path: root/lngamma.c

/* mpfr_lngamma -- lngamma function

Copyright 2005, 2006, 2007 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.

This file is part of the MPFR Library.

The 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 2.1 of the License, or (at your
option) any later version.

The 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 MPFR Library; see the file COPYING.LIB.  If not, 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"

/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!

   t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
   thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
   Taking the coefficient of degree n+1 > 1, we get:
   0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
   which gives:
   B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).

   Let C[n] = B[n]*(n+1)!.
   Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!,  k=0..n-1),
   which proves that the C[n] are integers.
*/
static mpz_t*
bernoulli (mpz_t *b, unsigned long n)
{
  if (n == 0)
    {
      b = (mpz_t *) (*__gmp_allocate_func) (sizeof (mpz_t));
      mpz_init_set_ui (b[0], 1);
    }
  else
    {
      mpz_t t;
      unsigned long k;

      b = (mpz_t *) (*__gmp_reallocate_func)
        (b, n * sizeof (mpz_t), (n + 1) * sizeof (mpz_t));
      mpz_init (b[n]);
      /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!,  k=0..n-1) */
      mpz_init_set_ui (t, 2 * n + 1);
      mpz_mul_ui (t, t, 2 * n - 1);
      mpz_mul_ui (t, t, 2 * n);
      mpz_mul_ui (t, t, n);
      mpz_div_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
                               for k=n-1 */
      mpz_mul (b[n], t, b[n-1]);
      for (k = n - 1; k-- > 0;)
        {
          mpz_mul_ui (t, t, 2 * k + 1);
          mpz_mul_ui (t, t, 2 * k + 2);
          mpz_mul_ui (t, t, 2 * k + 2);
          mpz_mul_ui (t, t, 2 * k + 3);
          mpz_div_ui (t, t, 2 * (n - k) + 1);
          mpz_div_ui (t, t, 2 * (n - k));
          mpz_addmul (b[n], t, b[k]);
        }
      /* take into account C[1] */
      mpz_mul_ui (t, t, 2 * n + 1);
      mpz_div_2exp (t, t, 1);
      mpz_sub (b[n], b[n], t);
      mpz_neg (b[n], b[n]);
      mpz_clear (t);
    }
  return b;
}

/* 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 double
mpfr_gamma_alpha (mp_prec_t p)
{
  if (p <= 100)
    return 0.6;
  else if (p <= 200)
    return 0.8;
  else if (p <= 500)
    return 0.8;
  else if (p <= 1000)
    return 1.3;
  else if (p <= 2000)
    return 1.7;
  else if (p <= 5000)
    return 2.2;
  else if (p <= 10000)
    return 3.4;
  else /* heuristic fit from above */
    return 0.26 * (double) MPFR_INT_CEIL_LOG2 ((unsigned long) p);
}

/* 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[2n]/(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
static int
#define GAMMA_FUNC mpfr_gamma_aux
#else
int
#define GAMMA_FUNC mpfr_lngamma
#endif
GAMMA_FUNC (mpfr_ptr y, mpfr_srcptr z0, mp_rnd_t rnd)
{
  mp_prec_t precy, w; /* working precision */
  mpfr_t s, t, u, v, z;
  unsigned long m, k, maxm;
  mpz_t *B;
  int inexact, compared;
  mp_exp_t err_s, err_t;
  unsigned long Bm = 0; /* number of allocated B[] */
  unsigned long oldBm;
  double d;
  MPFR_SAVE_EXPO_DECL (expo);

#ifndef IS_GAMMA
  /* special cases */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z0)))
    {
      if (MPFR_IS_NAN (z0) || MPFR_IS_NEG (z0))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else /* lngamma(+Inf) = lngamma(+0) = +Inf */
        {
          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 (z0))
    {
      MPFR_SAVE_EXPO_MARK (expo);
      if (mpfr_get_si (z0, GMP_RNDZ) % 2 == 0 || mpfr_integer_p (z0))
        {
          MPFR_SAVE_EXPO_FREE (expo);
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      MPFR_SAVE_EXPO_FREE (expo);
    }
#endif

  precy = MPFR_PREC(y);

  compared = mpfr_cmp_ui (z0, 1);

