Fixed-point arithmetic
Encyclopedia
In computing
, a fixed-point number representation is a real data type
for a number that has a fixed number of digits after (and sometimes also before) the radix point
(after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating point
number representation.
Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor
has no floating point unit
(FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Most low-cost embedded
microprocessor
s and microcontroller
s do not have an FPU.
that is scaled by a specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1230000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.
The scaling factor is usually a power of 10 (for human convenience) or a power of 2 (for computational efficiency). However, other scaling factors may be used occasionally, e.g. a time value in hours may be represented as a fixed-point type with a scale factor of 1/3600 to obtain values with one-second accuracy.
The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type, multiplied by the scaling factor; and similarly for the minimum value. For example, consider a fixed-point type represented as a binary integer with b bits in two's complement
format, with a scaling factor of 1/2f (that is, the last f bits are fraction bits): the minimum representable value is −2b-1/2f and the maximum value is (2b-1-1)/2f.
. The rounding rules and methods are usually part of the language's specification.
To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow
occurs (i.e. provided that the sum of the two integers fits in the underlying integer type.) If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.
To multiply two fixed-point numbers, it suffices to multiply the two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. This operation involves no rounding. For example, multiplying the numbers 123 scaled by 1/1000 (0.123) and 25 scaled by 1/10 (2.5) yields the integer 123×25 = 3075 scaled by (1/1000)×(1/10) = 1/10000, that is 3075/10000 = 0.3075. If the two operands belong to the same fixed-point type, and the result is also to be represented in that type, then the product of the two integers must be explicitly multiplied by the common scaling factor; in this case the result may have to be rounded, and overflow may occur. For example, if the common scaling factor is 1/100, multiplying 1.23 by 0.25 entails multiplying 123 by 25 to yield 3075 with an intermediate scaling factor if 1/10000. This then must be multiplied by 1/100 to yield either 31 (0.31) or 30 (0.30), depending on the rounding method used, to result in a final scale factor of 1/100.
To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors. The first division involves rounding in general. For example, division of 3456 scaled by 1/100 (34.56) by 1234 scaled by 1/1000 (1.234) yields the integer 3456÷1234 = 3 (rounded) with scale factor (1/100)/(1/1000) = 10, that is, 30. One can obtain a more accurate result by first converting the dividend
to a more precise type: in the same example, converting 3456 scaled by 1/100 (34.56) to 3456000 scaled by 1/100000, before dividing by 1234 scaled by 1/1000 (1.234), would yield 3456000÷1234 = 2801 (rounded) with scaling factor (1/100000)/(1/1000) = 1/100, that is 28.01 (instead of 290). If both operands and the desired result are represented in the same fixed-point type, then the quotient of the two integers must be explicitly divided by the common scaling factor.
In terms of a quicker and easier programming approach, it is sometimes considered compilers should cast the two integer elements of a division to a floating point value, to avoid unexpected losses of precision.
Binary fixed-point types are most commonly used, because the rescaling operations can be implemented as fast bit shifts. Binary fixed-point numbers can represent fractional powers of two exactly, but, like binary floating-point numbers, cannot exactly represent fractional powers of ten. If exact fractional powers of ten are desired, then a decimal format should be used. For example, one-tenth (0.1) and one-hundredth (0.01) can be represented only approximately by binary fixed-point or binary floating-point representations, while they can be represented exactly in decimal fixed-point or decimal floating-point representations. These representations may be encoded in many ways, including BCD
.
s, there is possibility for information loss. For instance, the result of fixed point multiplication could potentially have as many bits as the sum of the number of bits in the two operands. In order to fit the result into the same number of bits as the operands, the answer must be rounded
or truncated. If this is the case, the choice of which bits to keep is very important. When multiplying two fixed point numbers with the same format, for instance with
integer bits, and 
fractional bits, the answer could have up to 
integer bits, and 
fractional bits.
For simplicity, fixed-point multiply procedures use the same result format as the operands. This has the effect of keeping the middle bits; the I-number of least significant integer bits, and the Q-number of most significant fractional bits. Fractional bits lost below this value represent a precision loss which is common in fractional multiplication. If any integer bits are lost, however, the value will be radically inaccurate.
