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What Every Programmer Should Know About Floating-Point Arithmetic
-brazil- writes "Every programmer forum gets a steady stream of novice questions about numbers not 'adding up.' Apart from repetitive explanations, SOP is to link to a paper by David Goldberg which, while very thorough, is not very accessible for novices. To alleviate this, I wrote The Floating-Point Guide, as a floating-point equivalent to Joel Spolsky's excellent introduction to Unicode. In doing so, I learned quite a few things about the intricacies of the IEEE 754 standard, and just how difficult it is to compare floating-point numbers using an epsilon. If you find any errors or omissions, you can suggest corrections."
This article should mention strictfp in the section on Java.
If you think that every language except Java implements IEEE-754 to the letter, you are sadly mistakenly. That fact is Java can be used just fine for floating point work in most applications.
No, irrationality has nothing to do with it. It's a matter of numeric systems, i.e. binary vs. decimal. For example, 0.2 is a rational number. Express it in binary floating point and you'll see the problem: 2/10 is 1/5 is 1/101 in binary. Let's calculate the mantissa: 1/101=110011001100... (long division: 1/5->2/5->4/5->8/5=1,r3->6/5=1,r1->2/5->4/5->8/5...)
All numeric systems have this problem. It keeps tripping up programmers because of the conversion between them. Nobody would expect someone to write down 1/3 as a decimal number, but because people keep forgetting that computers use binary floating point numbers, they do expect them not to make rounding errors with numbers like 0.2.
to floating point, please, everyone should've read Everything you ever wanted to know about C types and part 2 (which explains fp too).
this will save a lot of time & questions to most beginning (and maybe mediocre) programmers.
Java have a strictfp keyword for strict IEEE-754 arithmetic.
The non-trivial problems with floating-point really only turn up in the kind of calculations where *any* format would have the same or worse problems (most scientific computing simply *cannot* be done in integers, as they overflow too easily).
Floating-point is an excellent tool, you just have to know what it can and cannot do.
The illegal we do immediately. The unconstitutional takes a little longer.
--Henry Kissinger
People interested into floating point math will find some very interesting materials and horror stories in the documents collected at the home page of professor William Kahan, the man behind IEEE754 standard.
According to my personal experience the paper by David Goldberg cited in the post isn't that difficult after all. Plenty of interesting materials can also be found in the Oppenheim & Shafer textbook about digital signal processing.
I think the original poster was referring to this piece by the father of floating point, William Kahan, and Joe Darcy
"How Java's Floating-Point Hurts Everyone Everywhere"
http://www.eecs.berkeley.edu/~wkahan/JAVAhurt.pdf
Internal arithmetic always includes the exact solution, but only the rarest circumstances does it actually give the exact solution. For example, an acceptable interval answer for 1/3 would be [0.33,0.34]. That interval includes the exact answer, but does not express it.