In the 1960s, I. Bennett Goldberg and David W. Matula published papers relating floating-point number systems of different bases, showing the conditions under which conversions between them round-trip; that is, when conversion to another base and back returns the original number. Independently, both authors derived the formula that specifies the number of significant digits required for round-trip conversions.
In his paper “27 Bits Are Not Enough for 8-Digit Accuracy”, Goldberg shows the formula in the context of decimal to binary floating-point conversions. He starts with a simple example — a 7-bit binary floating-point system — and shows that it does not have enough precision to round-trip all 2-digit decimal floating-point numbers. I took his example and put it into diagrams, giving you a high level view of what governs round-trip conversions. I also extended his example to show that the same concept applies to binary to decimal floating-point round-trips.
The well-known digit counts for round-trip conversions to and from IEEE 754 floating-point are dictated by these same principles.
(Updated June 22, 2015: added a tenth display form, “decimal integer times a power of ten”.)
In the strictest sense, converting a decimal number to binary floating-point means putting it in IEEE 754 format — a multi-byte structure composed of a sign field, an exponent field, and a significand field. Viewing it in this raw form (binary or hex) is useful, but there are other forms that are more enlightening.
I’ve written an online converter that takes a decimal number as input, converts it to floating-point, and then displays its exact floating-point value in ten forms (including the two raw IEEE forms). I will show examples of these forms in this article.
I recently wrote that App Inventor represents its numbers in floating-point. I’ve since discovered something curious about integers. When typed into math blocks, they are represented in floating-point; but when generated through calculations, they are represented as arbitrary-precision integers — big integers.
After discovering that App Inventor represents numbers in floating-point, I wanted to see how it handled some edge case decimal/floating-point conversions. In one group of tests, I gave it numbers that were converted to floating-point incorrectly in other programming languages (I include the famous PHP and Java numbers). In another group of tests, I gave it numbers that, when converted to floating-point and back, demonstrate the rounding algorithm used when printing halfway cases. It turns out that App Inventor converts all examples correctly, and prints numbers using the round-half-to-even rule.
I am exploring App Inventor, an Android application development environment for novice programmers. I am teaching it to my kids, as an introductory step towards “real” app development. While playing with it I wondered: are its numbers implemented in decimal? No, they aren’t. They are implemented in double-precision binary floating-point. I put together a few simple block programs to demonstrate this, and to expose the usual floating-point “gotchas”.
GCC was recently fixed so that its decimal to floating-point conversions are done correctly; it now calls the MPFR function mpfr_strtofr() instead of using its own algorithm. However, GCC still does its conversion in two steps: first it converts to an intermediate precision (160 or 192 bits), and then it rounds that result to a target precision (53 bits for double-precision floating-point). That is double rounding — how does it avoid double rounding errors? It uses round-to-odd rounding on the intermediate result.
I don’t write about computer programming, but I do write extensively about how computers work — in particular, about how they do arithmetic with binary numbers. For your “hour of code” I’d like to introduce you to binary numbers and binary addition. I’ve selected several of my articles for you to read, and I’ve written some exercises you can try on my online calculators.
This fix, which will be available in version 4.9, scraps the old algorithm and replaces it with a call to MPFR function mpfr_strtofr(). I tested the fix on version 4.8.1, replacing its copy of gcc/real.c with the fixed one. I found no incorrect conversions.
While running some of GCC’s string to double conversion testcases I discovered a bug in David Gay’s strtod(): it converts some very small subnormal numbers incorrectly. Unlike numbers 2-1075 or smaller, which should convert to zero under round-to-nearest/ties-to-even rounding, numbers between 2-1075 and 2-1074 should convert to 2-1074, the smallest number representable in double-precision binary floating-point. strtod() correctly converts the former to 0, but it incorrectly converts the latter to 0 as well.
Water tested the conversion of 2-1075 — in retrospect an obvious corner case I should have tried — and found that it converted incorrectly to 0x0.0000000000001p-1022. That’s 2-1074, the smallest double-precision value. It should have converted to 0, under round-to-nearest/ties-to-even rounding.
(Update 11/13/13: This bug has been fixed for version 2.19.)