打印浮点数的精确值

Question

How does a program (MySQL is an example) store a float like 0.9 and then return it to me as 0.9? Why does it not return 0.899...?

The issue I am currently experiencing is retrieving floating point values from MySQL with C++ and then reprinting those values.

Answer 1

What's happening, in a nutshell, is that the function which converts the floating-point approximation of 0.9 to decimal text is actually coming up with a value like 0.90000....0123 or 0.89999....9573 . This gets rounded to 0.90000...0 . And then these trailing zeros are trimmed off so you get a tidy looking 0.9 .

Although floating-point numbers are inexact, and often do not use base 10 internally, they can in fact precisely save and recover a decimal representation. For instance, an IEEE 754 64 bit representation has enough precision to preserve 15 decimal digits. This is often mapped to the C language type double , and that language has the constant DBL_DIG , which will be 15 when double is this IEEE type.

If a decimal number with 15 digits or less is converted to double, it can be coverted back to exactly that number. The conversion routine just has to round it off at 15 digits; of course if the conversion routine uses, say, 40 digits, there will be messy trailing digits representing the error between the floating-point value and the original number. The more digits you print, the more accurately rendered is that error.

There is also the opposite problem: given a floating-point object, can it be printed into decimal such that the resulting decimal can be scanned back to reproduce that object? For an IEEE 64 bit double, the number of decimal digits required for that is 17.

Answer 2

There are software libraries, like Gnu MP that implement arbitrary precision arithmetic, that calculate floating point numbers to specified precision. Using Gnu MP you can, for example, add 0.3 to 0.6, and get exactly 0.9. No more, no less.

Database servers do pretty much the same thing.

For normal, run of the mill applications, native floating point arithmetic is fast, and it's good enough. But database servers typically have plenty of spare CPU cycles. Their limiting factors will not be available CPU, but things like available I/O bandwidth. They'll have plenty of CPU cycles to burn on executing complicated arbitrary precision arithmetic calculations.

Answer 3

There are a number of algorithms for rounding floating point numbers in a way that will result in the same internal representation when read back in. For an overview of the subject, with links to papers with full details of the algorithms, see

Printing Floating-Point Numbers