Fixed-point arithmetic

:This article is about fixed-point arithmetic in computing. For mathematical fixed points, see fixed point (mathematics).

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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.

Related Topics:
Computing - Real data type

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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.

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Fixed-point can exactly represent decimal fractions while still employing the base 2 arithmetic that is

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efficient in most of today's computers.

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Most floating point representations in computers use base 2 values, which cannot exactly represent

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most fractions that are easily represented in base 10.

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For example, one-tenth (.1) and one-hundredth (.01) can be represented only approximately

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by base-2 floating point representations, while they can be represented exactly in fixed-point representations — one simply stores the data values multiplied by the appropriate power of 10.

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As long as the numeric value uses only the number of digits specified after the

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decimal point, fixed-point values

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can exactly represent all values up to its maximum value (determined by the number of bits

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in its representation).

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This is in contrast to floating-point representations, which include

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an automatically-managed exponent but cannot represent as many digits accurately (given the

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same number of bits in its representation).

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A common use for fixed-point numbers is for storing monetary values, where the inexact values

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of floating-point numbers are often a liability.

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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

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improved performance necessary for an application. Historically,

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fixed-point representations were the norm for decimal data types (for example, in PL/I or COBOL).

Related Topics:
PL/I - COBOL

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The Ada programming language includes built-in support

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for both fixed-point and floating-point.

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Very few computer languages include built-in support for fixed 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.

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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.

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However, if they are needed, fixed-point numbers can be implemented even

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in programming languages like

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C and

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C++ that do not include such support built-in.

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Here are some examples of fixed-point representations still in common use:

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• 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 and Toast are software libraries that decode the Ogg Vorbis and GSM Full Rate audio formats respectively. These codecs use fixed-point arithmetic 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 produce output in real time.
• All 3D graphics engines on Sony's original PlayStation and Nintendo's Game Boy Advance and Nintendo DS video game systems use fixed-point arithmetic for the same reason as Tremor and Toast: to gain throughput on an architecture without an FPU.
 
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