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Calculations where numbers' precision is only limited by computer memory
digits of precision are potentially limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Fixed-precision arithmetic, also referred to as finite-precision arithmetic, is arithmetic on numbers that are represented in a fixed number of digits
Fixed-precision_arithmetic
Computer format for representing real numbers
Arbitrary Precision Numbers JTC1/SC22/WG14 (2008), status of TR 18037: Embedded C GCC wiki, Fixed-Point Arithmetic Support Using GCC, section 5.13 Fixed-Point
Fixed-point_arithmetic
Computer approximation for real numbers
computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of digits in
Floating-point_arithmetic
Combinational digital circuit
In computing, an arithmetic logic unit (ALU) is a combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers
Arithmetic_logic_unit
Form of entropy encoding used in data compression
Arithmetic coding (AC) is a form of entropy coding used in lossless data compression. Normally, a string of characters is represented using a fixed number
Arithmetic_coding
Pattern-recognition performance metrics
Accuracy is a weighted arithmetic mean of Precision and Inverse Precision (weighted by Bias) as well as a weighted arithmetic mean of Recall and Inverse
Precision_and_recall
32-bit computer number format
represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum
Single-precision floating-point format
Single-precision_floating-point_format
128-bit computer number format
the value of extra-precise arithmetic and the price of implementing it to run fast; very soon two more bytes of precision will become tolerable, and ultimately
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
IEEE standard for floating-point arithmetic
computers did not have full 60-bit adders, so integer arithmetic was limited to 48 bits of precision from the floating-point unit. Exception processing from
IEEE_754
Implementation of arithmetic operations
includes: Fixed-size arithmetic "Integer arithmetic", which in practice is modular arithmetic by a power of 2. Fixed-point arithmetic Modular arithmetic Multi-modular
Computer_arithmetic
16-bit computer number format
software for machine learning and neural networks often use half-precision arithmetic to accelerate computations. In these applications, the distinct bfloat16
Half-precision floating-point format
Half-precision_floating-point_format
support arbitrary-precision arithmetic. Software that supports arbitrary precision computations: bc the POSIX arbitrary-precision arithmetic language that
List of arbitrary-precision arithmetic software
List_of_arbitrary-precision_arithmetic_software
Strategies to make sure approximate calculations stay close to accurate
than fixed-length format floating-point instructions. When high performance is not a requirement, but high precision is, variable length arithmetic can
Floating-point error mitigation
Floating-point_error_mitigation
Computer arithmetic error
In computer programming, an integer overflow occurs when an arithmetic operation on integers attempts to create a numeric value that is outside of the
Integer_overflow
Replacing a number with a simpler value
computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms
Rounding
Type of arithmetic where output is limited to a fixed range of values
Saturation arithmetic is a version of arithmetic in which all operations, such as addition and multiplication, are limited to a fixed range between a minimum
Saturation_arithmetic
Spiral with constant distance from itself
(the specific arithmetic spiral of Archimedes). It is the locus corresponding to the locations over time of a point moving away from a fixed point with a
Archimedean_spiral
C library for arbitrary-precision floating-point arithmetic
copying the ideas from the ANSI/IEEE-754 standard for fixed-precision floating-point arithmetic (correct rounding and exceptions, in particular). More
GNU_MPFR
Binary representation for signed numbers
property makes the system simpler to implement, especially for higher-precision arithmetic. Additionally, unlike ones' complement systems, two's complement
Two's_complement
Data type approximating a real number
calculated to any desired precision. Rational number are used, for example, in Interpress from Xerox Corporation. A fixed-point data type uses the same
Real_data_type
Variant of floating-point numbers in computers
moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called tapered arithmetic) […]
Tapered_floating_point
Method in computer arithmetic
floating point (BFP) is a method used to provide an arithmetic approaching floating point while using a fixed-point processor. BFP assigns a group of significands
Block_floating_point
Upper bound on rounding error in floating-point arithmetic
