I recently upgraded my eight year old site to use a responsive design, making it mobile-friendly. I’ve kept the basic look of the site, although I retired the header image. The site is more readable, both on desktop and mobile devices.
Every double-precision floating-point number can be specified with 17 significant decimal digits or less. A simple way to generate this 17-digit number is to round the full-precision decimal value of the double to 17 digits. For example, the double-precision value 0x1.6d4c11d09ffa1p-3, which in decimal is 1.783677474777478899614635565740172751247882843017578125 x 10-1, can be recovered from the decimal floating-point literal 1.7836774747774789e-1. The extra digits are unnecessary, since they will only take you to the same double.
On the other hand, an arbitrary, arbitrarily long decimal literal rounded or truncated to 17 digits may not convert to the double-precision value it’s supposed to. This is a subtle point, one that has even tripped up implementers of widely used decimal to floating-point conversion routines (glibc strtod() and Visual C++ strtod(), for example).
How many decimal digits of precision does a binary floating-point number have?
For example, does an IEEE single-precision binary floating-point number, or float as it’s known, have 6-8 digits? 7-8 digits? 6-9 digits? 6 digits? 7 digits? 7.22 digits? 6-112 digits? (I’ve seen all those answers on the Web.)
I’ve always thought Java was one of the languages that prints the shortest decimal strings that round-trip back to floating-point. I was wrong.
Any double-precision floating-point number can be identified with at most 17 significant decimal digits. This means that if you convert a floating-point number to a decimal string, round it (to nearest) to 17 digits, and then convert that back to floating-point, you will recover the original floating-point number. In other words, the conversion will round-trip.
Sometimes (many) fewer than 17 digits will serve to round-trip; it is often desirable to find the shortest such string. Some programming languages generate shortest decimal strings, but many do not.1 If your language does not, you can attempt this yourself using brute force, by rounding a floating-point number to increasing length decimal strings and checking each time whether conversion of the string round-trips. For double-precision, you’d start by rounding to 15 digits, then if necessary to 16 digits, and then finally, if necessary, to 17 digits.
There is an interesting anomaly in this process though, one that I recently learned about from Mark Dickinson on stackoverflow.com: in rare cases, it’s possible to overlook the shortest decimal string that round-trips. Mark described the problem in the context of single-precision binary floating-point, but it applies to double-precision binary floating-point as well — or any precision binary floating-point for that matter. I will look at this anomaly in the context of double-precision floating-point, and give a detailed analysis of its cause.
PHP’s base_convert() is a useful function that converts integers between any pair of bases, 2 through 36. However, you might hesitate to use it after reading this vague and mysterious warning in its documentation:
base_convert() may lose precision on large numbers due to properties related to the internal “double” or “float” type used.
The truth is that it works perfectly for integers up to a certain maximum — you just have to know what that is. I will show you this maximum value in each of the 35 bases, and how to check if the values you are using are within this limit.
I’ve previously written a Windows program (in C++) and an Android app (in Java) to turn my Fretlight guitar into a binary clock. I’ve now written a Python program to do the same, running under Raspbian Linux on a Raspberry Pi computer. I will show you the code and tell you how to run it.
The 2015 edition of Hour of Code includes a new blocks-based, Star Wars themed coding lesson. In one of the exercises — a simple sprite-based game — you are asked to code a loop that adds 100 to your score every time R2-D2 encounters a Rebel Pilot. But instead of 100, I plugged in a floating-point number; I got the expected “unexpected” results.
About a year ago Bruce Dawson informed me that Microsoft is fixing their decimal to floating-point conversion routine in the next release of Visual Studio; I finally made the time to test the new code. I installed Visual Studio Community 2015 Release Candidate and ran my old C++ testcases. The good news: all of the individual conversion errors that I wrote about are fixed. The bad news: many errors remain.
I have been writing about the spacing of decimal and binary floating-point numbers, and about how their relative spacing determines whether numbers round-trip between those two bases. I’ve stated an inequality that captures the required spacing, and from it I have derived formulas that specify the number of digits required for round-trip conversions. I have not proven that this inequality holds, but I will prove “half” of it here. (I’m looking for help to prove the other half.)
In my article “7 Bits Are Not Enough for 2-Digit Accuracy” I showed how the relative spacing of decimal and binary floating-point numbers dictates when all conversions between those two bases round-trip. There are two formulas that capture this relationship, and I will derive them in this article. I will also show that it takes one more digit (or bit) of precision to round-trip a floating-point number than it does to round-trip an integer of equal significance.