diff --git a/diffrax/_misc.py b/diffrax/_misc.py
index 5a5a3b67..c10b37eb 100644
--- a/diffrax/_misc.py
+++ b/diffrax/_misc.py
@@ -73,17 +73,22 @@ def fill_forward(
 
 
 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
+    positive_between = (t0 < t1) & (t0 <= t) & (t <= t1)
+    negative_between = (t1 < t0) & (t <= t0) & (t1 <= t)
+    return jnp.where(positive_between | negative_between, jnp.clip(out, 0, 1), out)
 
 
 def adjoint_rms_seminorm(x: tuple[PyTree, PyTree, PyTree, PyTree]) -> RealScalarLike: