diff --git a/acb.h b/acb.h
index f1ab6d6e..1ac3da68 100644
--- a/acb.h
+++ b/acb.h
@@ -780,6 +780,7 @@ void acb_polylog_si(acb_t w, slong s, const acb_t z, slong prec);
 
 void acb_agm1(acb_t m, const acb_t z, slong prec);
 void acb_agm1_cpx(acb_ptr m, const acb_t z, slong len, slong prec);
+void acb_agm(acb_t res, const acb_t a, const acb_t b, slong prec);
 
 #define ACB_LAMBERTW_LEFT 2
 #define ACB_LAMBERTW_MIDDLE 4
diff --git a/acb/agm.c b/acb/agm.c
new file mode 100644
index 00000000..229c7ef9
--- /dev/null
+++ b/acb/agm.c
@@ -0,0 +1,152 @@
+/*
+    Copyright (C) 2020 Fredrik Johansson
+
+    This file is part of Arb.
+
+    Arb is free software: you can redistribute it and/or modify it under
+    the terms of the GNU Lesser General Public License (LGPL) as published
+    by the Free Software Foundation; either version 2.1 of the License, or
+    (at your option) any later version.  See <http://www.gnu.org/licenses/>.
+*/
+
+#include "acb.h"
+
+void mag_agm(mag_t res, const mag_t x, const mag_t y);
+
+static void
+agm_helper(acb_t res, const acb_t a, const acb_t b, slong prec)
+{
+    if (acb_rel_accuracy_bits(b) >= acb_rel_accuracy_bits(a))
+    {
+        acb_div(res, a, b, prec);
+        acb_agm1(res, res, prec);
+        acb_mul(res, res, b, prec);
+    }
+    else
+    {
+        acb_div(res, b, a, prec);
+        acb_agm1(res, res, prec);
+        acb_mul(res, res, a, prec);
+    }
+}
+
+void
+acb_agm(acb_t res, const acb_t a, const acb_t b, slong prec)
+{
+    acb_t t, u, v;
+
+    if (!acb_is_finite(a) || !acb_is_finite(b))
+    {
+        acb_indeterminate(res);
+        return;
+    }
+
+    if (acb_is_zero(a) || acb_is_zero(b))
+    {
+        acb_zero(res);
+        return;
+    }
+
+    if (arb_is_zero(acb_imagref(a)) && arb_is_zero(acb_imagref(b)))
+    {
+        if (arb_is_nonnegative(acb_realref(a)) && arb_is_nonnegative(acb_realref(b)))
+        {
+            arb_agm(acb_realref(res), acb_realref(a), acb_realref(b), prec);
+            arb_zero(acb_imagref(res));
+            return;
+        }
+    }
+
+    if (acb_contains_zero(a) || acb_contains_zero(b))
+    {
+        mag_t ra, rb;
+
+        mag_init(ra);
+        mag_init(rb);
+
+        acb_get_mag(ra, a);
+        acb_get_mag(rb, b);
+        mag_agm(ra, ra, rb);
+        acb_zero(res);
+        acb_add_error_mag(res, ra);
+
+        mag_clear(ra);
+        mag_clear(rb);
+
+        return;
+    }
+
+    acb_init(t);
+
+    acb_add(t, a, b, prec);
+    acb_mul_2exp_si(t, t, -1);
+
+    /* a ~= -b; bound magnitude */
+    if (acb_contains_zero(t))
+    {
+        mag_t ra, rb;
+
+        mag_init(ra);
+        mag_init(rb);
+
+        acb_get_mag(ra, a);
+        acb_get_mag(rb, b);
+        mag_mul(rb, ra, rb);
+        mag_sqrt(rb, rb);
+
+        acb_get_mag(ra, t);
+        mag_agm(ra, ra, rb);
+
+        acb_zero(res);
+        acb_add_error_mag(res, ra);
+
+        mag_clear(ra);
+        mag_clear(rb);
+
+        acb_clear(t);
+        return;
+    }
+
+    /* Do the initial step with the optimal square root, reducing to agm1 */
+
+    acb_init(u);
+    acb_init(v);
+
+    acb_mul(u, a, b, prec);
+
+    /* we can compute either square root here; avoid the branch cut */
+    if (arf_sgn(arb_midref(acb_realref(u))) >= 0)
+    {
+        acb_sqrt(u, u, prec);
+    }
+    else
+    {
+        acb_neg(u, u);
+        acb_sqrt(u, u, prec);
+        acb_mul_onei(u, u);
+    }
+
+    acb_div(v, t, u, prec);
+
+    if (arb_is_nonnegative(acb_realref(v)))
+    {
+        agm_helper(res, t, u, prec);
+    }
+    else if (arb_is_negative(acb_realref(v)))
+    {
+        acb_neg(u, u);
+        agm_helper(res, t, u, prec);
+    }
+    else
+    {
+        agm_helper(v, t, u, prec);
+        acb_neg(u, u);
+        agm_helper(res, t, u, prec);