Some operations, like divide, often have built-in result limiting so that any positive overflow results in the largest possible number that can be represented by the current format. Likewise, negative overflow results in the largest negative number represented by the current format. This built in limiting is often referred to as saturation.
Some processors support a hardware overflow flag
that can generate an exception
on the occurrence of an overflow, but it is usually too late to salvage the proper result at this point.
and C++
, which do not commonly include such support.
A common use of fixed-point BCD numbers is for storing monetary values, where the inexact values of binary floating-point numbers are often a liability. 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 (binary and decimal) and floating-point. JOVIAL and Coral 66 also provide both floating- and fixed-point types.
ISO/IEC TR 18037 specifies fixed-point data types for the C programming language; vendors are expected to implement the language extensions for fixed point arithmetic in coming years. Fixed-point support is implemented in GCC
.
Almost all relational database
s, and the SQL
query language, support fixed-point decimal arithmetic and storage of numbers. PostgreSQL
has a special numeric type for exact storage of numbers with up to 1000 digits.
Computing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
, a fixed-point number representation is a real data type
Real data type
A real data type is a data type used in a computer program to represent an approximation of a real number.Because the real numbers are not countable, computers cannot represent them exactly using a finite amount of information....
for a number that has a fixed number of digits after (and sometimes also before) the radix point
Radix point
In mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
(after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating point
Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
number representation.
Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor
Central processing unit
The central processing unit is the portion of a computer system that carries out the instructions of a computer program, to perform the basic arithmetical, logical, and input/output operations of the system. The CPU plays a role somewhat analogous to the brain in the computer. The term has been in...
has no floating point unit
Floating point unit
A floating-point unit is a part of a computer system specially designed to carry out operations on floating point numbers. Typical operations are addition, subtraction, multiplication, division, and square root...
(FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Most low-cost embedded
Embedded system
An embedded system is a computer system designed for specific control functions within a larger system. often with real-time computing constraints. It is embedded as part of a complete device often including hardware and mechanical parts. By contrast, a general-purpose computer, such as a personal...
microprocessor
Microprocessor
A microprocessor incorporates the functions of a computer's central processing unit on a single integrated circuit, or at most a few integrated circuits. It is a multipurpose, programmable device that accepts digital data as input, processes it according to instructions stored in its memory, and...
s and microcontroller
Microcontroller
A microcontroller is a small computer on a single integrated circuit containing a processor core, memory, and programmable input/output peripherals. Program memory in the form of NOR flash or OTP ROM is also often included on chip, as well as a typically small amount of RAM...
s do not have an FPU.
Representation
A value of a fixed-point data type is essentially an integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
that is scaled by a specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1230000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.
The scaling factor is usually a power of 10 (for human convenience) or a power of 2 (for computational efficiency). However, other scaling factors may be used occasionally, e.g. a time value in hours may be represented as a fixed-point type with a scale factor of 1/3600 to obtain values with one-second accuracy.
The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type, multiplied by the scaling factor; and similarly for the minimum value. For example, consider a fixed-point type represented as a binary integer with b bits in two's complement
Two's complement
The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two...
format, with a scaling factor of 1/2f (that is, the last f bits are fraction bits): the minimum representable value is −2b-1/2f and the maximum value is (2b-1-1)/2f.
Operations
To convert a number from a fixed point type with scaling factor R to another type with scaling factor S, the underlying integer must be multiplied by R and divided by S; that is, multiplied by the ratio R/S. Thus, for example, to convert the value 1.23 = 123/100 from a type with scaling factor R=1/100 to one with scaling factor S=1/1000, the underlying integer 123 must be multiplied by (1/100)/(1/1000) = 10, yielding the representation 1230/1000. If S does not divide R (in particular, if the new scaling factor R is less than the original S), the new integer will have to be roundedRounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
. The rounding rules and methods are usually part of the language's specification.
