In my article “Quick and Dirty Decimal to Floating-Point Conversion” I presented a small C program that converts a decimal string to a double-precision binary floating-point number. The number it produces, however, is not necessarily the closest — or so-called *correctly rounded* — double-precision binary floating-point number. This is not a failing of the algorithm; mathematically speaking, the algorithm is correct. The flaw comes in its implementation in limited precision binary floating-point arithmetic.

The quick and dirty program is implemented in native C, so it’s limited to double-precision floating-point arithmetic (although on some systems, extended precision may be used). Higher precision arithmetic — in fact, arbitrary precision arithmetic — is needed to ensure that all decimal inputs are converted correctly. I will demonstrate the need for high precision by analyzing three examples, all taken from Vern Paxson’s paper “A Program for Testing IEEE Decimal–Binary Conversion”.

Continue reading “Decimal to Floating-Point Needs Arbitrary Precision”