; + } + + /* m <= maxm ensures that 2*m*(2*m+1) <= ULONG_MAX */ + maxm = 1UL << (GMP_NUMB_BITS / 2 - 1); + + /* s:(1+u)^15, t:(1+u)^2, t <= 3/128 */ + + for (m = 2; MPFR_GET_EXP(v) + (mpfr_exp_t) w >= MPFR_GET_EXP(s); m++) + { + mpfr_mul (t, t, u, MPFR_RNDN); /* (1+u)^(10m-14) */ + if (m <= maxm) + { + mpfr_mul_ui (t, t, 2*(m-1)*(2*m-3), MPFR_RNDN); + mpfr_div_ui (t, t, 2*m*(2*m-1), MPFR_RNDN); + mpfr_div_ui (t, t, 2*m*(2*m+1), MPFR_RNDN); + } + else + { + mpfr_mul_ui (t, t, 2*(m-1), MPFR_RNDN); + mpfr_mul_ui (t, t, 2*m-3, MPFR_RNDN); + mpfr_div_ui (t, t, 2*m, MPFR_RNDN); + mpfr_div_ui (t, t, 2*m-1, MPFR_RNDN); + mpfr_div_ui (t, t, 2*m, MPFR_RNDN); + mpfr_div_ui (t, t, 2*m+1, MPFR_RNDN); + } + /* (1+u)^(10m-8) */ + /* invariant: t=1/(2m)/(2m-1)/z^(2m-1)/(2m+1)! */ + if (Bm <= m) + { + B = mpfr_bernoulli_internal (B, m); /* B[2m]*(2m+1)!, exact */ + Bm ++; + } + mpfr_mul_z (v, t, B[m], MPFR_RNDN); /* (1+u)^(10m-7) */ + MPFR_ASSERTD(MPFR_GET_EXP(v) <= - (2 * m + 3)); + mpfr_add (s, s, v, MPFR_RNDN); + } + /* m <= 1/2*Pi*e*z ensures that |v[m]| < 1/2^(2m+3) */ + MPFR_ASSERTD ((double) m <= 4.26 * mpfr_get_d (z, MPFR_RNDZ)); + + /* We have sum([(1+u)^(10m-7)-1]*1/2^(2m+3), m=2..infinity) + <= 1.46*u for u <= 2^(-3). + We have 0 < lngamma(z) - [(z - 1/2) ln(z) - z + 1/2 ln(2 Pi)] < 0.021 + for z >= 4, thus since the initial s >= 0.85, the different values of + s differ by at most one binade, and the total rounding error on s + in the for-loop is bounded by 2*(m-1)*ulp(final_s). + The error coming from the v's is bounded by + 1.46*2^(-w) <= 2*ulp(final_s). + Thus the total error so far is bounded by [(1+u)^15-1]*s+2m*ulp(s) + <= (2m+47)*ulp(s). + Taking into account the truncation error (which is bounded by the last + term v[] according to 6.1.42 in A&S), the bound is (2m+48)*ulp(s). + */ + + /* add 1/2*log(2*Pi) and subtract log(z0*(z0+1)*...*(z0+k-1)) */ + mpfr_const_pi (v, MPFR_RNDN); /* v = Pi*(1+u) */ + mpfr_mul_2ui (v, v, 1, MPFR_RNDN); /* v = 2*Pi * (1+u) */ + if (k) + { + unsigned long l; + mpfr_set (t, z0, MPFR_RNDN); /* t = z0*(1+u) */ + for (l = 1; l < k; l++) + { + mpfr_add_ui (u, z0, l, MPFR_RNDN); /* u = (z0+l)*(1+u) */ + mpfr_mul (t, t, u, MPFR_RNDN); /* (1+u)^(2l+1) */ + } + /* now t: (1+u)^(2k-1) */ + /* instead of computing log(sqrt(2*Pi)/t), we compute + 1/2*log(2*Pi/t^2), which trades a square root for a square */ + mpfr_mul (t, t, t, MPFR_RNDN); /* (z0*...*(z0+k-1))^2, (1+u)^(4k-1) */ + mpfr_div (v, v, t, MPFR_RNDN); + /* 2*Pi/(z0*...*(z0+k-1))^2 (1+u)^(4k+1) */ + } +#ifdef IS_GAMMA + err_s = MPFR_GET_EXP(s); + mpfr_exp (s, s, MPFR_RNDN); + /* If s is +Inf, we compute exp(lngamma(z0)). */ + if (mpfr_inf_p (s)) + { + inexact = mpfr_explgamma (y, z0, &expo, s, t, rnd); + if (inexact) + goto end0; + else + goto ziv_next; + } + /* before the exponential, we have s = s0 + h where + |h| <= (2m+48)*ulp(s), thus exp(s0) = exp(s) * exp(-h). + For |h| <= 1/4, we have |exp(h)-1| <= 1.2*|h| thus + |exp(s) - exp(s0)| <= 1.2 * exp(s) * (2m+48)* 2^(EXP(s)-w). */ + d = 1.2 * (2.0 * (double) m + 48.0); + /* the error on s is bounded by d*2^err_s * 2^(-w) */ + mpfr_sqrt (t, v, MPFR_RNDN); + /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1), + thus t = sqrt(v0)*(1+u)^(2k+3/2). */ + mpfr_mul (s, s, t, MPFR_RNDN); + /* the error on input s is bounded by (1+u)^(d*2^err_s), + and that on t is (1+u)^(2k+3/2), thus the + total error is (1+u)^(d*2^err_s+2k+5/2) */ + err_s += __gmpfr_ceil_log2 (d); + err_t = __gmpfr_ceil_log2 (2.0 * (double) k + 2.5); + err_s = (err_s >= err_t) ? err_s + 1 : err_t + 1; +#else + mpfr_log (t, v, MPFR_RNDN); + /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1), + thus log(v) = log(v0) + (4k+1)*log(1+u). Since |log(1+u)/u| <= 1.07 + for |u| <= 2^(-3), the absolute error on log(v) is bounded by + 1.07*(4k+1)*u, and the rounding error by ulp(t). */ + mpfr_div_2ui (t, t, 1, MPFR_RNDN); + /* the error on t is now bounded by ulp(t) + 0.54*(4k+1)*2^(-w). + We have sqrt(2*Pi)/(z0*(z0+1)*...*(z0+k-1)) <= sqrt(2*Pi)/k! <= 0.5 + since k>=3, thus t <= -0.5 and ulp(t) >= 2^(-w). + Thus the error on t is bounded by (2.16*k+1.54)*ulp(t). */ + err_t = MPFR_GET_EXP(t) + (mpfr_exp_t) + __gmpfr_ceil_log2 (2.2 * (double) k + 1.6); + err_s = MPFR_GET_EXP(s) + (mpfr_exp_t) + __gmpfr_ceil_log2 (2.0 * (double) m + 48.0); + mpfr_add (s, s, t, MPFR_RNDN); /* this is a subtraction in fact */ + /* the final error in ulp(s) is + <= 1 + 2^(err_t-EXP(s)) + 2^(err_s-EXP(s)) + <= 2^(1+max(err_t,err_s)-EXP(s)) if err_t <> err_s + <= 2^(2+max(err_t,err_s)-EXP(s)) if err_t = err_s */ + err_s = (err_t == err_s) ? 1 + err_s : ((err_t > err_s) ? err_t : err_s); + err_s += 1 - MPFR_GET_EXP(s); +#endif + if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err_s, precy, rnd))) + break; +#ifdef IS_GAMMA + ziv_next: +#endif + MPFR_ZIV_NEXT (loop, w); + } + +#ifdef IS_GAMMA + end0: +#endif + oldBm = Bm; + while (Bm--) + mpz_clear (B[Bm]); + (*__gmp_free_func) (B, oldBm * sizeof (mpz_t)); + + end: + if (inexact == 0) + inexact = mpfr_set (y, s, rnd); + MPFR_ZIV_FREE (loop); + + mpfr_clear (s); + mpfr_clear (t); + mpfr_clear (u); + mpfr_clear (v); + mpfr_clear (z); + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (y, inexact, rnd); +} + +#ifndef IS_GAMMA + +int +mpfr_lngamma (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) +{ + int inex; + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), + ("y[%Pu]=%.*Rg inexact=%d", + mpfr_get_prec (y), mpfr_log_prec, y, inex)); + + /* special cases */ + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) + { + if (MPFR_IS_NAN (x) || MPFR_IS_NEG (x)) + { + MPFR_SET_NAN (y); + MPFR_RET_NAN; + } + else /* lngamma(+Inf) = lngamma(+0) = +Inf */ + { + if (MPFR_IS_ZERO (x)) + mpfr_set_divby0 (); + MPFR_SET_INF (y); + MPFR_SET_POS (y); + MPFR_RET (0); /* exact */ + } + } + + /* if x < 0 and -2k-1 <= x <= -2k, then lngamma(x) = NaN */ + if (MPFR_IS_NEG (x) && (unit_bit (x) == 0 || mpfr_integer_p (x))) + { + MPFR_SET_NAN (y); + MPFR_RET_NAN; + } + + inex = mpfr_lngamma_aux (y, x, rnd); + return inex; +} + +int +mpfr_lgamma (mpfr_ptr y, int *signp, mpfr_srcptr x, mpfr_rnd_t rnd) +{ + int inex; + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), + ("y[%Pu]=%.