How is floating point stored? When does it matter?

Question

In follow up to this question, it appears that some numbers cannot be represented by floating point at all, and instead are approximated.

How are floating point numbers stored?

Is there a common standard for the different sizes?

What kind of gotchas do I need to watch out for if I use floating point?

Are they cross-language compatible (ie, what conversions do I need to deal with to send a floating point number from a python program to a C program over TCP/IP)?

Answer 1

As mentioned, the Wikipedia article on IEEE 754 does a good job of showing how floating point numbers are stored on most systems.

Now, here are some common gotchas:

• The biggest is that you almost never want to compare two floating point numbers for equality (or inequality). You'll want to use greater than/less than comparisons instead.
• The more operations you do on a floating point number, the more significant rounding errors can become.
• Precision is limited by the size of the fraction, so you may not be able to correctly add numbers that are separated by several orders of magnitude. (For example, you won't be able to add 1E-30 to 1E30.)

Answer 2

The standard is IEEE 754.

Of course, there are other means to store numbers when IEE754 isn't good enough. Libraries like Java's BigDecimal are available for most platforms and map well to SQL's number type. Symbols can be used for irrational numbers, and ratios that can't be accurately represented in binary or decimal floating point can be stored as a ratio.