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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
IEEE standard for floating-point arithmetic
standard. The standard defines: arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and
IEEE_754
64-bit computer number format
additional floating-point formats, including 32-bit base-2 single precision and, more recently, base-10 representations (decimal floating point). One of
Double-precision floating-point format
Double-precision_floating-point_format
Computer approximation for real numbers
floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating
Floating-point_arithmetic
Computer format for representing real numbers
support for decimal floating point in some programming languages, like C# and Python, has removed most of the need for decimal fixed-point support. In
Fixed-point_arithmetic
32-bit computer number format
32-bit base-2 floating-point variable has a maximum finite value of (2 − 2−23) × 2127 ≈ 3.4028235 × 1038. All integers with seven or fewer decimal digits, and
Single-precision floating-point format
Single-precision_floating-point_format
64-bit computer number format
In computing, decimal64 is a decimal floating-point computer number format that occupies 8 bytes (64 bits) in computer memory. The format was formally
Decimal64 floating-point format
Decimal64_floating-point_format
128-bit computer number format
In computing, quadruple precision (or quad precision) is a binary floating-point–based computer number format that occupies 16 bytes (128 bits) with precision
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
Efficient system of binary encoding for decimal digits used in decimal floating point
ISO/IEC/IEEE 60559:2011 standards for decimal floating point. Like Chen–Ho encoding, DPD encoding classifies each decimal digit into one of two ranges, depending
Densely_packed_decimal
128-bit computer number format
In computing, decimal128 is a decimal floating-point number format defined by the IEEE 754 Standard for Floating-Point Arithmetic. It is one of the standard's
Decimal128 floating-point format
Decimal128_floating-point_format
Number representation
included IEEE binary floating-point units which conform to the IEEE 754 Standard for Floating-Point Arithmetic. IEEE decimal floating-point was added to IBM
IBM hexadecimal floating-point
IBM_hexadecimal_floating-point
Computer operating on base-10 numbers
directly support decimal is IBM's Power ISA, which added support for IEEE 754-2008 decimal floating-point starting with Power ISA 2.05. Decimal integer support
Decimal_computer
The IEEE 754-2008 standard includes decimal floating-point number formats in which the significand and the exponent (and the payloads of NaNs) can be encoded
Binary_integer_decimal
System of digitally encoding numbers
instruction sets (e.g., ARM; x86 in long mode). However, decimal fixed-point and decimal floating-point formats are still important and continue to be used
Binary-coded_decimal
Concise notation for large or small numbers
(modulus) of the significand m is at least 1 but less than 10. Decimal floating point is a computer arithmetic system closely related to scientific notation
Scientific_notation
Numerical symbol
symbol can be called a decimal mark, decimal marker, or decimal sign. Symbol-specific names are also used; decimal point and decimal comma refer to a dot
Decimal_separator
Soviet ternary computer
interpreters—IP-2 (floating-point, 8 decimal digits), IP-3 (floating-point, 6 decimal digits), IP-4 (complex numbers, 8 decimal digits), IP-5 (floating-point, 12 decimal
Setun
Number in base-10 numeral system
binary-coded decimal, especially in database implementations, but there are other decimal representations in use (including decimal floating point such as
Decimal
German 1940s computer
operations and memory cells. Numbers were entered and output as decimal floating-point even though the internal working was in binary. The machine had
Z4_(computer)
C programming language standard, current revision
IEEE 754 standard for floating-point arithmetic, with extended binary floating-point arithmetic and (optional) decimal floating-point arithmetic. The constexpr
C23_(C_standard_revision)
32-bit computer number format
In computing, decimal32 is a decimal floating-point computer numbering format that occupies 4 bytes (32 bits) in computer memory. Like the binary16 and
Decimal32 floating-point format
Decimal32_floating-point_format
Line of mainframe computers
Enterprise PL/I, XL C, and the z/OS Java BigDecimal class can exploit hardware decimal floating point. The System z10 processor adds numerous new instructions
IBM_System_z10
Strategies to make sure approximate calculations stay close to accurate
definition, floating-point error cannot be eliminated, and, at best, can only be managed. Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic
Floating-point error mitigation
Floating-point_error_mitigation
Denormalized floating-point numbers near zero
exist in the IEEE binary floating-point formats, but they do exist in some other formats, including the IEEE decimal floating-point formats. Some systems
Subnormal_number
Computer instruction set architecture
IEEE 754 compliant floating-point operations are supported, including additional fused multiply–add (FMA) and decimal floating-point instructions. There
Power_ISA
IEEE standard for radix-independent floating-point arithmetic
floating-point standardisation. IEEE 854 arithmetic was first commercially implemented in the HP-71B handheld computer, which used decimal floating point
IEEE_854-1987
256-bit computer number format
(approximately 71 decimal digits: log10(2237) ≈ 71.344). The bits are laid out as follows: The octuple-precision binary floating-point exponent is encoded
Octuple-precision floating-point format
Octuple-precision_floating-point_format
C programming language standard draft planned for release in 2029
"Floating-point exception for Macro Replacements". N3287 - "Nonsensical Parenthetical in Mathematics Specification". N3291 - "Decimal Floating-Point Number
C29_(C_standard_revision)
IEEE floating-point value. Computing: 1×10−6176 is the smallest non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point
Orders_of_magnitude_(numbers)
Differentiating positive and negative zero
also be possible), or −1.0 × 0.0, or simply as −0.0. In IEEE 754 decimal floating-point formats, a negative zero is represented by an exponent being any