To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow
Arithmetic overflow
The term arithmetic overflow or simply overflow has the following meanings.# In a computer, the condition that occurs when a calculation produces a result that is greater in magnitude than that which a given register or storage location can store or represent.# In a computer, the amount by which a...
occurs (i.e. provided that the sum of the two integers fits in the underlying integer type.) If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.
To multiply two fixed-point numbers, it suffices to multiply the two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. This operation involves no rounding. For example, multiplying the numbers 123 scaled by 1/1000 (0.123) and 25 scaled by 1/10 (2.5) yields the integer 123×25 = 3075 scaled by (1/1000)×(1/10) = 1/10000, that is 3075/10000 = 0.3075. If the two operands belong to the same fixed-point type, and the result is also to be represented in that type, then the product of the two integers must be explicitly multiplied by the common scaling factor; in this case the result may have to be rounded, and overflow may occur. For example, if the common scaling factor is 1/100, multiplying 1.23 by 0.25 entails multiplying 123 by 25 to yield 3075 with an intermediate scaling factor if 1/10000. This then must be multiplied by 1/100 to yield either 31 (0.31) or 30 (0.30), depending on the rounding method used, to result in a final scale factor of 1/100.
To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors. The first division involves rounding in general. For example, division of 3456 scaled by 1/100 (34.56) by 1234 scaled by 1/1000 (1.234) yields the integer 3456÷1234 = 3 (rounded) with scale factor (1/100)/(1/1000) = 10, that is, 30. One can obtain a more accurate result by first converting the dividend
Dividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
to a more precise type: in the same example, converting 3456 scaled by 1/100 (34.56) to 3456000 scaled by 1/100000, before dividing by 1234 scaled by 1/1000 (1.234), would yield 3456000÷1234 = 2801 (rounded) with scaling factor (1/100000)/(1/1000) = 1/100, that is 28.01 (instead of 290). If both operands and the desired result are represented in the same fixed-point type, then the quotient of the two integers must be explicitly divided by the common scaling factor.
In terms of a quicker and easier programming approach, it is sometimes considered compilers should cast the two integer elements of a division to a floating point value, to avoid unexpected losses of precision.
Binary vs. decimal
The two most common fixed-point types are decimal and binary. Decimal fixed-point types have a scaling factor that is a power of ten, for binary fixed-point types it is a power of two.Binary fixed-point types are most commonly used, because the rescaling operations can be implemented as fast bit shifts. Binary fixed-point numbers can represent fractional powers of two exactly, but, like binary floating-point numbers, cannot exactly represent fractional powers of ten. If exact fractional powers of ten are desired, then a decimal format should be used. For example, one-tenth (0.1) and one-hundredth (0.01) can be represented only approximately by binary fixed-point or binary floating-point representations, while they can be represented exactly in decimal fixed-point or decimal floating-point representations. These representations may be encoded in many ways, including BCD
Binary-coded decimal
In computing and electronic systems, binary-coded decimal is a digital encoding method for numbers using decimal notation, with each decimal digit represented by its own binary sequence. In BCD, a numeral is usually represented by four bits which, in general, represent the decimal range 0 through 9...
.
Notation
There are various notations used to represent word length and radix point in a binary fixed-point number. In the following list, f represents the number of fractional bits, m the number of magnitude or integer bits, s the number of sign bits, and b the total number of bits.- Qf: The "Q" prefix. For example, Q15 represents a number with 15 fractional bits. This notation is ambiguous since it does not specify the word length, however it is usually assumed that the word length is either 16 or 32 bits depending on the target processor in use.
- Qm.f: The unambiguous form of the "Q" notation. Since the entire word is a 2's complement integer, a sign bit is implied. For example, Q1.30 describes a number with 1 integer bit and 30 fractional bits stored as a 32-bit 2's complement integer.
- fxm.b: The "fx" prefix is similar to the above, but uses the word length as the second item in the dotted pair. For example, fx1.16 describes a number with 1 magnitude bit and 15 fractional bits in a 16 bit word.
- s:m:f: Yet other notations include a sign bit, such as this one used in the PS2PlayStation 2The PlayStation 2 is a sixth-generation video game console manufactured by Sony as part of the PlayStation series. Its development was announced in March 1999 and it was first released on March 4, 2000, in Japan...
