Supplementary material to “ Accelerating Correctly Rounded Floating - Point Division When the Divisor is Known in Advance ”

• If m > n, the exact quotient of two n-bit numbers cannot be an m-bit number. • Let x, y ∈ Mn. x = y ⇒ |x/y − 1| ≥ 2−n. We call a breakpoint a value z where the rounding changes, that is, if t1 and t2 are real numbers satisfying t1 < z < t2 and ◦t is the rounding mode, then ◦t(t1) < ◦t(t2). For “directed” rounding modes (i.e., towards +∞, −∞ or 0), the breakpoints are the FP numbers. For rounding to the nearest mode, they are the exact middle of two consecutive FP numbers. For a ∈ Mn, we define a as its successor in Mn, that is, a = min{b ∈ Mn, b > a}, and a− as the predecessor of a, that is, a− = max{b ∈ Mn, b < a}. The next result gives a lower bound on the distance between a breakpoint (in roundto-nearest mode) and the quotient of two FP numbers.