diff --git a/array_api_strict/_elementwise_functions.py b/array_api_strict/_elementwise_functions.py
index b39bd86..6454764 100644
--- a/array_api_strict/_elementwise_functions.py
+++ b/array_api_strict/_elementwise_functions.py
@@ -300,14 +300,35 @@ def clip(
     if min is not None and max is not None and np.any(min > max):
         raise ValueError("min must be less than or equal to max")
 
-    result = np.clip(x._array, min, max)
     # Note: NumPy applies type promotion, but the standard specifies the
-    # return dtype should be the same as x
-    if result.dtype != x.dtype._np_dtype:
-        # TODO: I'm not completely sure this always gives the correct thing
-        # for integer dtypes. See https://github.com/numpy/numpy/issues/24976
-        result = result.astype(x.dtype._np_dtype)
-    return Array._new(result)
+    # return dtype should be the same as x We do this instead of just
+    # downcasting the result of np.clip() to handle some corner cases better
+    # (e.g., avoiding uint64 -> float64 promotion).
+
+    # Note: cases where min or max overflow (integer) or round (float) in the
+    # wrong direction when downcasting to x.dtype are unspecified. This code
+    # just does whatever NumPy does when it downcasts in the assignment, but
+    # other behavior could be preferred, especially for integers. For example,
+    # this code produces:
+
+    # >>> clip(asarray(0, dtype=int8), asarray(128, dtype=int16), None)
+    # -128
+
+    # but an answer of 0 might be preferred. See
+    # https://github.com/numpy/numpy/issues/24976 for more discussion on this issue.
+    min_shape = () if min is None else min.shape
+    max_shape = () if max is None else max.shape
+    result_shape = np.broadcast_shapes(x.shape, min_shape, max_shape)
+    out = asarray(np.broadcast_to(x._array, result_shape), copy=True)._array
+    if min is not None:
+        a = np.broadcast_to(min, result_shape)
+        ia = (out < a) | np.isnan(a)
+        out[ia] = a[ia]
+    if max is not None:
+        b = np.broadcast_to(max, result_shape)
+        ib = (out > b) | np.isnan(b)
+        out[ib] = b[ib]
+    return Array._new(out)
 
 def conj(x: Array, /) -> Array:
     """