We show for the first time how to reduce the cost of performing specific geometric constructions by using rounded arithmetic instead of exact arithmetic. Exploiting a property of floating point arithmetic called monotonicity, a new technique, double precision geometry, can replace exact arithmetic with rounded arithmetic in any efficient algorithm for computing the set of intersections of a set of lines or line segments. This technique reduces the complexity of any such line or segment arrangement algorithm by a constant factor. In addition, double precision geometry reduces by a factor of N the complexity of rendering segment arrangements on a 2N by 2N integer grid such that output segments have grid points as endpoints.