Fixed-point arithmetic
In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after the decimal (or binary or hexadecimal) point. For example, a fixed-point number with 4 digits after the decimal point could be used to store numbers such as 1.3467, 281243.3234 and 0.1000, but would round 1.0301789 to 1.0302 and 0.0000654 to 0.0001.
Fixed-point can exactly represent decimal fractions while still employing the base 2 arithmetic efficient in most of today's computers. Most floating point representations in computers use base 2 values, which cannot exactly represent most fractions that are easily represented in base 10. For example, one-tenth (.1) and one-hundredth (.01) can only be represented approximately by floating point representations, while they can be represented exactly in fixed-point representations.
As long as the numeric value uses only the number of digits specified after the decimal point, fixed-point values can exactly represent all values up to its maximum value (determined by the number of bits in its representation). This is in contrast to floating-point representations, which include an automatically-managed exponent but cannot represent as many digits accurately (given the same number of bits in its representation).
A common use for fixed-point numbers is for storing monetary values, where the "inexact" values of floating-point numbers are often a liability. Fixed-point representations are also sometimes used if either the executing processor does not have any floating point unit (FPU) or if fixed-point provides an improved performance necessary for an application. Historically, fixed-point representations were the norm for decimal data types (for example, in PL/I or COBOL). The Ada programming language includes built-in support for both fixed-point and floating-point.
Many computer languages do not include built-in support for floating point values, because for most applications, floating-point representations are fast enough and accurate enough. Floating-point representations are more flexible than fixed-point representations, because they can handle a wider dynamic range. Floating-point representations are also slightly easier to use, because they do not require programmers to specify the number of digits after the decimal point.
However, if they are needed, fixed-point numbers can be implemented even in programming languages like C and C++ that do not include such support built-in.
Here are some last-gasp examples of fixed-point representations:
- GnuCash is an application for tracking money. It is written in C and switched from a floating-point representation of money to a fixed-point implementation as of version 1.6. This change was made to avoid the potential rounding errors of floating-point representations.
- Tremor is a software library that decodes the Ogg Vorbis audio format. Tremor uses fixed-point because many audio decoding hardware devices do not have an FPU (to save money) and audio decoding requires enough performance that a software implementation of floating-point on low-speed devices would not work well.
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