Correctly rounded multiplication by arbitrary precision constants

Jean-Michel Muller (1, 2) , Nicolas Brisebarre (2, 3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 ARENAIRE - Computer arithmetic
3 LArAl - Laboratoire d'Arithmétique et d'Algèbre

We introduce an algorithm for multiplying a floating-point number $x$ by a constant $C$ that is not exactly representable in floating-point arithmetic. Our algorithm uses a multiplication and a fused multiply and add instruction. We give methods for checking whether, for a given value of $C$ and a given floating-point format, our algorithm returns a correctly rounded result for any $x$. When it does not, our methods give the values $x$ for which it does not.

Keywords: computer arithmetic; floating-point arithmetic; multiplication; fused multiply-add

Domains: Computer Science [cs]; Other [cs.OH]

Jean-Michel Muller, Nicolas Brisebarre. Correctly rounded multiplication by arbitrary precision constants. ARITH'17, 17th IEEE Symposium on Computer Arithmetic, EEE Computer Society Technical Committee on VLSI, Jun 2005, Cape Cod, United States. ⟨10.1109/ARITH.2005.13⟩. ⟨ensl-00000010⟩