GSGameSharkGameShark is the brand name of a line of video game cheat cartridges and other products for a variety of console video game systems and Windows based computers. Currently, the brand name is owned by Mad Catz, who actively markets GameShark products for the PlayStation, Xbox, Nintendo, and Sega game...
User's Guide. It also differs from conventional usage by using a colon instead of a period as the separator. For example, in this notation, 0:8:0 represents an unsigned 8-bit integer.
Precision loss and overflow
Because fixed point operations can produce results that have more bits than the operandOperand
In mathematics, an operand is the object of a mathematical operation, a quantity on which an operation is performed.-Example :The following arithmetic expression shows an example of operators and operands:3 + 6 = 9\;...
s, there is possibility for information loss. For instance, the result of fixed point multiplication could potentially have as many bits as the sum of the number of bits in the two operands. In order to fit the result into the same number of bits as the operands, the answer must be rounded
Rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
or truncated. If this is the case, the choice of which bits to keep is very important. When multiplying two fixed point numbers with the same format, for instance with
integer bits, and fractional bits, the answer could have up to integer bits, and fractional bits.For simplicity, fixed-point multiply procedures use the same result format as the operands. This has the effect of keeping the middle bits; the I-number of least significant integer bits, and the Q-number of most significant fractional bits. Fractional bits lost below this value represent a precision loss which is common in fractional multiplication. If any integer bits are lost, however, the value will be radically inaccurate.
Some operations, like divide, often have built-in result limiting so that any positive overflow results in the largest possible number that can be represented by the current format. Likewise, negative overflow results in the largest negative number represented by the current format. This built in limiting is often referred to as saturation.
Some processors support a hardware overflow flag
Overflow flag
In computer processors, the overflow flag is usually a single bit in a system status register used to indicate when an arithmetic overflow has occurred in an operation, indicating that the signed two's-complement result would not fit in the number of bits used for the operation...
that can generate an exception
Exception
Exception may refer to:* An action that is not part of ordinary operations or standards* Exception handling, in programming languages** or a programming interrupt itself of which exception handling is meant to deal with....
on the occurrence of an overflow, but it is usually too late to salvage the proper result at this point.
Implementations
Very few computer languages include built-in support for fixed point values, because for most applications, binary or decimal floating-point representations are usually simpler to use and accurate enough. Floating-point representations are easier to use than fixed-point representations, because they can handle a wider dynamic range and do not require programmers to specify the number of digits after the radix point. However, if they are needed, fixed-point numbers can be implemented even in programming languages like CC (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
and C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...
, which do not commonly include such support.
A common use of fixed-point BCD numbers is for storing monetary values, where the inexact values of binary floating-point numbers are often a liability. Historically, fixed-point representations were the norm for decimal data types; for example, in PL/I
PL/I
PL/I is a procedural, imperative computer programming language designed for scientific, engineering, business and systems programming applications...
or COBOL
COBOL
COBOL is one of the oldest programming languages. Its name is an acronym for COmmon Business-Oriented Language, defining its primary domain in business, finance, and administrative systems for companies and governments....
. The Ada programming language includes built-in support for both fixed-point (binary and decimal) and floating-point. JOVIAL and Coral 66 also provide both floating- and fixed-point types.
ISO/IEC TR 18037 specifies fixed-point data types for the C programming language; vendors are expected to implement the language extensions for fixed point arithmetic in coming years. Fixed-point support is implemented in GCC
GNU Compiler Collection
The GNU Compiler Collection is a compiler system produced by the GNU Project supporting various programming languages. GCC is a key component of the GNU toolchain...
.
Almost all relational database
Database
A database is an organized collection of data for one or more purposes, usually in digital form. The data are typically organized to model relevant aspects of reality , in a way that supports processes requiring this information...
s, and the SQL
SQL
SQL is a programming language designed for managing data in relational database management systems ....
query language, support fixed-point decimal arithmetic and storage of numbers. PostgreSQL
PostgreSQL
PostgreSQL, often simply Postgres, is an object-relational database management system available for many platforms including Linux, FreeBSD, Solaris, MS Windows and Mac OS X. It is released under the PostgreSQL License, which is an MIT-style license, and is thus free and open source software...
has a special numeric type for exact storage of numbers with up to 1000 digits.