+        acb_union(res, res, v, prec);
+    }
+
+    acb_clear(t);
+    acb_clear(u);
+    acb_clear(v);
+}
+
diff --git a/acb/agm1.c b/acb/agm1.c
index b0f65213..1bb3fa98 100644
--- a/acb/agm1.c
+++ b/acb/agm1.c
@@ -12,6 +12,8 @@
 #include "acb.h"
 #include "acb_poly.h"
 
+void mag_agm(mag_t res, const mag_t x, const mag_t y);
+
 /* Checks that |arg(z)| <= 3 pi / 4 */
 static int
 acb_check_arg(const acb_t z)
@@ -96,6 +98,21 @@ acb_agm_close_taylor(acb_t res, acb_t z, acb_t z2,
     acb_mul(res, res, aplusb, prec);
 }
 
+static void
+acb_agm1_around_zero(acb_t res, const acb_t z, slong prec)
+{
+    mag_t a, b;
+    mag_init(a);
+    mag_init(b);
+    mag_one(a);
+    acb_get_mag(b, z);
+    mag_agm(a, a, b);
+    acb_zero(res);
+    acb_add_error_mag(res, a);
+    mag_clear(a);
+    mag_clear(b);
+}
+
 void
 acb_agm1_basecase(acb_t res, const acb_t z, slong prec)
 {
@@ -129,16 +146,7 @@ acb_agm1_basecase(acb_t res, const acb_t z, slong prec)
 
     if (!acb_check_arg(z))
     {
-        mag_t one;
-        mag_init(one);
-        mag_init(err);
-        mag_one(one);
-        acb_get_mag(err, z);
-        mag_max(err, err, one);
-        acb_zero(res);
-        acb_add_error_mag(res, err);
-        mag_clear(err);
-        mag_clear(one);
+        acb_agm1_around_zero(res, z, prec);
         return;
     }
 
@@ -417,28 +425,73 @@ acb_agm1_deriv_right(acb_t Mz, acb_t Mzp, const acb_t z, slong prec)
 }
 
 void
-acb_agm1(acb_t m, const acb_t z, slong prec)
+acb_agm1(acb_t res, const acb_t z, slong prec)
 {
-    if (arf_sgn(arb_midref(acb_realref(z))) >= 0)
+    if (acb_is_zero(z))
     {
-        acb_agm1_basecase(m, z, prec);
+        acb_zero(res);
+    }
+    else if (!acb_is_finite(z))
+    {
+        acb_indeterminate(res);
+    }
+    else if (acb_contains_zero(z))
+    {
+        acb_agm1_around_zero(res, z, prec);
+    }
+    else if (arf_sgn(arb_midref(acb_realref(z))) >= 0)
+    {
+        acb_agm1_basecase(res, z, prec);
     }
     else if (acb_equal_si(z, -1))
     {
-        acb_zero(m);
+        acb_zero(res);
     }
     else
     {
-        /* use M(z) = (z+1)/2 * M(2 sqrt(z) / (z+1)) */
+        /* use M(1,z) = M((z+1)/2, sqrt(z))
+                      = (z+1)/2 * M(1, 2 sqrt(z) / (z+1))
+                      = sqrt(z) * M(1, (z+1) / (2 sqrt(z)) */
         acb_t t;
+
         acb_init(t);
         acb_add_ui(t, z, 1, prec);
-        acb_sqrt(m, z, prec);
-        acb_div(m, m, t, prec);
-        acb_mul_2exp_si(m, m, 1);
-        acb_agm1_basecase(m, m, prec);
-        acb_mul(m, m, t, prec);
-        acb_mul_2exp_si(m, m, -1);
+        acb_mul_2exp_si(t, t, -1);
+
+        if (acb_contains_zero(t))
+        {
+            mag_t ra, rb;
+
+            mag_init(ra);
+            mag_init(rb);
+
+            acb_get_mag(ra, t);
+            acb_get_mag(rb, z);
+            mag_sqrt(rb, rb);
+
+            mag_agm(ra, ra, rb);
+
+            acb_zero(res);
+            acb_add_error_mag(res, ra);
+
+            mag_clear(ra);
+            mag_clear(rb);
+        }
+        else if (acb_rel_accuracy_bits(t) > acb_rel_accuracy_bits(z))
+        {
+            acb_sqrt(res, z, prec);
+            acb_div(res, res, t, prec);
+            acb_agm1_basecase(res, res, prec);
+            acb_mul(res, res, t, prec);
+        }
+        else
+        {
+            acb_sqrt(res, z, prec);
+            acb_div(t, t, res, prec);
+            acb_agm1_basecase(t, t, prec);
+            acb_mul(res, res, t, prec);
+        }
+
         acb_clear(t);
     }
 }
diff --git a/acb/test/t-agm.c b/acb/test/t-agm.c
new file mode 100644
index 00000000..a28d892b
--- /dev/null
+++ b/acb/test/t-agm.c
@@ -0,0 +1,435 @@
+/*
+    Copyright (C) 2020 Fredrik Johansson
+
+    This file is part of Arb.
