Double rounding is when a number is rounded twice, first from n0 digits to n1 digits, and then from n1 digits to n2 digits. Double rounding is often harmless, giving the same result as rounding once, directly from n0 digits to n2 digits. However, sometimes a doubly rounded result will be incorrect, in which case we say that a double rounding error has occurred.
For example, consider the 6-digit decimal number 7.23496. Rounded directly to 3 digits — using round-to-nearest, round half to even rounding — it’s 7.23; rounded first to 5 digits (7.2350) and then to 3 digits it’s 7.24. The value 7.24 is incorrect, reflecting a double rounding error.
In a computer, double rounding occurs in binary floating-point arithmetic; the typical example is a calculated result that’s rounded to fit into an x87 FPU extended precision register and then rounded again to fit into a double-precision variable. But I’ve discovered another context in which double rounding occurs: conversion from a decimal floating-point literal to a single-precision floating-point variable. The double rounding is from full-precision binary to double-precision, and then from double-precision to single-precision.
In this article, I’ll show example conversions in C that are tainted by double rounding errors, and how attaching the ‘f’ suffix to floating-point literals prevents them — in gcc C at least, but not in Visual C++!