an arithmetic operation on floating-point numbers such as addition or multiplication, and (3) ∘ {\displaystyle \circ } is the infinite precision operation
Machine_epsilon
Algorithmic technique
stable) summation method by a fixed algorithm in fixed precision (i.e. not those that use arbitrary-precision arithmetic, nor algorithms whose memory and
Pairwise_summation
First edition of the IEEE 754 floating-point standard
properties of IEEE 754 floating point numbers Fixed-point arithmetic Precision: The number of decimal digits precision is calculated via number_of_mantissa_bits
IEEE_754-1985
Branch of elementary mathematics
use arbitrary-precision arithmetic, for which the precision of calculations is only restricted by the computer's memory. Forms of arithmetic can also be
Arithmetic
Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
Measure of computer performance
was used as early as 1970 as well. Note that besides integer (or fixed-point) arithmetics, examples of integer operation include data movement (A to B) or
Floating point operations per second
Floating_point_operations_per_second
Algorithm in numerical analysis
stable) summation method by a fixed algorithm in fixed precision (i.e. not those that use arbitrary-precision arithmetic, nor algorithms whose memory and
Kahan_summation_algorithm
Computer architecture bit width
Quadruple precision (128 bits) floating-point numbers can store 113-bit fixed-point numbers or integers accurately without losing precision (thus 64-bit
128-bit_computing
Computational error due to rounding numbers
given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due
Round-off_error
Data type for storing floating-point numbers in base-10
decimal floats as an extension to C and C++. Arbitrary-precision arithmetic Floating-point arithmetic Floating-point error mitigation "Floating-point numeric
Decimal_data_type
Discrete Fourier transform algorithm
Rader–Brenner algorithm, are intrinsically less stable. In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with
Fast_Fourier_transform
Number format for specifying provision
temp -= (b >> 1); */ } return (int16_t)(temp / b); } Fixed-point arithmetic Floating-point arithmetic "Appendix A.2". TMS320C64x DSP Library Programmer's
Q_(number_format)
Measure of angles
measurement system, BAMS) is a measure of angles using binary numbers and fixed-point arithmetic, in which a full turn is represented by the value 1. These representation
Binary_angular_measurement
Data types supported by the C programming language
methods of processing of data elements. The C language provides basic arithmetic types, such as integer and real number types, and syntax to build array
C_data_types
Variant of floating-point numbers in computers
introduced Type III unums (posits), for fixed floating-point-like values and valids for interval arithmetic. In March 2022, a standard was ratified and
Unum_(number_format)
Central computer component that executes instructions
full integer range needed would be cost-prohibitive. See Arbitrary-precision arithmetic for more details on purely software-supported arbitrary-sized integers
Central_processing_unit
Number in base-10 numeral system
generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations. Many ancient cultures calculated with numerals
Decimal
Number functioning as an exponent
applying a scale factor to the real value. Similarly, because hardware arithmetic has a fixed width (commonly 16, 32, or 64 bits, depending on the data type)
Scale factor (computer science)
Scale_factor_(computer_science)
Method for division with remainder
of Y to make division by Y a simple right shift. As a concrete fixed-point arithmetic example, for 32-bit unsigned integers, division by 3 can be replaced
Division_algorithm
Decimal representation of real numbers in computing
Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal
Decimal_floating_point
fraction m/n where m and n are two integer numbers, either with a fixed or arbitrary precision. Depending on the language, the denominator n may be constrained
Rational_data_type
Computer program to render and display many kinds of fractals
arithmetic (also known as fixed-point arithmetic), for faster rendering on computers without math coprocessors. Since then, floating-point arithmetic
Fractint
System of digitally encoding numbers
calculation that fixed-point decimal arithmetic provides. Denser packings of BCD exist which avoid the storage penalty and also need no arithmetic operations
Binary-coded_decimal
Second edition of the IEEE 754 floating-point standard
IEEE 754r) is a revision of the IEEE 754 standard for floating-point arithmetic. It was published in August 2008 and is a significant revision to, and
IEEE_754-2008_revision
Procedural, imperative computer programming language
variables): The arithmetic type comprises these attributes: The base, scale, precision and scale factor of the Picture-for-arithmetic type is encoded
