Easy Accurate Reading and Writing of Floating-Point Numbers

Presented here are algorithms for converting between (decimal) scientific-notation and (binary) IEEE-754 double-precision floating-point numbers. These algorithms are much simpler than those previously published. The values are stable under repeated conversions between the formats. The scientific representations generated have only the minimum number of mantissa digits needed to convert back to the original binary values. Also presented is an algorithm for printing IEEE-754 double-precision floating-point numbers with a specified number of decimal digits of precision. For the specified number of digits, the decimal numbers produced are the closest possible to the binary values. Introduction Articles from Steele and White[SW90], Clinger[Cli90], and Burger and Dybvig[BD96] establish that binary floating-point numbers can be converted into and out of decimal representations without losing accuracy using a minimum number of (decimal) significant digits. The lossless algorithms from these papers all require high-precision integer calculations, although not for every conversion. In How to Read Floating-Point Numbers Accurately[Cli90] Clinger astutely observes that successive rounding operations do not have the same effect as a single rounding operation. This is the crux of the difficulty with both reading and writing floating-point numbers. But instead of constructing his algorithm to do a single rounding operation, Clinger and the other authors follow Matula[Mat68, Mat70] in doing successive integer divisions and remainders. Both Steel and White[SW90] and Clinger[Cli90] claim that the input and output problems are fundamentally different from each other because the floating-point format has a fixed precision while the decimal representation does not. Yet, bignum rounding divisions accomplish accurate conversions in both directions. The algorithms from How to Print Floating-point Numbers Accurately[SW90] and Printing floatingpoint numbers quickly and accurately[BD96] are iterative and complicated. The read and write algorithms presented here do at most 2 and 4 bignum divisions, respectively. Over the range of IEEE-754[IEE85] doubleprecision numbers, the largest intermediate bignum used by presented algorithms is 339 decimal digits (1126 bits). According to Steele and White[SW90], the largest bignum used by their algorithm is 1050 bits. These are not large for bignums, being orders of magnitude smaller than the smallest precisions which get speed benefits from FFT multiplication. 1 Digilant, 100 North Washington Street Suite 502, Boston, MA 02114. Email: [email protected]