Scaling Newton-Raphson division iterations to avoid double rounding
Jean‐Michel Muller · 2010
When performing divisions using Newton-Raphson (or similar) iterations on a processor with a floating-point fused multiply-add instruction, one must sometimes scale the iterations, to avoid over/underflow and/or loss of accuracy. This may lead to double-roundings, resulting in output values that may not be correctly rounded when the quotient is in the subnormal range. We show how to avoid this problem. 1 introduction Throughout the paper, we assume a radix-2, precision-p, floating-point system that is compliant with the IEEE 754-2008 Standard for Floating-Point Arithmetic [4]. We also assume that a fused multiply-add (FMA) instruction is available. That instruction evaluates expressions of the form xy + z with one final rounding only. We also assume that the ambient rounding mode is round to nearest (this is the only one for which the problem we are dealing with, namely double rounding, occurs).