PL/I
Mathematical function, inverse of an exponential function
power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. Newton's method, an
Logarithm
Algorithmic runtime requirements for common math procedures
figures assume that arithmetic with individual elements has complexity O(1), as is the case with fixed-precision floating-point arithmetic or operations on
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Internal representation of numeric values in a digital computer
operations, which provide rounding errors of a different form. Arbitrary-precision arithmetic Binary-coded decimal Binary-to-text encoding Binary number Gray code
Computer_number_format
Part of a number in scientific notation
idea of floating-point arithmetic in his Essays on Automatics, where he proposed the format n; m, showing the need for a fixed-sized significand as currently
Significand
General-purpose programming language
executable form was not entirely machine language; rather, floating-point arithmetic, sub-scripting, input/output, and function references were interpreted
Fortran
Computer representation of real numbers
alternative to fixed-point and floating-point number systems. Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal
Logarithmic_number_system
Datum of integral data type
maxsize Rust: i32::MAX, i32::MIN, etc. Turing: maxint Arbitrary-precision arithmetic Binary-coded decimal (BCD) C data types Integer overflow Signed number
Integer_(computer_science)
Numerical calculations carrying along derivatives
computed automatically, accurately to working precision, and using at most a small constant factor of more arithmetic operations than the original program. Automatic
Automatic_differentiation
Determining where a point is in relation to a coplanar polygon
the Jordan curve theorem. If implemented on a computer with finite-precision arithmetic, the results may be incorrect if the point lies very close to that
Point_in_polygon
Instruction set architecture
paired for double precision numbers. Odd numbered registers cannot be used for arithmetic or branching, just as part of a double precision register pair,
MIPS_architecture
Offset of the exponent field of floating-point numbers
6.8.5 Exponent Representation". In Sumner, Frank H. (ed.). Design of Arithmetic Units for Digital Computers. Macmillan Computer Science Series (1 ed.)
Exponent_bias
Floating-point values coded as few bits
makes sense as a float (it would just be a signed number). fixed-point arithmetic half-precision floating-point format bfloat16 floating-point format G.711
Minifloat
Group of 32-bit RISC processor cores
integer divide, and saturation arithmetic instructions. The Cortex-M4 adds DSP instructions and an optional single-precision floating-point unit (VFPv4-SP)
ARM_Cortex-M
Algorithms for calculating square roots
enough if guess - next_guess < Decimal(f"1e-{precision}"): break guess = next_guess else: raise ArithmeticError(f"Heron method did not converge within
Square_root_algorithms
Attribute of data
integers ranging in value from −2,147,483,648 to 2,147,483,647, with arithmetic operations that wrap on overflow. In Rust this 32-bit integer type is
Data_type
Network Time Protocol implementation
64-bit floating point arithmetic and uses 64-bit fixed point operations only when necessary to preserve the ultimate precision, about 232 picoseconds
Ntpd
Intuitive grasp of numbers
and grammatical number of the Pirahã language – Muran language Plant arithmetic – Form of plant intelligence Subitizing – Perception of object number
Number_sense
Class of mathematical expression
preserves the sign of the result in case of arithmetic underflow. For example, using single-precision IEEE arithmetic, if x = − 2 − 149 {\displaystyle \textstyle
Division_by_zero
General-purpose programming language
that result would then be compared with c. Python uses arbitrary-precision arithmetic for all integer operations. The Decimal type/class in the decimal
Python_(programming_language)
Part of a computer system
feature. In 1963, the GE-235 featured an "Auxiliary Arithmetic Unit" for floating point and double-precision calculations. Historically, some systems implemented
Floating-point_unit
Formula that provides the solutions to a quadratic equation
idealized arithmetic of real numbers, but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number
Quadratic_formula
Technique for selecting hash functions
one may replace summation by exclusive or. In practice, if double-precision arithmetic is available, this is instantiated with the multiply-shift hash family
Universal_hashing
Arithmetic operation
denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The
Addition
Amount of computational work that a computer system performs