#ifndef IS_GAMMA
  if (compared == 0 || (compared > 0 && mpfr_cmp_ui (z0, 2) == 0))
    return mpfr_set_ui (y, 0, GMP_RNDN);  /* lngamma(1 or 2) = +0 */
#endif

  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);

  MPFR_SAVE_EXPO_MARK (expo);

  if (compared < 0)
    {
      mp_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);
      while (1)
        {
          w += MPFR_INT_CEIL_LOG2 (w) + 13;
          MPFR_ASSERTD(w >= 3);
          mpfr_set_prec (s, w);
          mpfr_set_prec (t, w);
          mpfr_set_prec (u, w);
          mpfr_set_prec (v, w);
          mpfr_ui_sub (s, 2, z0, GMP_RNDD); /* s = (2-z0) * (1+2u) */
          mpfr_const_pi (t, GMP_RNDN);      /* t = Pi * (1+u) */
          mpfr_lngamma (u, s, GMP_RNDN); /* lngamma(2-x) */
          /* Let s = (2-z0) + h. By construction, -(2-z0)*(2u) <= h <= 0.
             We have lngamma(s) = lngamma(2-z0) + h*Psi(z), z in [2-z0+h,2-z0].
             Since 2-z0+h >= 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). */
          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, GMP_RNDN); /* Pi*(2-x), (1+u)^4 */
          err_s = MPFR_GET_EXP(s); /* 2-x <= 2^err_s */
          mpfr_sin (s, s, GMP_RNDN); /* sin(Pi*(2-x)) */
          /* the error on s is bounded by 1/2*ulp(s) + [(1+u)^4-1]*(2-x)
             <= 1/2*ulp(s) + 5*2^(-w)*(2-x) for w >= 3 */
          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 ulps, thus the relative
             error is bounded by 2^(err_s+1) */
          err_s ++; /* relative error */

          mpfr_sub_ui (v, z0, 1, GMP_RNDN); /* v = (x-1) * (1+u) */
          mpfr_mul (v, v, t, GMP_RNDN); /* v = Pi*(x-1) * (1+u)^3 */
          mpfr_div (v, v, s, GMP_RNDN); /* Pi*(x-1)/sin(Pi*(2-x)) */
          /* (1+u)^(4+2^err_s+1) */
          err_s = (err_s <= 2) ? 3 + (err_s / 2) : err_s + 1;
          MPFR_ASSERTD(MPFR_IS_POS(v));
          mpfr_log (v, v, GMP_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, GMP_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, GMP_RNDN, GMP_RNDZ, precy
                                  + (rnd == GMP_RNDN)))
                goto end;
            }
        }
    }

  /* 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);

  do
    {
      w += MPFR_INT_CEIL_LOG2 (w) + 13;
      MPFR_ASSERTD (w >= 3);

      mpfr_set_prec (s, 53);
      /* we need z >= w*log(2)/(2*Pi) to get an absolute error less than 2^(-w)
         but the optimal value is about 0.155665*w */
      /* FIXME: replace double by mpfr_t types. */
      mpfr_set_d (s, mpfr_gamma_alpha (precy) * (double) w, GMP_RNDU);
      if (mpfr_cmp (z0, s) < 0)
        {
          mpfr_sub (s, s, z0, GMP_RNDU);
          k = mpfr_get_ui (s, GMP_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, GMP_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, GMP_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_2exp (t, z, 1, GMP_RNDN); /* t = 2z * (1+t5) */
      mpfr_sub_ui (t, t, 1, GMP_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, GMP_RNDN); /* (2z-1)*log(z) * (1+t7)^6 */
      mpfr_div_2exp (s, s, 1, GMP_RNDN); /* (z-1/2)*log(z) * (1+t7)^6 */
      mpfr_sub (s, s, z, GMP_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, GMP_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, GMP_RNDN); /* 1/(12z) * (1+u)^2, t <= 3/128 */
      mpfr_set (v, t, GMP_RNDN);        /* (1+u)^2, v < 2^(-5) */
      mpfr_add (s, s, v, GMP_RNDN);     /* (1+u)^15 */

      mpfr_mul (u, u, u, GMP_RNDN); /* 1/z^2 * (1+u)^3 */

      if (Bm == 0)
        {
          B = bernoulli ((mpz_t *) 0, 0);
          B = bernoulli (B, 1);
          Bm = 2;
        }