+
+    Arb is free software: you can redistribute it and/or modify it under
+    the terms of the GNU Lesser General Public License (LGPL) as published
+    by the Free Software Foundation; either version 2.1 of the License, or
+    (at your option) any later version.  See <http://www.gnu.org/licenses/>.
+*/
+
+#include "acb.h"
+
+#define NUM_TESTS 256
+const double agm_testdata[NUM_TESTS][6] = {
+    {-1, -1, -1, -1, -1.00000000000000, -1.00000000000000},
+    {-1, -1, -1, 0, -1.04916052873278, -0.478155746088161},
+    {-1, -1, -1, 1, -1.19814023473559, 0.000000000000000},
+    {-1, -1, -1, 2, -1.35980258211696, 0.412918513270561},
+    {-1, -1, 0, -1, -0.478155746088161, -1.04916052873278},
+    {-1, -1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {-1, -1, 0, 1, -0.778004078788958, 0.188411067988680},
+    {-1, -1, 0, 2, -0.966415146777638, 0.589593010800278},
+    {-1, -1, 1, -1, 0.000000000000000, -1.19814023473559},
+    {-1, -1, 1, 0, 0.188411067988680, -0.778004078788958},
+    {-1, -1, 1, 1, 0.000000000000000, 0.000000000000000},
+    {-1, -1, 1, 2, -0.574659643809835, 0.810208383866421},
+    {-1, -1, 2, -1, 0.412918513270561, -1.35980258211696},
+    {-1, -1, 2, 0, 0.589593010800278, -0.966415146777638},
+    {-1, -1, 2, 1, 0.810208383866421, -0.574659643809835},
+    {-1, -1, 2, 2, 1.08423239187061, -0.238299975053003},
+    {-1, 0, -1, -1, -1.04916052873278, -0.478155746088161},
+    {-1, 0, -1, 0, -1.00000000000000, 0.000000000000000},
+    {-1, 0, -1, 1, -1.04916052873278, 0.478155746088161},
+    {-1, 0, -1, 2, -1.13815978557677, 0.897764105638205},
+    {-1, 0, 0, -1, -0.599070117367796, -0.599070117367796},
+    {-1, 0, 0, 0, 0.000000000000000, 0.000000000000000},
+    {-1, 0, 0, 1, -0.599070117367796, 0.599070117367796},
+    {-1, 0, 0, 2, -0.742646422399778, 1.01005159361958},
+    {-1, 0, 1, -1, -0.188411067988680, -0.778004078788958},
+    {-1, 0, 1, 0, 0.000000000000000, 0.000000000000000},
+    {-1, 0, 1, 1, -0.188411067988680, 0.778004078788958},
+    {-1, 0, 1, 2, -0.363823969691055, 1.15631976909117},
+    {-1, 0, 2, -1, 0.178866979357132, -0.985872907297986},
+    {-1, 0, 2, 0, 0.422966208408802, 0.661266183461805},
+    {-1, 0, 2, 1, 0.178866979357132, 0.985872907297986},
+    {-1, 0, 2, 2, -0.0126420344317752, 1.32391495927282},
+    {-1, 1, -1, -1, -1.19814023473559, 0.000000000000000},
+    {-1, 1, -1, 0, -1.04916052873278, 0.478155746088161},
+    {-1, 1, -1, 1, -1.00000000000000, 1.00000000000000},
+    {-1, 1, -1, 2, -1.02005412633766, 1.47134936286465},
+    {-1, 1, 0, -1, -0.778004078788958, -0.188411067988680},
+    {-1, 1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {-1, 1, 0, 1, -0.478155746088161, 1.04916052873278},
+    {-1, 1, 0, 2, -0.571004782644619, 1.52731627482094},
+    {-1, 1, 1, -1, 0.000000000000000, 0.000000000000000},
+    {-1, 1, 1, 0, 0.188411067988680, 0.778004078788958},
+    {-1, 1, 1, 1, 0.000000000000000, 1.19814023473559},
+    {-1, 1, 1, 2, -0.136827548397369, 1.62780548487271},
+    {-1, 1, 2, -1, 0.810208383866421, 0.574659643809835},
+    {-1, 1, 2, 0, 0.589593010800278, 0.966415146777638},
+    {-1, 1, 2, 1, 0.412918513270561, 1.35980258211696},
+    {-1, 1, 2, 2, 0.267405171219803, 1.75269801601936},
+    {-1, 2, -1, -1, -1.35980258211696, 0.412918513270561},
+    {-1, 2, -1, 0, -1.13815978557677, 0.897764105638205},
+    {-1, 2, -1, 1, -1.02005412633766, 1.47134936286465},
+    {-1, 2, -1, 2, -1.00000000000000, 2.00000000000000},
+    {-1, 2, 0, -1, -0.985872907297986, 0.178866979357132},
+    {-1, 2, 0, 0, 0.000000000000000, 0.000000000000000},
+    {-1, 2, 0, 1, -0.420983902077155, 1.47839688785392},
+    {-1, 2, 0, 2, -0.493004461502931, 2.02900451838781},
+    {-1, 2, 1, -1, 0.574659643809835, 0.810208383866421},
+    {-1, 2, 1, 0, 0.363823969691055, 1.15631976909117},
+    {-1, 2, 1, 1, 0.136827548397369, 1.62780548487271},
+    {-1, 2, 1, 2, 0.000000000000000, 2.11638793788003},
+    {-1, 2, 2, -1, 0.963366302386887, 0.963366302386887},
+    {-1, 2, 2, 0, 0.777111977967075, 1.37110999239961},
+    {-1, 2, 2, 1, 0.599070117367796, 1.79721035210339},
+    {-1, 2, 2, 2, 0.455159607692816, 2.23571522629432},
+    {0, -1, -1, -1, -0.478155746088161, -1.04916052873278},
+    {0, -1, -1, 0, -0.599070117367796, -0.599070117367796},
+    {0, -1, -1, 1, -0.778004078788958, -0.188411067988680},
+    {0, -1, -1, 2, -0.985872907297986, 0.178866979357132},
+    {0, -1, 0, -1, 0.000000000000000, -1.00000000000000},
+    {0, -1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {0, -1, 0, 1, 0.000000000000000, 0.000000000000000},
+    {0, -1, 0, 2, 0.661266183461805, 0.422966208408802},
+    {0, -1, 1, -1, 0.478155746088161, -1.04916052873278},
+    {0, -1, 1, 0, 0.599070117367796, -0.599070117367796},
+    {0, -1, 1, 1, 0.778004078788958, -0.188411067988680},
+    {0, -1, 1, 2, 0.985872907297986, 0.178866979357132},
+    {0, -1, 2, -1, 0.897764105638205, -1.13815978557677},
+    {0, -1, 2, 0, 1.01005159361958, -0.742646422399778},
+    {0, -1, 2, 1, 1.15631976909117, -0.363823969691055},
+    {0, -1, 2, 2, 1.32391495927282, -0.0126420344317752},
+    {0, 0, -1, -1, 0.000000000000000, 0.000000000000000},
+    {0, 0, -1, 0, 0.000000000000000, 0.000000000000000},
+    {0, 0, -1, 1, 0.000000000000000, 0.000000000000000},
+    {0, 0, -1, 2, 0.000000000000000, 0.000000000000000},
+    {0, 0, 0, -1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 0, 0, 0.000000000000000, 0.000000000000000},
+    {0, 0, 0, 1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 0, 2, 0.000000000000000, 0.000000000000000},
+    {0, 0, 1, -1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 1, 0, 0.000000000000000, 0.000000000000000},
+    {0, 0, 1, 1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 1, 2, 0.000000000000000, 0.000000000000000},
+    {0, 0, 2, -1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 2, 0, 0.000000000000000, 0.000000000000000},
+    {0, 0, 2, 1, 0.000000000000000, 0.000000000000000},
+    {0, 0, 2, 2, 0.000000000000000, 0.000000000000000},
+    {0, 1, -1, -1, -0.778004078788958, 0.188411067988680},
+    {0, 1, -1, 0, -0.599070117367796, 0.599070117367796},
+    {0, 1, -1, 1, -0.478155746088161, 1.04916052873278},
+    {0, 1, -1, 2, -0.420983902077155, 1.47839688785392},
+    {0, 1, 0, -1, 0.000000000000000, 0.000000000000000},
+    {0, 1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {0, 1, 0, 1, 0.000000000000000, 1.00000000000000},
+    {0, 1, 0, 2, 0.000000000000000, 1.45679103104691},
+    {0, 1, 1, -1, 0.778004078788958, 0.188411067988680},
+    {0, 1, 1, 0, 0.599070117367796, 0.599070117367796},
+    {0, 1, 1, 1, 0.478155746088161, 1.04916052873278},
+    {0, 1, 1, 2, 0.420983902077155, 1.47839688785392},
+    {0, 1, 2, -1, 1.15631976909117, 0.363823969691055},
+    {0, 1, 2, 0, 1.01005159361958, 0.742646422399778},
+    {0, 1, 2, 1, 0.897764105638205, 1.13815978557677},
+    {0, 1, 2, 2, 0.820905313474926, 1.53147158277780},
+    {0, 2, -1, -1, -0.966415146777638, 0.589593010800278},
+    {0, 2, -1, 0, -0.742646422399778, 1.01005159361958},
+    {0, 2, -1, 1, -0.571004782644619, 1.52731627482094},
+    {0, 2, -1, 2, -0.493004461502931, 2.02900451838781},
+    {0, 2, 0, -1, 0.661266183461805, 0.422966208408802},
+    {0, 2, 0, 0, 0.000000000000000, 0.000000000000000},
+    {0, 2, 0, 1, 0.000000000000000, 1.45679103104691},
+    {0, 2, 0, 2, 0.000000000000000, 2.00000000000000},
+    {0, 2, 1, -1, 0.966415146777638, 0.589593010800278},
+    {0, 2, 1, 0, 0.742646422399778, 1.01005159361958},
+    {0, 2, 1, 1, 0.571004782644619, 1.52731627482094},
+    {0, 2, 1, 2, 0.493004461502931, 2.02900451838781},
+    {0, 2, 2, -1, 1.37110999239961, 0.777111977967075},
+    {0, 2, 2, 0, 1.19814023473559, 1.19814023473559},
+    {0, 2, 2, 1, 1.05542091981433, 1.64623191253177},
+    {0, 2, 2, 2, 0.956311492176322, 2.09832105746556},
+    {1, -1, -1, -1, 0.000000000000000, -1.19814023473559},
+    {1, -1, -1, 0, -0.188411067988680, -0.778004078788958},
+    {1, -1, -1, 1, 0.000000000000000, 0.000000000000000},
+    {1, -1, -1, 2, 0.574659643809835, 0.810208383866421},
+    {1, -1, 0, -1, 0.478155746088161, -1.04916052873278},
+    {1, -1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {1, -1, 0, 1, 0.778004078788958, 0.188411067988680},
+    {1, -1, 0, 2, 0.966415146777638, 0.589593010800278},
+    {1, -1, 1, -1, 1.00000000000000, -1.00000000000000},
+    {1, -1, 1, 0, 1.04916052873278, -0.478155746088161},
+    {1, -1, 1, 1, 1.19814023473559, 0.000000000000000},
+    {1, -1, 1, 2, 1.35980258211696, 0.412918513270561},
+    {1, -1, 2, -1, 1.47134936286465, -1.02005412633766},
+    {1, -1, 2, 0, 1.52731627482094, -0.571004782644619},
+    {1, -1, 2, 1, 1.62780548487271, -0.136827548397369},
+    {1, -1, 2, 2, 1.75269801601936, 0.267405171219803},
+    {1, 0, -1, -1, 0.188411067988680, -0.778004078788958},
+    {1, 0, -1, 0, 0.000000000000000, 0.000000000000000},
+    {1, 0, -1, 1, 0.188411067988680, 0.778004078788958},
+    {1, 0, -1, 2, 0.363823969691055, 1.15631976909117},
+    {1, 0, 0, -1, 0.599070117367796, -0.599070117367796},
+    {1, 0, 0, 0, 0.000000000000000, 0.000000000000000},
+    {1, 0, 0, 1, 0.599070117367796, 0.599070117367796},
+    {1, 0, 0, 2, 0.742646422399778, 1.01005159361958},
+    {1, 0, 1, -1, 1.04916052873278, -0.478155746088161},
+    {1, 0, 1, 0, 1.00000000000000, 0.000000000000000},
+    {1, 0, 1, 1, 1.04916052873278, 0.478155746088161},
+    {1, 0, 1, 2, 1.13815978557677, 0.897764105638205},
+    {1, 0, 2, -1, 1.47839688785392, -0.420983902077155},
+    {1, 0, 2, 0, 1.45679103104691, 0.000000000000000},
+    {1, 0, 2, 1, 1.47839688785392, 0.420983902077155},
+    {1, 0, 2, 2, 1.53147158277780, 0.820905313474926},
+    {1, 1, -1, -1, 0.000000000000000, 0.000000000000000},
+    {1, 1, -1, 0, -0.188411067988680, 0.778004078788958},
+    {1, 1, -1, 1, 0.000000000000000, 1.19814023473559},
+    {1, 1, -1, 2, 0.136827548397369, 1.62780548487271},
+    {1, 1, 0, -1, 0.778004078788958, -0.188411067988680},
+    {1, 1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {1, 1, 0, 1, 0.478155746088161, 1.04916052873278},
+    {1, 1, 0, 2, 0.571004782644619, 1.52731627482094},
+    {1, 1, 1, -1, 1.19814023473559, 0.000000000000000},
+    {1, 1, 1, 0, 1.04916052873278, 0.478155746088161},
+    {1, 1, 1, 1, 1.00000000000000, 1.00000000000000},
+    {1, 1, 1, 2, 1.02005412633766, 1.47134936286465},
+    {1, 1, 2, -1, 1.62780548487271, 0.136827548397369},
+    {1, 1, 2, 0, 1.52731627482094, 0.571004782644619},
+    {1, 1, 2, 1, 1.47134936286465, 1.02005412633766},
+    {1, 1, 2, 2, 1.45679103104691, 1.45679103104691},
+    {1, 2, -1, -1, -0.574659643809835, 0.810208383866421},
+    {1, 2, -1, 0, -0.363823969691055, 1.15631976909117},
+    {1, 2, -1, 1, -0.136827548397369, 1.62780548487271},
+    {1, 2, -1, 2, 0.000000000000000, 2.11638793788003},
+    {1, 2, 0, -1, 0.985872907297986, 0.178866979357132},
