From fb6e85669cc76104aae26db530c03f0bffab9ae7 Mon Sep 17 00:00:00 2001
From: Fredrik Johansson <fredrik.johansson@gmail.com>
Date: Fri, 22 Mar 2013 23:48:18 +0100
Subject: [PATCH] fast gamma for rational and integer arguments

---
 doc/source/fmprb.rst |  2 ++
 fmprb.h              |  1 +
 fmprb/gamma.c        | 42 ++++++++++++++++++++++++++
 fmprb/gamma_fmpq.c   | 70 ++++++++++++++++++++++++++++++++++++++++++++
 4 files changed, 115 insertions(+)
 create mode 100644 fmprb/gamma_fmpq.c

diff --git a/doc/source/fmprb.rst b/doc/source/fmprb.rst
index acd19c54..fa8161e7 100644
--- a/doc/source/fmprb.rst
+++ b/doc/source/fmprb.rst
@@ -698,6 +698,8 @@ Gamma function
 
 .. function:: void fmprb_gamma(fmprb_t y, const fmprb_t x, long prec)
 
+.. function:: void fmprb_gamma_fmpq(fmprb_t y, const fmpq_t x, long prec)
+
     Sets `y = \Gamma(x)`, the gamma function.
 
 .. function:: void fmprb_rgamma(fmprb_t y, const fmprb_t x, long prec)
diff --git a/fmprb.h b/fmprb.h
index fbf3f164..ff1c5e72 100644
--- a/fmprb.h
+++ b/fmprb.h
@@ -321,6 +321,7 @@ void fmprb_agm(fmprb_t z, const fmprb_t x, const fmprb_t y, long prec);
 void fmprb_lgamma(fmprb_t y, const fmprb_t x, long prec);
 void fmprb_rgamma(fmprb_t y, const fmprb_t x, long prec);
 void fmprb_gamma(fmprb_t y, const fmprb_t x, long prec);
+void fmprb_gamma_fmpq(fmprb_t y, const fmpq_t x, long prec);
 
 void fmprb_digamma(fmprb_t y, const fmprb_t x, long prec);
 
diff --git a/fmprb/gamma.c b/fmprb/gamma.c
index 4a24e418..27527a1a 100644
--- a/fmprb/gamma.c
+++ b/fmprb/gamma.c
@@ -33,6 +33,48 @@ _fmprb_gamma(fmprb_t y, const fmprb_t x, long prec, int inverse)
     long r, n, wp;
     fmprb_t t, u, v;
 
+    if (fmprb_is_exact(x))
+    {
+        const fmpr_struct * mid = fmprb_midref(x);
+
+        if (fmpr_is_special(mid))
+        {
+            if (!inverse && fmpr_is_pos_inf(mid))
+            {
+                fmprb_set(y, x);
+            }
+            else if (fmpr_is_nan(mid) || fmpr_is_neg_inf(mid) || !inverse)
+            {
+                fmpr_nan(fmprb_midref(y));
+                fmpr_pos_inf(fmprb_radref(y));
+            }
+            else
+            {
+                fmprb_zero(y);
+            }
+            return;
+        }
+        else
+        {
+            const fmpz exp = *fmpr_expref(mid);
+            const fmpz man = *fmpr_manref(mid);
+
+            /* fast gamma(n), gamma(n/2) or gamma(n/4) */
+            if (!COEFF_IS_MPZ(exp) && (exp >= -2) &&
+                ((double) fmpz_bits(&man) + exp < prec))
+            {
+                fmpq_t a;
+                fmpq_init(a);
+                fmpr_get_fmpq(a, mid);
+                fmprb_gamma_fmpq(y, a, prec + 2 * inverse);
+                if (inverse)
+                    fmprb_ui_div(y, 1, y, prec);
+                fmpq_clear(a);
+                return;
+            }
+        }
+    }
+
     wp = prec + FLINT_BIT_COUNT(prec);
 
     gamma_stirling_choose_param_fmprb(&reflect, &r, &n, x, 1, 0, wp);
diff --git a/fmprb/gamma_fmpq.c b/fmprb/gamma_fmpq.c
new file mode 100644
index 00000000..286d05ca
--- /dev/null
+++ b/fmprb/gamma_fmpq.c
@@ -0,0 +1,70 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB 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 General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2013 Fredrik Johansson
+
+******************************************************************************/
+
+#include "fmprb.h"
+#include "gamma.h"
+
+void
+fmprb_gamma_fmpq(fmprb_t y, const fmpq_t x, long prec)
+{
+    fmpz p, q;
+
+    p = *fmpq_numref(x);
+    q = *fmpq_denref(x);
+
+    if ((q == 1 || q == 2 || q == 3 || q == 4 || q == 6) && !COEFF_IS_MPZ(p))
+    {
+        if (q == 1)
+        {
+            if (p <= 0)
+            {
+                fmpr_nan(fmprb_midref(y));
+                fmpr_pos_inf(fmprb_radref(y));
+                return;
+            }
+
+            if (p < 1200 || 1.44265 * (p*log(p) - p) < 15.0 * prec)
+            {
+                fmpz_t t;
+                fmpz_init(t);
+                fmpz_fac_ui(t, p - 1);
+                fmprb_set_round_fmpz(y, t, prec);
+                fmpz_clear(t);
+                return;
+            }
+        }
+
+        p = FLINT_ABS(p);
+
+        if (p < q * 500.0 || p < q * (500.0 + 0.1 * prec * sqrt(prec)))
+        {
+            gamma_fmpq_outward(y, x, prec);
+            return;
+        }
+    }
+
+    gamma_fmpq_stirling(y, x, prec);
+}
+