diff --git a/skyfield/searchlib.py b/skyfield/searchlib.py
index 0efcab37f..fc0b3ee69 100644
--- a/skyfield/searchlib.py
+++ b/skyfield/searchlib.py
@@ -66,7 +66,7 @@ def _find_discrete(ts, jd, f, epsilon, num):
             y = y.take(indices + 1)
             # Keep only the last of several zero crossings that might
             # possibly be separated by less than epsilon.
-            mask = concatenate(((diff(ends) > 3.0 * epsilon), (True,)))
+            mask = concatenate(((diff(ends) > epsilon), (True,)))
             ends = ends[mask]
             y = y[mask]
             break
diff --git a/skyfield/sgp4lib.py b/skyfield/sgp4lib.py
index 14139cde2..bc2eb29e1 100644
--- a/skyfield/sgp4lib.py
+++ b/skyfield/sgp4lib.py
@@ -272,7 +272,7 @@ def below_horizon_at(t):
         doublets = repeat(concatenate(((t0.tt,), jdmax, (t1.tt,))), 2)
         jdo = (doublets[:-1] + doublets[1:]) / 2.0
 
-        trs, rs = _find_discrete(t0.ts, jdo, below_horizon_at, half_second, 8)
+        trs, rs = _find_discrete(t0.ts, jdo, below_horizon_at, half_second/4, 8)
 
         jd = concatenate((jdmax, trs.tt))
         v = concatenate((ones, rs * 2))