+    {1, 2, 0, 0, 0.000000000000000, 0.000000000000000},
+    {1, 2, 0, 1, 0.420983902077155, 1.47839688785392},
+    {1, 2, 0, 2, 0.493004461502931, 2.02900451838781},
+    {1, 2, 1, -1, 1.35980258211696, 0.412918513270561},
+    {1, 2, 1, 0, 1.13815978557677, 0.897764105638205},
+    {1, 2, 1, 1, 1.02005412633766, 1.47134936286465},
+    {1, 2, 1, 2, 1.00000000000000, 2.00000000000000},
+    {1, 2, 2, -1, 1.79721035210339, 0.599070117367796},
+    {1, 2, 2, 0, 1.64623191253177, 1.05542091981433},
+    {1, 2, 2, 1, 1.54030226799894, 1.54030226799894},
+    {1, 2, 2, 2, 1.48529284479956, 2.02010318723916},
+    {2, -1, -1, -1, 0.412918513270561, -1.35980258211696},
+    {2, -1, -1, 0, 0.178866979357132, -0.985872907297986},
+    {2, -1, -1, 1, 0.810208383866421, 0.574659643809835},
+    {2, -1, -1, 2, 0.963366302386887, 0.963366302386887},
+    {2, -1, 0, -1, 0.897764105638205, -1.13815978557677},
+    {2, -1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {2, -1, 0, 1, 1.15631976909117, 0.363823969691055},
+    {2, -1, 0, 2, 1.37110999239961, 0.777111977967075},
+    {2, -1, 1, -1, 1.47134936286465, -1.02005412633766},
+    {2, -1, 1, 0, 1.47839688785392, -0.420983902077155},
+    {2, -1, 1, 1, 1.62780548487271, 0.136827548397369},
+    {2, -1, 1, 2, 1.79721035210339, 0.599070117367796},
+    {2, -1, 2, -1, 2.00000000000000, -1.00000000000000},
+    {2, -1, 2, 0, 2.02900451838781, -0.493004461502931},
+    {2, -1, 2, 1, 2.11638793788003, 0.000000000000000},
+    {2, -1, 2, 2, 2.23571522629432, 0.455159607692816},
+    {2, 0, -1, -1, 0.589593010800278, -0.966415146777638},
+    {2, 0, -1, 0, 0.422966208408802, 0.661266183461805},
+    {2, 0, -1, 1, 0.589593010800278, 0.966415146777638},
+    {2, 0, -1, 2, 0.777111977967075, 1.37110999239961},
+    {2, 0, 0, -1, 1.01005159361958, -0.742646422399778},
+    {2, 0, 0, 0, 0.000000000000000, 0.000000000000000},
+    {2, 0, 0, 1, 1.01005159361958, 0.742646422399778},
+    {2, 0, 0, 2, 1.19814023473559, 1.19814023473559},
+    {2, 0, 1, -1, 1.52731627482094, -0.571004782644619},
+    {2, 0, 1, 0, 1.45679103104691, 0.000000000000000},
+    {2, 0, 1, 1, 1.52731627482094, 0.571004782644619},
+    {2, 0, 1, 2, 1.64623191253177, 1.05542091981433},
+    {2, 0, 2, -1, 2.02900451838781, -0.493004461502931},
+    {2, 0, 2, 0, 2.00000000000000, 0.000000000000000},
+    {2, 0, 2, 1, 2.02900451838781, 0.493004461502931},
+    {2, 0, 2, 2, 2.09832105746556, 0.956311492176322},
+    {2, 1, -1, -1, 0.810208383866421, -0.574659643809835},
+    {2, 1, -1, 0, 0.178866979357132, 0.985872907297986},
+    {2, 1, -1, 1, 0.412918513270561, 1.35980258211696},
+    {2, 1, -1, 2, 0.599070117367796, 1.79721035210339},
+    {2, 1, 0, -1, 1.15631976909117, -0.363823969691055},
+    {2, 1, 0, 0, 0.000000000000000, 0.000000000000000},
+    {2, 1, 0, 1, 0.897764105638205, 1.13815978557677},
+    {2, 1, 0, 2, 1.05542091981433, 1.64623191253177},
+    {2, 1, 1, -1, 1.62780548487271, -0.136827548397369},
+    {2, 1, 1, 0, 1.47839688785392, 0.420983902077155},
+    {2, 1, 1, 1, 1.47134936286465, 1.02005412633766},
+    {2, 1, 1, 2, 1.54030226799894, 1.54030226799894},
+    {2, 1, 2, -1, 2.11638793788003, 0.000000000000000},
+    {2, 1, 2, 0, 2.02900451838781, 0.493004461502931},
+    {2, 1, 2, 1, 2.00000000000000, 1.00000000000000},
+    {2, 1, 2, 2, 2.02010318723916, 1.48529284479956},
+    {2, 2, -1, -1, 1.08423239187061, -0.238299975053003},
+    {2, 2, -1, 0, -0.0126420344317752, 1.32391495927282},
+    {2, 2, -1, 1, 0.267405171219803, 1.75269801601936},
+    {2, 2, -1, 2, 0.455159607692816, 2.23571522629432},
+    {2, 2, 0, -1, 1.32391495927282, -0.0126420344317752},