*Rg signp=%d inexact=%d", + mpfr_get_prec (y), mpfr_log_prec, y, *signp, inex)); + + *signp = 1; /* most common case */ + + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) + { + if (MPFR_IS_NAN (x)) + { + MPFR_SET_NAN (y); + MPFR_RET_NAN; + } + else + { + if (MPFR_IS_ZERO (x)) + mpfr_set_divby0 (); + *signp = MPFR_INT_SIGN (x); + MPFR_SET_INF (y); + MPFR_SET_POS (y); + MPFR_RET (0); + } + } + + if (MPFR_IS_NEG (x)) + { + if (mpfr_integer_p (x)) + { + MPFR_SET_INF (y); + MPFR_SET_POS (y); + mpfr_set_divby0 (); + MPFR_RET (0); + } + + if (unit_bit (x) == 0) + *signp = -1; + + /* For tiny negative x, we have gamma(x) = 1/x - euler + O(x), + thus |gamma(x)| = -1/x + euler + O(x), and + log |gamma(x)| = -log(-x) - euler*x + O(x^2). + More precisely we have for -0.4 <= x < 0: + -log(-x) <= log |gamma(x)| <= -log(-x) - x. + Since log(x) is not representable, we may have an instance of the + Table Maker Dilemma. The only way to ensure correct rounding is to + compute an interval [l,h] such that l <= -log(-x) and + -log(-x) - x <= h, and check whether l and h round to the same number + for the target precision and rounding modes. */ + if (MPFR_EXP(x) + 1 <= - (mpfr_exp_t) MPFR_PREC(y)) + /* since PREC(y) >= 1, this ensures EXP(x) <= -2, + thus |x| <= 0.25 < 0.4 */ + { + mpfr_t l, h; + int ok, inex2; + mpfr_prec_t w = MPFR_PREC (y) + 14; + mpfr_exp_t expl; + + while (1) + { + mpfr_init2 (l, w); + mpfr_init2 (h, w); + /* we want a lower bound on -log(-x), thus an upper bound + on log(-x), thus an upper bound on -x. */ + mpfr_neg (l, x, MPFR_RNDU); /* upper bound on -x */ + mpfr_log (l, l, MPFR_RNDU); /* upper bound for log(-x) */ + mpfr_neg (l, l, MPFR_RNDD); /* lower bound for -log(-x) */ + mpfr_neg (h, x, MPFR_RNDD); /* lower bound on -x */ + mpfr_log (h, h, MPFR_RNDD); /* lower bound on log(-x) */ + mpfr_neg (h, h, MPFR_RNDU); /* upper bound for -log(-x) */ + mpfr_sub (h, h, x, MPFR_RNDU); /* upper bound for -log(-x) - x */ + inex = mpfr_prec_round (l, MPFR_PREC (y), rnd); + inex2 = mpfr_prec_round (h, MPFR_PREC (y), rnd); + /* Caution: we not only need l = h, but both inexact flags + should agree. Indeed, one of the inexact flags might be + zero. In that case if we assume ln|gamma(x)| cannot be + exact, the other flag should be correct. We are conservative + here and request that both inexact flags agree. */ + ok = SAME_SIGN (inex, inex2) && mpfr_equal_p (l, h); + if (ok) + mpfr_set (y, h, rnd); /* exact */ + else + expl = MPFR_EXP (l); + mpfr_clear (l); + mpfr_clear (h); + if (ok) + return inex; + /* if ulp(log(-x)) <= |x| there is no reason to loop, + since the width of [l, h] will be at least |x| */ + if (expl < MPFR_EXP(x) + (mpfr_exp_t) w) + break; + w += MPFR_INT_CEIL_LOG2(w) + 3; + } + } + } + + inex = mpfr_lngamma_aux (y, x, rnd); + return inex; +} + +#endif |