Other
- libfixmathLibfixmathlibfixmath is a platform-independent fixed point maths library aimed at developers wanting to perform fast non-integer maths on platforms lacking a FPU...
is a recent open-source library for the manipulation of fixed point numbers, it currently supports Q16.16Q (number format)Q is a fixed point number format where the number of fractional bits is specified. For example, a Q15 number has 15 fractional bits; a Q1.14 number has 1 integer bit and 14 fractional bits...
and Q0.32Q (number format)Q is a fixed point number format where the number of fractional bits is specified. For example, a Q15 number has 15 fractional bits; a Q1.14 number has 1 integer bit and 14 fractional bits...
formats and provides an interface similar to math.hMath.hC mathematical operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. Most of the functions involve the use of floating point numbers. Different C standards provide different albeit backwards-compatible, sets of functions...
. - GnuCashGnuCashGnuCash is a free and open source accounting software program that implements a double-entry bookkeeping system. It was initially aimed at developing capabilities similar to Intuit, Inc.'s Quicken application, but also has features for small business accounting...
is an application for tracking money which is written in C. It switched from a floating-point representation of money to a fixed-point implementation as of version 1.6. This change was made to trade the less predictable rounding errors of floating-point representations for more control over rounding (for example, to the nearest centCent (currency)In many national currencies, the cent is a monetary unit that equals 1⁄100 of the basic monetary unit. Etymologically, the word cent derives from the Latin word "centum" meaning hundred. Cent also refers to a coin which is worth one cent....
). - TremorTremor (software)Tremor by the Xiph.Org Foundation is a fixed-point version of the Vorbis decoder for those platforms without floating point operations.It is a software library that decodes the Vorbis audio format. It is free software released under the New BSD license...
and Toast are software libraries that decode the Ogg VorbisVorbisVorbis is a free software / open source project headed by the Xiph.Org Foundation . The project produces an audio format specification and software implementation for lossy audio compression...
and GSM Full RateFull RateFull Rate or FR or GSM-FR or GSM 06.10 was the first digital speech coding standard used in the GSM digital mobile phone system. The bit rate of the codec is 13 kbit/s, or 1.625 bits/audio sample...
audio formats respectively. These codecs use fixed-point arithmetic because many audio decoding hardware devices do not have an FPU (partly to save money, but primarily to save power - integer units are much smaller in silicon area than an FPU) 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 graphics3D computer graphics3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
engines on Sony's original PlayStationPlayStationThe is a 32-bit fifth-generation video game console first released by Sony Computer Entertainment in Japan on December 3, .The PlayStation was the first of the PlayStation series of consoles and handheld game devices. The PlayStation 2 was the console's successor in 2000...
, Sega's SaturnSega SaturnThe is a 32-bit fifth-generation video game console that was first released by Sega on November 22, 1994 in Japan, May 11, 1995 in North America, and July 8, 1995 in Europe...
, Nintendo's Game Boy AdvanceGame Boy AdvanceThe is a 32-bit handheld video game console developed, manufactured, and marketed by Nintendo. It is the successor to the Game Boy Color. It was released in Japan on March 21, 2001; in North America on June 11, 2001; in Australia and Europe on June 22, 2001; and in the People's Republic of China...
(only 2D2D computer graphics2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them...
), Nintendo DSNintendo DSThe is a portable game console produced by Nintendo, first released on November 21, 2004. A distinctive feature of the system is the presence of two separate LCD screens, the lower of which is a touchscreen, encompassed within a clamshell design, similar to the Game Boy Advance SP...
(2D and 3D) and GP2X WizGP2X WizThe GP2X Wiz is an open-source, Linux-based handheld video game console and portable media player developed by South Korean company GamePark Holdings. It was released on May 12, 2009, and was also the first console from both Game Park and Game Park Holdings to be also released outside South Korea...
video game systems use fixed-point arithmetic for the same reason as Tremor and Toast: to gain throughput on an architecture without an FPU. - The OpenGL ESOpenGL ESOpenGL for Embedded Systems is a subset of the OpenGL 3D graphics application programming interface designed for embedded systems such as mobile phones, PDAs, and video game consoles. OpenGL ES is managed by the not-for-profit technology consortium, the Khronos Group, Inc.- Versions :Several...