      /* m <= maxm ensures that 2*m*(2*m+1) <= ULONG_MAX */
      maxm = 1UL << (BITS_PER_MP_LIMB / 2 - 1);

      /* s:(1+u)^15, t:(1+u)^2, t <= 3/128 */

      for (m = 2; MPFR_GET_EXP(v) + (mp_exp_t) w >= MPFR_GET_EXP(s); m++)
        {
          mpfr_mul (t, t, u, GMP_RNDN); /* (1+u)^(10m-14) */
          if (m <= maxm)
            {
              mpfr_mul_ui (t, t, 2*(m-1)*(2*m-3), GMP_RNDN);
              mpfr_div_ui (t, t, 2*m*(2*m-1), GMP_RNDN);
              mpfr_div_ui (t, t, 2*m*(2*m+1), GMP_RNDN);
            }
          else
            {
              mpfr_mul_ui (t, t, 2*(m-1), GMP_RNDN);
              mpfr_mul_ui (t, t, 2*m-3, GMP_RNDN);
              mpfr_div_ui (t, t, 2*m, GMP_RNDN);
              mpfr_div_ui (t, t, 2*m-1, GMP_RNDN);
              mpfr_div_ui (t, t, 2*m, GMP_RNDN);
              mpfr_div_ui (t, t, 2*m+1, GMP_RNDN);
            }
          /* (1+u)^(10m-8) */
          /* invariant: t=1/(2m)/(2m-1)/z^(2m-1)/(2m+1)! */
          if (Bm <= m)
            {
              B = bernoulli (B, m); /* B[2m]*(2m+1)!, exact */
              Bm ++;
            }
          mpfr_mul_z (v, t, B[m], GMP_RNDN); /* (1+u)^(10m-7) */
          MPFR_ASSERTD(MPFR_GET_EXP(v) <= - (2 * m + 3));
          mpfr_add (s, s, v, GMP_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, GMP_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, GMP_RNDN); /* v = Pi*(1+u) */
      mpfr_mul_2exp (v, v, 1, GMP_RNDN); /* v = 2*Pi * (1+u) */
      if (k)
        {
          unsigned long l;
          mpfr_set (t, z0, GMP_RNDN); /* t = z0*(1+u) */
          for (l = 1; l < k; l++)
            {
              mpfr_add_ui (u, z0, l, GMP_RNDN); /* u = (z0+l)*(1+u) */
              mpfr_mul (t, t, u, GMP_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, GMP_RNDN); /* (z0*...*(z0+k-1))^2, (1+u)^(4k-1) */
          mpfr_div (v, v, t, GMP_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, GMP_RNDN);
      /* 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, GMP_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, GMP_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, GMP_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_2exp (t, t, 1, GMP_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) + (mp_exp_t)
        __gmpfr_ceil_log2 (2.2 * (double) k + 1.6);
      err_s = MPFR_GET_EXP(s) + (mp_exp_t)
        __gmpfr_ceil_log2 (2.0 * (double) m + 48.0);
      mpfr_add (s, s, t, GMP_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
    }
  while (MPFR_UNLIKELY (!MPFR_CAN_ROUND (s, w - err_s, precy, rnd)));

  oldBm = Bm;
  while (Bm--)
    mpz_clear (B[Bm]);
  (*__gmp_free_func) (B, oldBm * sizeof (mpz_t));

 end:
  inexact = mpfr_set (y, s, rnd);

  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);
}