use fixed-point arithmetic to calculate the load average for efficiency and simplicity. This however limits the achievable update-rate and precision. Given
Load_(computing)
Number representing a continuous quantity
floating-point arithmetic, often a 64-bit representation with around 16 decimal digits of precision. Real numbers satisfy the usual rules of arithmetic, but floating-point
Real_number
Guidance and navigation computer used in Apollo spacecraft
programs. Interpreted code, which featured double precision trigonometric, scalar and vector arithmetic (16 and 24-bit), even an MXV (matrix × vector) instruction
Apollo_Guidance_Computer
Statistical law in machine learning
per token (logarithm of perplexity) for language modeling; Accuracy, precision, recall, and F1 score for classification tasks; Mean squared error (MSE)
Neural_scaling_law
networking library Mpg123 — MP3 audio decoding library MPIR — multiple-precision arithmetic library MsQuic — Microsoft implementation of the QUIC transport protocol
List_of_C_software_and_tools
Value for unrepresentable data
and symbolic computation or other extensions to basic floating-point arithmetic. In floating-point calculations, NaN is not the same as infinity, although
NaN
some of the digits might just be removed i.e. truncated Fixed-point arithmetic § Precision loss and overflow Data loss "What Is Meant By Data Truncation
Data_truncation
Computer algebra system
implemented are general functions such as f(x), arbitrary-precision and interval arithmetic, as well as matrices. Mathomatic is capable of solving, differentiating
Mathomatic
GPU microarchitecture by Nvidia
Render output units A Tensor core is a mixed-precision FPU specifically designed for matrix arithmetic. Volta is also reported to be included in the
Volta_(microarchitecture)
Missile guidance computer
fixed point, 2's complement Logic levels: 0 V for logical 0 (false), -10 V for logical 1 (true) Data word length (bits): 11 or 24 (double precision)
D-17B
Multi-chip CPU by IBM implementing the POWER instruction set architecture
POWER1's fixed-point register file, an arithmetic logic unit (ALU) for general instructions, and a dedicated fixed-point multiply and divide unit. It also
POWER1
Numbers significantly larger than those used regularly
from the usual axioms of set theory. Arbitrary-precision arithmetic – Calculations where numbers' precision is only limited by computer memory Dirac large
Large_numbers
Early floating-point math coprocessor
"single" precision) floating-point, and 16-bit or 32-bit ("single" or "double" precision) fixed-point calculation of 14 different arithmetic and trigonometric
Intel_8231/8232
Computer algebra system
depending on these enhancements may not work on Maxima, and bugs which were fixed in Macsyma may still be present in Maxima, and vice versa. Maxima participated
Maxima_(software)
Root-finding method
approximate arithmetic is used, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available
Secant_method
RISC instruction set architecture
and an upgrade from 80-bit "extended-precision" floating-point arithmetic to 128-bit "quad-precision" arithmetic. SPARC V8 served as the basis for IEEE
SPARC
Computing using random bit streams
Processing involves sending a fixed number of bits instead of a stream. One of the advantages of this approach is that the precision is improved. To see why
Stochastic_computing
Computation model defining an abstract machine
Aiken (1937). However: … the emphasis is on programming a fixed iterable sequence of arithmetical operations. The fundamental importance of conditional iteration
Turing_machine
Emerging class of microprocessor
like video processing units, they may have a focus on low precision fixed point arithmetic for image processing. They are distinct from GPUs, which contain
Vision_processing_unit
Number with a real and an imaginary part
this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More
Complex_number
Mathematical proof at least partially generated by computer
computing numerically yet with mathematical rigour. One uses set-valued arithmetic and inclusion principle[clarification needed] in order to ensure that
Computer-assisted_proof
GPU microarchitecture designed by Nvidia
from higher numerical precisions (i.e., FP16) to lower precisions that are faster to perform (i.e., FP8) when the loss in precision is deemed acceptable
Hopper_(microarchitecture)
See more about arbitrary-size/precision numbers below. Both languages offer library-defined arbitrary-precision arithmetic types for arbitrary-size integers
Comparison of C Sharp and Java
Comparison_of_C_Sharp_and_Java
Open-source CPU instruction set architecture
instructions (set F) include single-precision arithmetic and also comparison-branches similar to the integer arithmetic. It requires an additional set of
RISC-V
Series of scientific calculators by Texas Instruments
