diff --git a/arb/bin.c b/arb/bin.c
index dc5d0cdb3..d75284683 100644
--- a/arb/bin.c
+++ b/arb/bin.c
@@ -1,5 +1,6 @@
 /*
     Copyright (C) 2013 Fredrik Johansson
+    Copyright (C) 2021, 2022 Albin Ahlbäck
 
     This file is part of Arb.
 
@@ -10,42 +11,106 @@
 */
 
 #include "arb.h"
+#include "flint/fmpz.h"
+
+#define LOG2_BIN(n, k)                      \
+    (-1.3257480647361592                    \
+     + ((n) + .5) * log2(n)                 \
+     - ((k) + .5) * log2(k)                 \
+     - ((n) - (k) + .5) * log2((n) - (k)))  \
+
+static void
+_arb_bin_ui(arb_t x, const arb_t n, ulong k, slong prec)
+{
+    arb_t t, u;
+
+    arb_init(t);
+    arb_init(u);
+
+    arb_sub_ui(t, n, k - 1, prec);
+    arb_rising_ui(t, t, k, prec);
+    arb_fac_ui(u, k, prec);
+    arb_div(x, t, u, prec);
+
+    arb_clear(t);
+    arb_clear(u);
+}
 
 void
 arb_bin_ui(arb_t x, const arb_t n, ulong k, slong prec)
 {
     if (k == 0)
-    {
         arb_one(x);
-    }
     else if (k == 1)
-    {
         arb_set_round(x, n, prec);
-    }
-    else
+    else if (arb_is_int(n)
+            && arb_is_nonnegative(n)
+            && ARF_EXP(arb_midref(n)) <= FLINT_BITS) /* fits in ulong */
     {
-        arb_t t, u;
-
-        arb_init(t);
-        arb_init(u);
-
-        arb_sub_ui(t, n, k - 1, prec);
-        arb_rising_ui(t, t, k, prec);
-        arb_fac_ui(u, k, prec);
-        arb_div(x, t, u, prec);
-
-        arb_clear(t);
-        arb_clear(u);
+        fmpz_t tmp;
+        fmpz_init(tmp);
+        arb_get_unique_fmpz(tmp, n);
+        arb_bin_uiui(x, fmpz_get_ui(tmp), k, prec);
     }
+    else
+        _arb_bin_ui(x, n, k, prec);
 }
 
 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);
+    k = FLINT_MIN(k, n - k);
+
+#if LONG_MAX == WORD_MAX
+    if (n <= (UWORD(1) << 12)
+            || k <= (UWORD(1) << 7)
+#if FLINT_BITS == 64
+            || (n <= (UWORD(1) << 32) && k <= (UWORD(1) << 8))
+#else
+            || (n <= UWORD_MAX && k <= (UWORD(1) << 8))
+#endif
+            || (n <= (UWORD(1) << 22) && k <= (UWORD(1) << 9))
+            || (n <= (UWORD(1) << 16) && k <= (UWORD(1) << 10))
+            || ((double) prec) < LOG2_BIN(n, k))
+    {
+        mpz_t mres;
+        mpz_init(mres);
+        flint_mpz_bin_uiui(mres, n, k);
+        arb_set_round_mpz(x, mres, prec);
+        mpz_clear(mres);
+    }
+#else
+    if (n <= (UWORD(1) << 12)
+            || k <= (UWORD(1) << 7)
+            || (n < UWORD_MAX && k <= (UWORD(1) << 8))
+            || (n <= (UWORD(1) << 22) && k <= (UWORD(1) << 9))
+            || (n <= (UWORD(1) << 16) && k <= (UWORD(1) << 10)))
+    {
+        mpz_t mres;
+        mpz_init(mres);
+        flint_mpz_bin_uiui(mres, n, k);
+        arb_set_round_mpz(x, mres, prec);
+        mpz_clear(mres);
+    }
+    else if (k < (UWORD(1) << 32) && ((double) prec) < LOG2_BIN(n, k))
+    {
+        mpz_t mres;
+        __mpz_struct mn = {1, 1, NULL};
+        mpz_init(mres);
+        mn._mp_d = &n;
+        mpz_bin_ui(mres, &mn, k);
+        arb_set_round_mpz(x, mres, prec);
+        mpz_clear(mres);
+    }
+#endif
+    else
+    {
+        arf_struct atmp;
+        mag_struct mtmp = {0, 0};
+        arb_struct tmp = {atmp, mtmp};
+        arf_init_set_ui(&atmp, n);
+        arb_bin_ui(x, &tmp, k, prec);
+    }
 }
 
+#undef LOG2_BIN
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)