diff --git a/doc/source/mag.rst b/doc/source/mag.rst
index bf0c5757..fcd3e24e 100644
--- a/doc/source/mag.rst
+++ b/doc/source/mag.rst
@@ -164,6 +164,14 @@ Arithmetic
 
     Sets *z* to a lower bound for `x^e`.
 
+.. function:: void mag_sqrt(mag_t z, const mag_t x)
+
+    Sets *z* to an upper bound for `\sqrt{x}`.
+
+.. function:: void mag_rsqrt(mag_t z, const mag_t x)
+
+    Sets *z* to an upper bound for `1/\sqrt{x}`.
+
 Fast, unsafe versions
 -------------------------------------------------------------------------------
 
diff --git a/mag.h b/mag.h
index 68f053c7..e3d3dcd5 100644
--- a/mag.h
+++ b/mag.h
@@ -649,7 +649,6 @@ double mag_d_log_lower_bound(double x);
 
 void mag_log1p(mag_t z, const mag_t x);
 
-/* TODO: test */
 void mag_pow_ui(mag_t z, const mag_t x, ulong e);
 void mag_pow_ui_lower(mag_t z, const mag_t x, ulong e);
 
@@ -663,7 +662,6 @@ void mag_bernoulli_div_fac_ui(mag_t z, ulong n);
 /* TODO: test */
 void mag_set_fmpz_2exp_fmpz_lower(mag_t z, const fmpz_t man, const fmpz_t exp);
 
-/* TODO: document */
 void mag_sqrt(mag_t y, const mag_t x);
 void mag_rsqrt(mag_t y, const mag_t x);
 
diff --git a/mag/pow_ui.c b/mag/pow_ui.c
index 96095986..974b8939 100644
--- a/mag/pow_ui.c
+++ b/mag/pow_ui.c
@@ -28,7 +28,7 @@
 void
 mag_pow_ui(mag_t z, const mag_t x, ulong e)
 {
-    if (mag_is_inf(z))
+    if (mag_is_inf(x))
     {
         mag_inf(z);
     }
@@ -74,7 +74,7 @@ mag_pow_ui_lower(mag_t z, const mag_t x, ulong e)
         else
             mag_mul_lower(z, x, x);
     }
-    else if (mag_is_inf(z))
+    else if (mag_is_inf(x))
     {
         mag_inf(z);
     }
diff --git a/mag/test/t-pow_ui.c b/mag/test/t-pow_ui.c
new file mode 100644
index 00000000..cef36b0e
--- /dev/null
+++ b/mag/test/t-pow_ui.c
@@ -0,0 +1,101 @@
+/*=============================================================================
+
+    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) 2014 Fredrik Johansson
+
+******************************************************************************/
+
+#include "mag.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("pow_ui....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 100000; iter++)
+    {
+        fmpr_t x, y, w;
+        mag_t xb, yb;
+        ulong e;
+
+        fmpr_init(x);
+        fmpr_init(y);
+        fmpr_init(w);
+
+        mag_init(xb);
+        mag_init(yb);
+
+        mag_randtest_special(xb, state, 80);
+        mag_randtest_special(yb, state, 80);
+        e = n_randtest(state);
+
+        mag_get_fmpr(x, xb);
+
+        fmpr_pow_sloppy_ui(y, x, e, 2 * MAG_BITS, FMPR_RND_UP);
+        if (fmpr_is_nan(y))
+            fmpr_pos_inf(y);
+
+        mag_pow_ui(yb, xb, e);
+        mag_get_fmpr(w, yb);
+
+        MAG_CHECK_BITS(xb)
+        MAG_CHECK_BITS(yb)
+
+        if (!(fmpr_cmpabs(y, w) <= 0))
+        {
+            printf("FAIL\n\n");
+            printf("e = %lu\n\n", e);
+            printf("x = "); fmpr_print(x); printf("\n\n");
+            printf("y = "); fmpr_print(y); printf("\n\n");
+            printf("w = "); fmpr_print(w); printf("\n\n");
+            abort();
+        }
+
+        mag_pow_ui(xb, xb, e);
+
+        if (!mag_equal(xb, yb))
+        {
+            printf("FAIL (aliasing)\n\n");
+            printf("e = %lu\n\n", e);
+            mag_print(xb); printf("\n\n");
+            mag_print(yb); printf("\n\n");
+            abort();
+        }
+
+        fmpr_clear(x);
+        fmpr_clear(y);
+        fmpr_clear(w);
+
+        mag_clear(xb);
+        mag_clear(yb);
+    }
+
+    flint_randclear(state);
+    flint_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+
diff --git a/mag/test/t-pow_ui_lower.c b/mag/test/t-pow_ui_lower.c
new file mode 100644
index 00000000..1aa75914
--- /dev/null
+++ b/mag/test/t-pow_ui_lower.c
@@ -0,0 +1,101 @@
+/*=============================================================================
+
+    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) 2014 Fredrik Johansson
+
+******************************************************************************/
+
+#include "mag.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("pow_ui_lower....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 100000; iter++)
+    {
+        fmpr_t x, y, w;
+        mag_t xb, yb;
+        ulong e;
+
+        fmpr_init(x);
+        fmpr_init(y);
+        fmpr_init(w);
+
+        mag_init(xb);
+        mag_init(yb);
+
+        mag_randtest_special(xb, state, 80);
+        mag_randtest_special(yb, state, 80);
+        e = n_randtest(state);
+
+        mag_get_fmpr(x, xb);
+
+        fmpr_pow_sloppy_ui(y, x, e, 2 * MAG_BITS, FMPR_RND_DOWN);
+        if (fmpr_is_nan(y))
+            fmpr_pos_inf(y);
+
+        mag_pow_ui_lower(yb, xb, e);
+        mag_get_fmpr(w, yb);
+
+        MAG_CHECK_BITS(xb)
+        MAG_CHECK_BITS(yb)
+
+        if (!(fmpr_cmpabs(w, y) <= 0))
+        {
+            printf("FAIL\n\n");
+            printf("e = %lu\n\n", e);
+            printf("x = "); fmpr_print(x); printf("\n\n");
+            printf("y = "); fmpr_print(y); printf("\n\n");
+            printf("w = "); fmpr_print(w); printf("\n\n");
+            abort();
+        }
+
+        mag_pow_ui_lower(xb, xb, e);
+
+        if (!mag_equal(xb, yb))
+        {
+            printf("FAIL (aliasing)\n\n");
+            printf("e = %lu\n\n", e);
+            mag_print(xb); printf("\n\n");
+            mag_print(yb); printf("\n\n");
+            abort();
+        }
+
+        fmpr_clear(x);
+        fmpr_clear(y);
+        fmpr_clear(w);
+
+        mag_clear(xb);
+        mag_clear(yb);
+    }
+
+    flint_randclear(state);
+    flint_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+