the mantissa with a two-digit exponent, and calculates with 12-digit precision internally. TI-36 SOLAR was based on 1985 version of TI-35 PLUS, but incorporates
TI-36
Programmable machine that processes data
machine that can be programmed to automatically carry out sequences of arithmetic or logical operations (computation). Modern digital electronic computers
Computer
GPU microarchitecture by Nvidia
multiply-add (FMA) instruction for both single and double precision arithmetic. Up to 16 double precision fused multiply-add operations can be performed per
Fermi_(microarchitecture)
Computer chip instruction set extension
(65 unique mnemonics using 70 encodings), most of which work on single precision floating-point data. SIMD instructions can greatly increase performance
Streaming_SIMD_Extensions
Notation for expressing numbers
commonly used. For very large integers, for example, the GNU Multiple Precision Arithmetic Library (GMP) uses bases 232 or 264—grouping binary digits by 32
Numeral_system
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
Boy/Male
Indian, Sanskrit
Firmly Fixed
Girl/Female
Tamil
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Firmly fixed
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya
Firmly Fixed
Girl/Female
Hindu
Fixed zodiac without precession
Girl/Female
Bengali, Indian, Kannada, Marathi
Firmly Fixed
Boy/Male
Indian, Sanskrit
Firmly Fixed
Girl/Female
Tamil
Fixed
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Fixed
Girl/Female
Hindu, Indian, Marathi
Directed; Fixed
Boy/Male
Hindu, Indian, Kannada, Telugu
Fixed
Girl/Female
Tamil
Fixed
Girl/Female
Hindu
Fixed
Girl/Female
Hindu, Indian, Marathi, Sanskrit, Telugu
Immovable; Fixed; Quiet
Boy/Male
Arabic, Muslim
Firm; Fixed; The Greatness
Boy/Male
African, Australian, Nigerian
God has Fixed it
Boy/Male
Indian, Sanskrit
Well Fixed
Boy/Male
Indian, Sanskrit
Fixed
Girl/Female
Gujarati, Indian
Firmly Fixed
Boy/Male
Biblical American Egyptian Hebrew
Put, who puts, fixed'.
Girl/Female
Tamil
Nirayana | நீராயநா
Fixed zodiac without precession
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
Boy/Male
African
Chaste.
Surname or Lastname
English
English : regional name for someone from the county of Cheshire in northwestern England, the name of which is recorded in Domesday Book as Cestrescire, from the name of the county seat, Chester, + Old English scīr ‘district’, ‘division’.
Boy/Male
Hindu, Indian, Japanese, Tamil
Honest; Truth; Child of Bliss
Female
English
Slovak form of English Alice, ALICA means "noble sort."
Surname or Lastname
English
English : from Middle English pertriche ‘partridge’ (via Old French and Latin from Greek perdix), either a metonymic occupational name for a hunter of the bird or a nickname for someone with some fancied resemblance to it, or a habitational name for someone living at a house distinguished by the sign of a partridge. This surname has been established in Ireland since the 17th century. As an American family name, it has probably absorbed some cases of other European surnames with the same meaning, e.g. Italian Pernice.
Boy/Male
Hindu
Boy/Male
Indian, Punjabi, Sikh
Victory of Beauty
Boy/Male
English
Made of Oak
Boy/Male
Assamese, Hindu, Indian, Kannada, Telugu
Nil
Girl/Female
Muslim
Gift of Allah, Concern, Solicitude
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
FIXED PRECISION-ARITHMETIC
a.
Fixed; solidified.
v. t.
To supply with food; to victual; as, to provision a garrison.
a.
Fixed; settled.
a.
Cutting off; (Logic) exactly limiting by cutting off all that is not absolutely relative to the purpose; as, precisive censure; precisive abstraction.
imp. & p. p.
of Fix
n.
Foresight of consequences, and provision against them; prevision; premeditation.
a.
Motionless; fixed.
n.
An overprecise person; one rigidly or ceremoniously exact in the observance of rules; a formalist; -- formerly applied to the English Puritans.
n.
A precisian.
a.
Securely placed or fastened; settled; established; firm; imovable; unalterable.
a.
Firmly fixed or established; fast fixed; firm.
n.
The act of revising; reexamination for correction; review; as, the revision of a book or writing, or of a proof sheet; a revision of statutes.
n.
Foresight; foreknowledge; prescience.
a.
Depending on will or discretion; not governed by any fixed rules; as, an arbitrary decision; an arbitrary punishment.
n.
Want of precision.
n.
An account or report of a conclusion, especially of a legal adjudication or judicial determination of a question or cause; as, a decision of arbitrators; a decision of the Supreme Court.
a.
Repaired by foxing; as, foxed boots.
n.
The quality or state of being precise; exact limitation; exactness; accuracy; strict conformity to a rule or a standard; definiteness.
n.
The quality of being decided; prompt and fixed determination; unwavering firmness; as, to manifest great decision.
a.
Stable; non-volatile.