1.x specification includes a fixed point profile, as it's an API aimed for embedded systems, which don't always have an FPU. - TeX font metricTeX font metricTeX font metric is a font file format used by the TeX typesetting system. It is a font metric format, not an outline font format like TrueType, because it provides only the information necessary to typeset the font such as each character's width, height and depth. The actual glyphs are stored...
files use 32-bit signed fixed-point numbers, with 12 bits to the left of the decimal, extensively. - The dc and bcBc programming languagebc, for bench calculator, is "an arbitrary precision calculator language" with syntax similar to the C programming language. bc is typically used as either a mathematical scripting language or as an interactive mathematical shell....
programs are arbitrary precisionArbitrary-precision arithmeticIn computer science, arbitrary-precision arithmetic indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most ALU hardware, which typically...
calculators, but only keep track of a (user-specified) fixed number of fractional digits. - VisSimVisSimVisSim is a visual block diagram language for simulation of dynamical systems and model based design of embedded systems. It is developed by Visual Solutions of Westford, Massachusetts....
A visually programmed block diagram language that supports a fixed-point block set to allow simulation and automatic code generation of fixed-point operations. Both word size and radix point can be specified on an operator basis. - FractintFractintFractInt is a freeware program that can render and display many kinds of fractals.-Name:Its name comes from the words fractal and integer, since the first versions of it computed fractals by using only integer arithmetic , which led to much faster rendering on x86 computers without math coprocessors...
represents numbers as Q2.29Q (number format)Q is a fixed point number format where the number of fractional bits is specified. For example, a Q15 number has 15 fractional bits; a Q1.14 number has 1 integer bit and 14 fractional bits...
fixed-point numbers, to speed up drawing on old PCs with 386Intel 80386The Intel 80386, also known as the i386, or just 386, was a 32-bit microprocessor introduced by Intel in 1985. The first versions had 275,000 transistors and were used as the central processing unit of many workstations and high-end personal computers of the time...
or 486SXIntel 80486SXThe Intel's i486SX was a modified Intel 486DX microprocessor with its floating-point unit disconnected. All early 486SX chips were actually i486DX chips with a defective FPU...
processors, which lacked an FPU. - Doom was the last first-person shooterFirst-person shooterFirst-person shooter is a video game genre that centers the gameplay on gun and projectile weapon-based combat through first-person perspective; i.e., the player experiences the action through the eyes of a protagonist. Generally speaking, the first-person shooter shares common traits with other...
title by id SoftwareId SoftwareId Software is an American video game development company with its headquarters in Richardson, Texas. The company was founded in 1991 by four members of the computer company Softdisk: programmers John Carmack and John Romero, game designer Tom Hall, and artist Adrian Carmack...
to use a 16.16 fixed point representation for all of its non-integer computations, including map system, geometry, rendering, player movement etc. This was done in order for the game to be playable on 386 and 486SX CPUs without a FPU. For compatibility reasons, this representation is still used in modern Doom source portDoom source portA Doom source port is a source port of id Tech 1, the game engine used by the video game Doom. The term usually denotes a modification made by Doom fans, as opposed to any of the official Doom versions produced by id Software or affiliated companies.-Doom source release:The source code for the Doom...
s.
External links
- Fixed-Point Arithmetic - An Introduction Representing and implementing fixed-point arithmetic in digital signal processing, by Randy Yates
- A Calculated Look at Fixed-Point Arithmetic (archive.org)
- A Calculated Look at Fixed-Point Arithmetic (PDF)
- Working with floating point parameters in an integer world How to represent floating point values in a fixed point processor, by Dinu Madau
- Fixed Point Representation And Fractional Math
- Wikibooks Embedded Systems-Embedded System Basics-Fixed-Point Arithmetic