diff --git a/diffrax/_misc.py b/diffrax/_misc.py
index ac61b813..1548d5b2 100644
--- a/diffrax/_misc.py
+++ b/diffrax/_misc.py
@@ -73,18 +73,36 @@ def fill_forward(
     return ys
 
 
+@jax.custom_jvp
+def _clip(x, cond):
+    return cast(jax.Array, jnp.where(cond, jnp.clip(x, 0, 1), x))
+
+
+@_clip.defjvp
+def _clip_jvp(inputs, tangents):
+    (x, cond) = inputs
+    (t_x, _) = tangents
+    return _clip(x, cond), t_x
+
+
 def linear_rescale(t0, t, t1) -> Array:
-    """Calculates (t - t0) / (t1 - t0), assuming t0 <= t <= t1.
+    """Calculates (t - t0) / (t1 - t0).
 
     Specially handles the edge case t0 == t1:
         - zero is returned;
         - gradients through all three arguments are zero.
+        - output conditionally clipped to be in [0,1] to protect
+        from floating point errors.
     """
 
     cond = t0 == t1
     numerator = cast(Array, jnp.where(cond, 0, t - t0))
     denominator = cast(Array, jnp.where(cond, 1, t1 - t0))
-    return numerator / denominator
+    out = numerator / denominator
+    # We need to clip due to GPU computations sometimes giving `(x/x) > 1`
+    # https://github.com/jax-ml/jax/issues/24807
+    clip_cond = ((t0 <= t) & (t <= t1)) | ((t1 <= t) & (t <= t0))
+    return _clip(out, clip_cond)
 
 
 def adjoint_rms_seminorm(x: tuple[PyTree, PyTree, PyTree, PyTree]) -> RealScalarLike: