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 nine forms (including the two raw IEEE forms). I will show examples of these forms in this article.

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While testing my new decimal to floating-point converter I discovered a bug in old territory: PHP incorrectly converts the number 2.2250738585072012e-308.

<?php printf("%.17g",2.2250738585072012e-308); ?>

This prints 2.2250738585072009E-308; it should print 2.2250738585072014e-308. (I verified that the internal double value is wrong; the printed value correctly represents it.)

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On Monday, Psy’s Gangnam Style video exceeded the limit of YouTube’s view counter; this is what Google had to say (hat tip: Digg):

“We never thought a video would be watched in numbers greater than a 32-bit integer (=2,147,483,647 views)…”

2,147,483,647 is 2^{31} – 1, the maximum positive value a 32-bit signed integer can contain.

Google has since fixed the counter, but they didn’t say how (32-bit unsigned integer? 64-bit integer?). (*Update:* By deduction from this Wall Street Journal article, Google is now using 64-bit signed integers — although the number they cite is 2^{63}, not 2^{63} – 1.)

The interesing thing is the “Easter egg” Google placed. If you hover your mouse over the counter, it spins like a slot machine; if you hold the mouse there long enough it will show a negative number. **But the negative number is not what I expected.** Is there a bug in the Easter egg?

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With a little research and some USB tracing, I wrote a Windows program — *and an Android app* — that turns my Fretlight guitar into a BCD mode binary clock!

My Fretlight BCD Clock (**click image for video**)

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To complete my exploration of numbers in App Inventor, I’ve written an app that converts integers between decimal and binary. It uses the standard algorithms, which I’ve just translated into blocks.

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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*.

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Variants of the question “Is floating point math broken?” are asked every day on Stackoverflow.com. I don’t think the questions will ever stop, not even by the year 2091 (that’s the year that popped into my head after just reading the gazillionth such question).

A representative floating-point question asked on stackoverflow.com in 2009, projected to be just as popular in 2091 (**click thumbnail to enlarge**)

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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.

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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”.

App Inventor Blocks To Display 0.1 to 17 Digits

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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.

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I want to contribute to the Hour of Code event happening now during Computer Science Education Week.

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.

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GCC, the GNU Compiler Collection, recently fixed this eight and a half year old bug: “GCC Bugzilla – Bug 21718: real.c rounding not perfect.” This bug was the cause of incorrect decimal string to binary floating-point conversions. I first wrote about it over three years ago, and then recently in September and October. I also just wrote a detailed description of GCC’s conversion algorithm last month.

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.

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