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”.
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.
For years I’ve followed, through RSS, floating-point related questions on stackoverflow.com. Every day it seems there is a question like “why does 19.24 plus 6.95 equal 26.189999999999998?” I decided to track these questions, to see if my sense of their frequency was correct. I found that, in the last 40 days, there were 18 such questions. That’s not one per day, but still — a lot!
I read about an interesting method for decimal/binary conversion in chapter two of Gerald R. Rising’s book “Inside Your Calculator”. Unlike the standard string-oriented conversion algorithms, the algorithms in his book perform arithmetic on an encoding I’ve dubbed “deci-binary”. Using this encoding, binary numbers are input and output as decimal strings consisting of only ones and zeros, exploiting built-in language facilities for decimal input and output. I will demonstrate this conversion process with Java code I have written.
Paul Bristow, a Boost.Math library author and reader of my blog, recently alerted me to a problem he discovered many years ago in Visual C++: some double-precision floating-point values fail to round-trip through a stringstream as a 17-digit decimal string. Interestingly, the 17-digit strings that C++ generates are not the problem; they are correctly rounded. The problem is that the conversion of those strings to floating-point is sometimes incorrect, off by one binary ULP.
I’ve previously discovered that Visual Studio makes incorrect decimal to floating-point conversions, and that Microsoft is OK with it — at least based on their response to my now deleted bug report. But incorrect decimal to floating-point conversions in this context seems like a problem that needs fixing. When you serialize a double to a 17-digit decimal string, shouldn’t you get the same double back later? Apparently Microsoft doesn’t think so, because Paul’s bug report has also been deleted.
The exact decimal equivalent of an arbitrary double-precision binary floating-point number is typically an unwieldy looking number, like this one:
In general, when you print a floating-point number, you don’t want to see all its digits; most of them are “garbage” in a sense anyhow. But how many digits do you need? You’d like a short string, yet you’d want it long enough so that it identifies the original floating-point number. A well-known result in computer science is that you need 17 significant decimal digits to identify an arbitrary double-precision floating-point number. If you were to round the exact decimal value of any floating-point number to 17 significant digits, you’d have a number that, when converted back to floating-point, gives you the original floating-point number; that is, a number that round-trips. For our example, that number is 0.10000000000000001.
But 17 digits is the worst case, which means that fewer digits — even as few as one — could work in many cases. The number required depends on the specific floating-point number. For our example, the short string 0.1 does the trick. This means that 0.1000000000000000055511151231257827021181583404541015625 and 0.10000000000000001 and 0.1 are the same, at least as far as their floating-point representations are concerned.
This is a companion article to “Number of Bits in a Decimal Integer”
If you have an integer expressed in decimal and want to know how many bits are required to express it in binary, you can perform a simple calculation. If you want to know how many bits are required to express a d-digit decimal integer in binary, you can perform other simple calculations for that.
What if you want to go in the opposite direction, that is, from binary to decimal? There are similar calculations for determining the number of decimal digits required for a specific binary integer or for a b-bit binary integer. I will show you these calculations, which are essentially the inverses of their decimal to binary counterparts.
Numbers are entered into computers as strings of text. These strings are converted to binary, into the numeric form understood by the computer’s hardware. Numbers with a decimal point — numbers we think of as real numbers — are converted into a format called binary floating-point. The procedure that converts decimal strings to binary floating-point — IEEE double-precision binary floating-point in particular — goes by the name strtod(), which stands for string to double.
Converting decimal strings to doubles correctly and efficiently is a surprisingly complex task; fortunately, David M. Gay did this for us when he wrote this paper and released this code over 20 years ago. (He maintains this reference copy of strtod() to this day.) His code appears in many places, including in the Python, PHP, and Java programming languages, and in the Firefox, Chrome, and Safari Web browsers.
I’ve spent considerable time reverse engineering strtod(); neither the paper nor the code are easy reads. I’ve written articles about how each of its major pieces work, and I’ve discovered bugs (as have a few of my readers) along the way. This article ties all of my strtod() research together.
Many new programmers become aware of binary floating-point after seeing their programs give odd results: “Why does my program print 0.10000000000000001 when I enter 0.1?”; “Why does 0.3 + 0.6 = 0.89999999999999991?”; “Why does 6 * 0.1 not equal 0.6?” Questions like these are asked every day, on online forums like stackoverflow.com.
The answer is that most decimals have infinite representations in binary. Take 0.1 for example. It’s one of the simplest decimals you can think of, and yet it looks so complicated in binary:
The bits go on forever; no matter how many of those bits you store in a computer, you will never end up with the binary equivalent of decimal 0.1.
I’ve shown you two ways to convert a bicimal to a fraction: the subtraction method and the direct method. In this article, I will show you a third method — a common method I call the series method — that uses the formula for infinite geometric series to create the fraction.