+    {2, 2, 0, 0, 0.000000000000000, 0.000000000000000},
+    {2, 2, 0, 1, 0.820905313474926, 1.53147158277780},
+    {2, 2, 0, 2, 0.956311492176322, 2.09832105746556},
+    {2, 2, 1, -1, 1.75269801601936, 0.267405171219803},
+    {2, 2, 1, 0, 1.53147158277780, 0.820905313474926},
+    {2, 2, 1, 1, 1.45679103104691, 1.45679103104691},
+    {2, 2, 1, 2, 1.48529284479956, 2.02010318723916},
+    {2, 2, 2, -1, 2.23571522629432, 0.455159607692816},
+    {2, 2, 2, 0, 2.09832105746556, 0.956311492176322},
+    {2, 2, 2, 1, 2.02010318723916, 1.48529284479956},
+    {2, 2, 2, 2, 2.00000000000000, 2.00000000000000},
+};
+
+int main()
+{
+    slong iter;
+    flint_rand_t state;
+
+    flint_printf("agm....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+
+    /* test particular values against table */
+    {
+        acb_t x, y, r, s;
+
+        acb_init(x);
+        acb_init(y);
+        acb_init(r);
+        acb_init(s);
+
+        for (iter = 0; iter < NUM_TESTS; iter++)
+        {
+            acb_set_d_d(x, agm_testdata[iter][0], agm_testdata[iter][1]);
+            acb_set_d_d(y, agm_testdata[iter][2], agm_testdata[iter][3]);
+            acb_set_d_d(s, agm_testdata[iter][4], agm_testdata[iter][5]);
+            mag_set_d(arb_radref(acb_realref(s)), 1e-13);
+            mag_set_d(arb_radref(acb_imagref(s)), 1e-13);
+
+            acb_agm(r, x, y, 64);
+
+            if (!acb_overlaps(r, s))
+            {
+                flint_printf("FAIL\n\n");
+                flint_printf("x = "); acb_printd(x, 15); flint_printf("\n\n");
+                flint_printf("y = "); acb_printd(y, 15); flint_printf("\n\n");
+                flint_printf("r = "); acb_printd(r, 15); flint_printf("\n\n");
+                flint_printf("s = "); acb_printd(s, 15); flint_printf("\n\n");
+                flint_abort();
+            }
+    }
+
+        acb_clear(x);
+        acb_clear(y);
+        acb_clear(r);
+        acb_clear(s);
+    }
+
+    /* self-consistency test */
+    for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
+    {
+        acb_t m1, m2;
+        acb_t x1, x2, y1, y2, t;
+        slong prec1, prec2;
+
+        prec1 = 2 + n_randint(state, 200);
+        prec2 = 2 + n_randint(state, 200);
+
+        acb_init(x1);
+        acb_init(x2);
+        acb_init(y1);
+        acb_init(y2);
+        acb_init(m1);
+        acb_init(m2);
+        acb_init(t);
+
+        acb_randtest(x1, state, prec1, 1 + n_randint(state, 100));
+        acb_randtest(y1, state, prec1, 1 + n_randint(state, 100));
+        acb_randtest(m1, state, prec1, 1 + n_randint(state, 100));
+        acb_randtest(m2, state, prec1, 1 + n_randint(state, 100));
+
+        if (n_randint(state, 2))
+        {
+            acb_randtest(t, state, prec2, 1 + n_randint(state, 200));
+            acb_mul(y1, x1, t, prec1);
+        }
+
+        if (n_randint(state, 2))
+            acb_swap(x1, y1);
+
+        if (n_randint(state, 2))
+        {
+            acb_set(x2, x1);
+        }
+        else
+        {
+            acb_randtest(t, state, prec2, 1 + n_randint(state, 200));
+            acb_add(x2, x1, t, prec2);
+            acb_sub(x2, x2, t, prec2);
+        }
+
+        if (n_randint(state, 2))
+        {
+            acb_set(y2, y1);
+        }
+        else
+        {
+            acb_randtest(t, state, prec2, 1 + n_randint(state, 200));
+            acb_add(y2, y1, t, prec2);
+            acb_sub(y2, y2, t, prec2);
+        }
+
+        acb_agm(m1, x1, y1, prec1);
+        acb_agm(m2, x2, y2, prec2);
+
+        if (!acb_overlaps(m1, m2))
+        {
+            flint_printf("FAIL (overlap)\n\n");
