diff --git a/maths/fast.hpp b/maths/fast.hpp
index e9dfe108..9a8c0ef5 100644
--- a/maths/fast.hpp
+++ b/maths/fast.hpp
@@ -21,9 +21,11 @@ namespace cruft::maths::fast {
         return std::fma (b, t, a) + std::fma (d, t, c) * t2;
     }
 
-    // approximates sin over the range [0; pi/4] using a polynomial.
+    // approximates sin over the range [-2pi, 2pi] using a polynomial
+    // approximation over [0; pi/4]
     //
     // the relative error should be on the order of 1e-6.
+    // the absolute error should be maximally around 1e-5
     //
     // sollya:
     //   display=hexadecimal;
@@ -38,15 +40,21 @@ namespace cruft::maths::fast {
     constexpr float
     sin (float x) noexcept
     {
-        CHECK_LT (x,  2 * pi<float>);
-        CHECK_GT (x, -2 * pi<float>);
+        CHECK_LE (x,  2 * pi<float>);
+        CHECK_GE (x, -2 * pi<float>);
 
-        // flip negative angles due to having an even function
-        float neg = x < 0 ? x *= -1, -1 : 1;
+        // Push values into the range [0, 2pi]
+        if (x < 0)
+            x += 2 * pi<float>;
 
-        // reduce the angle due to the period
-        int periods = static_cast<int> (x / 2 / pi<float>);
-        x -= periods * 2 * pi<float>;
+        // Values in the range [pi, 2pi] are the negation of [0, pi]
+        float neg = x > pi<float> ? -1 : 1;
+        if (x > pi<float>)
+            x -= pi<float>;
+
+        // Values in [0,pi] are symmetric about pi/2
+        if (x > pi<float> / 2)
+            x = pi<float> - x;
 
         // 1,3,5,7:      1, -0x1.555546p-3f,  0x1.11077ap-7f, -0x1.9954e8p-13f;
         // 1,3,5,7,9:    1, -0x1.555556p-3f,  0x1.11115cp-7f, -0x1.a0403ap-13f,  0x1.75dc26p-19f;