From 2e8b7f7b57293cf4b5a9da4ec6d2f4235bbd680b Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Albin=20Ahlb=C3=A4ck?= <albin.ahlback@gmail.com>
Date: Mon, 18 Oct 2021 17:39:41 +0200
Subject: [PATCH] Compute bin_uiui via Flint for small n

---
 arb/bin.c          | 26 ++++++++++++++++++++++----
 doc/source/arb.rst |  8 ++++++--
 2 files changed, 28 insertions(+), 6 deletions(-)

diff --git a/arb/bin.c b/arb/bin.c
index dc5d0cdb3..f21e7f9f3 100644
--- a/arb/bin.c
+++ b/arb/bin.c
@@ -1,5 +1,6 @@
 /*
     Copyright (C) 2013 Fredrik Johansson
+    Copyright (C) 2021 Albin Ahlbäck
 
     This file is part of Arb.
 
@@ -39,13 +40,30 @@ arb_bin_ui(arb_t x, const arb_t n, ulong k, slong prec)
     }
 }
 
+static
+void
+_arb_bin_uiui_small(arb_t x, ulong n, ulong k, slong prec)
+{
+    fmpz_t b;
+    fmpz_init(b);
+    fmpz_bin_uiui(b, n, k);
+    arb_set_round_fmpz(x, b, prec);
+    fmpz_clear(b);
+}
+
 void
 arb_bin_uiui(arb_t x, ulong n, ulong k, slong prec)
 {
     arb_t t;
-    arb_init(t);
-    arb_set_ui(t, n);
-    arb_bin_ui(x, t, k, prec);
-    arb_clear(t);
+
+    if (n < 3000)
+        _arb_bin_uiui_small(x, n, k, prec);
+    else
+    {
+        arb_init(t);
+        arb_set_ui(t, n);
+        arb_bin_ui(x, t, k, prec);
+        arb_clear(t);
+    }
 }
 
diff --git a/doc/source/arb.rst b/doc/source/arb.rst
index 039882bf9..9a4e09a34 100644
--- a/doc/source/arb.rst
+++ b/doc/source/arb.rst
@@ -1348,11 +1348,15 @@ Gamma function and factorials
 
 .. function:: void arb_bin_ui(arb_t z, const arb_t n, ulong k, slong prec)
 
-.. function:: void arb_bin_uiui(arb_t z, ulong n, ulong k, slong prec)
-
     Computes the binomial coefficient `z = {n \choose k}`, via the
     rising factorial as `{n \choose k} = (n-k+1)_k / k!`.
 
+.. function:: void arb_bin_uiui(arb_t z, ulong n, ulong k, slong prec)
+
+    Computes the binomial coefficient `z = {n \choose k}`. If `n` is small,
+    it is computed via Flint's implementation `fmpz_bin_uiui`. If `n` is big,
+    it is computed via rising factorial as `{n \choose k} = (n-k+1)_k / k!`.
+
 .. function:: void arb_gamma(arb_t z, const arb_t x, slong prec)
               void arb_gamma_fmpq(arb_t z, const fmpq_t x, slong prec)
               void arb_gamma_fmpz(arb_t z, const fmpz_t x, slong prec)