+            flint_printf("iter = %wd, prec1 = %wd, prec2 = %wd\n\n", iter, prec1, prec2);
+
+            flint_printf("x1 = "); acb_printd(x1, 30); flint_printf("\n\n");
+            flint_printf("y1 = "); acb_printd(y1, 30); flint_printf("\n\n");
+            flint_printf("x2 = "); acb_printd(x2, 30); flint_printf("\n\n");
+            flint_printf("y2 = "); acb_printd(y2, 30); flint_printf("\n\n");
+            flint_printf("m1 = "); acb_printd(m1, 30); flint_printf("\n\n");
+            flint_printf("m2 = "); acb_printd(m2, 30); flint_printf("\n\n");
+            flint_abort();
+        }
+
+        acb_div(t, x1, y1, prec1);
+        if (acb_is_finite(t) && (!acb_is_real(t) || !arb_is_negative(acb_realref(t))))
+        {
+            if (n_randint(state, 2))
+            {
+                acb_div(t, x1, y1, prec1);
+                acb_agm1(t, t, prec1);
+                acb_mul(t, t, y1, prec1);
+            }   
+            else
+            {
+                acb_div(t, y1, x1, prec1);
+                acb_agm1(t, t, prec1);
+                acb_mul(t, t, x1, prec1);
+            }
+
+            if (!acb_overlaps(t, m1))
+            {
+                flint_printf("FAIL (overlap 2)\n\n");
+                flint_printf("iter = %wd, prec1 = %wd, prec2 = %wd\n\n", iter, prec1, prec2);
+
+                flint_printf("x1 = "); acb_printd(x1, 30); flint_printf("\n\n");
+                flint_printf("y1 = "); acb_printd(y1, 30); flint_printf("\n\n");
+                flint_printf("m1 = "); acb_printd(m1, 30); flint_printf("\n\n");
+                flint_printf("t = "); acb_printd(t, 30); flint_printf("\n\n");
+                flint_abort();
+            }
+        }
+
+        acb_clear(x1);
+        acb_clear(x2);
+        acb_clear(y1);
+        acb_clear(y2);
+        acb_clear(m1);
+        acb_clear(m2);
+        acb_clear(t);
+    }
+
+    flint_randclear(state);
+    flint_cleanup();
+    flint_printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+
diff --git a/doc/source/acb.rst b/doc/source/acb.rst
index 47cd65a1..dc75484a 100644
--- a/doc/source/acb.rst
+++ b/doc/source/acb.rst
@@ -1049,6 +1049,16 @@ See :ref:`algorithms_agm` for implementation details.
     arithmetic-geometric mean at the point *z* truncated to length *len*, i.e.
     `M(z+x) \in \mathbb{C}[[x]]`.
 
+.. function:: void acb_agm(acb_t m, const acb_t x, const acb_t y, slong prec)
+
+    Sets *m* to the arithmetic-geometric mean of *x* and *y*. The square
+    roots in the AGM iteration are chosen so as to form the "optimal"
+    AGM sequence. This gives a well-defined function of *x* and *y* except
+    when `x / y` is a negative real number, in which case there are two
+    optimal AGM sequences. In that case, an arbitrary but consistent
+    choice is made (if a decision cannot be made due to inexact arithmetic,
+    the union of both choices is returned).
+
 Other special functions
 -------------------------------------------------------------------------------
 
diff --git a/doc/source/mag.rst b/doc/source/mag.rst
index 51880996..1bb83036 100644
--- a/doc/source/mag.rst
+++ b/doc/source/mag.rst
@@ -18,7 +18,7 @@ exact and therefore do not require separate *lower* versions.
 
 A common mistake is to forget computing a lower bound for the argument
 of a decreasing function that is meant to be bounded from above,
-or vice versa. For example, to compute and upper bound for `(x+1)/(y+1)`,
+or vice versa. For example, to compute an upper bound for `(x+1)/(y+1)`,
 the parameter *x* should initially be an upper bound while *y* should be
 a lower bound, and one should do::