## The Spacing of Decimal Floating-Point Numbers

In a computer, decimal floating-point numbers are converted to binary floating-point numbers for calculation, and binary floating-point numbers are converted to decimal floating-point numbers for display or storage. In general, these conversions are inexact; they are rounded, and rounding is governed by the spacing of numbers in each set.

Floating-point numbers are unevenly spaced, and the spacing varies with the base of the number system. Binary floating-point numbers have power of two sized gaps that change size at power of two boundaries. Decimal floating-point numbers are similarly spaced, but with power of ten sized gaps changing size at power of ten boundaries. In this article, I will discuss the spacing of decimal floating-point numbers.

## The Spacing of Binary Floating-Point Numbers

An IEEE 754 binary floating-point number is a number that can be represented in normalized binary scientific notation. This is a number like 1.00000110001001001101111 x 2-10, which has two parts: a significand, which contains the significant digits of the number, and a power of two, which places the “floating” radix point. For this example, the power of two turns the shorthand 1.00000110001001001101111 x 2-10 into this ‘longhand’ binary representation: 0.000000000100000110001001001101111.

The significands of IEEE binary floating-point numbers have a limited number of bits, called the precision; single-precision has 24 bits, and double-precision has 53 bits. The range of power of two exponents is also limited: the exponents in single-precision range from -126 to 127; the exponents in double-precision range from -1022 to 1023. (The example above is a single-precision number.)

Limited precision makes binary floating-point numbers discontinuous; there are gaps between them. Precision determines the number of gaps, and precision and exponent together determine the size of the gaps. Gap size is the same between consecutive powers of two, but is different for every consecutive pair.

## Nine Ways to Display a Floating-Point Number

(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.

## PHP Converts 2.2250738585072012e-308 Incorrectly

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.)

## Gangnam Style Video Overflows YouTube Counter

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 231 – 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 263, not 263 – 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?

## App Inventor Computes With Big Integers

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.

## Floating-Point Will Still Be Broken In the Year 2091

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).

## Testing Some Edge Case Conversions In App Inventor

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.

## Numbers In App Inventor Are Stored As Floating-Point

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 Avoids Double Rounding Errors With Round-To-Odd

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.

## real.c Rounding Is Perfect (GCC Now Converts Correctly)

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.