Signed_zero
General-purpose programming language
arithmetic for all integer operations. The Decimal type/class in the decimal module provides decimal floating-point numbers to a pre-defined arbitrary precision
Python_(programming_language)
First working programmable, fully automatic digital computer
Program memory: Punched celluloid tape Input: Decimal floating-point numbers Output: Decimal floating-point numbers Input and Output was facilitated by
Z3_(computer)
Floating-point number formats
of the x87 floating point: "An extended format as wide as we dared (80 bits) was included to serve the same support role as the 13 decimal internal format
Extended_precision
Part of a number in scientific notation
in the complex plane. The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10−2 power
Significand
Computer architecture bit width
128-bit accumulator, and its instruction set included 128-bit floating-point and packed decimal arithmetic. A CPU with 128-bit multimedia extensions was designed
128-bit_computing
Standards for the C programming language
60559:2011-compatible binary floating-point arithmetic ISO/IEC TS 18661-2:2015, on IEC 60559:2011-compatible decimal floating point arithmetic ISO/IEC TS 18661-3:2015
ANSI_C
Number of digits of a numeral system
same length. The first row in the tables is the base written in decimal. Floating-point arithmetic Mixed radix Polynomial Radix economy Radix sort Non-standard
Radix
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
32-bit_computing
IBM's 64-bit instruction set architecture implemented by its mainframe computers
(PSA) Cryptographic Facility IEEE Binary-floating-point instructions added by ESA/390 IEEE Decimal-floating-point instructions For information on when each
Z/Architecture
C programming language standard, 1999 revision
bounds checking. Work continues on technical reports addressing decimal floating point, additional mathematical special functions, and additional dynamic
C99
Data type for storing floating-point numbers in base-10
as floating-point numbers or fixed-point numbers. However, such representations typically restrict the denominator to a power of two. Most decimal fractions
Decimal_data_type
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
40-bit_computing
Form of text that defines C code
often one of the IEEE floating-point formats. Floating-point constants may be written in decimal notation, e.g. 1.23. Decimal scientific notation may
C_syntax
Open-source CPU instruction set architecture
the RISC-V ISA is a load–store architecture. Its floating-point instructions use IEEE 754 floating-point. Notable features of the RISC-V ISA include: instruction
RISC-V
General-purpose programming language
mathematical functions, integrated into C++17, ISO/IEC TR 24733:2011 on decimal floating-point arithmetic, ISO/IEC TS 18822:2015 on the standard filesystem library
C++
British computer scientist
C#, CLI), and IEEE (754 decimal floating-point). He retired from IBM in March 2010. Cowlishaw has worked on aspects of decimal arithmetic; his proposal
Mike_Cowlishaw
16-bit computer number format
(log10(211) ≈ 3.311 decimal digits, or 4 digits ± slightly less than 5 units in the last place). The half-precision binary floating-point exponent is encoded
Half-precision floating-point format
Half-precision_floating-point_format
Computer architecture bit width
registers are used to store several smaller numbers, such as eight 32-bit floating-point numbers, and a single instruction can operate on all these values in
256-bit_computing
Topics referred to by the same term
MVS Davidon-Fletcher-Powell formula in mathematical optimization Decimal floating point Defensive fighting position, a military term Department of Finance
DFP
Computer architecture bit width
floating point, where double-precision formats offered more than 10-digit accuracy. The 360s also included instructions for variable-length decimal arithmetic
36-bit_computing
2007 family of multiprocessors by IBM
integer units, two binary floating-point units, an AltiVec unit, and a novel decimal floating-point unit. The binary floating-point unit incorporates "many
POWER6
2014 family of multi-core microprocessors by IBM
Fixed-point unit (FXU) Vector and scalar unit (VSU) Decimal floating-point unit (DFU) Each core has sixteen execution pipelines: Two fixed-point pipelines
POWER8
Computer architecture bit width
4-bit value holds a single binary-coded decimal (BCD) digit, making it a natural size for directly processing decimal values. As a 4-bit value is generally
4-bit_computing
Programming tool for profiling, memory debugging and memory leak detection
MIPS64 little and big endian, for MIPS DSP ASE on MIPS32, for s390x Decimal Floating Point instructions, for POWER8 (Power ISA 2.07) instructions, for Intel
Valgrind
Large and powerful computer
prediction and nuclear simulations (only recently standardized decimal floating point, not used in supercomputers, are appropriate for monetary values
Mainframe_computer
2012 64-bit mainframe microprocessor by IBM
two integer units, two load–store units, one binary floating-point unit and one decimal floating point unit. The zEC12 chip can decode three instructions
IBM_zEC12
Relational database programming language
types (NUMERIC, DECIMAL, SMALLINT, INTEGER, BIGINT) Approximate numeric types (FLOAT, REAL, DOUBLE PRECISION) Decimal floating-point type (DECFLOAT) Datetime
SQL
Computer architecture bit width
byte-addressable memory with 8-bit bytes, as opposed to bit-addressable or decimal digit-addressable or word-addressable memory, although its general-purpose
8-bit_computing
Computer architecture bit width
multiple which could store a single ASCII character or two binary-coded decimal digits. The 16-bit word length thus became more common in the 1960s, especially
16-bit_computing
Computer architecture bit width
The maximum value of a signed 512-bit integer is 2511 − 1, written in decimal as 6,703,903,964,971,298,549,787,012,499,102,923,063,739,682
512-bit_computing
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
31-bit_computing
First pocket scientific calculator
named an IEEE Milestone in 2009. The calculator used a traditional floating decimal display for numbers that could be displayed in that format, but automatically
HP-35
Base-16 numeric representation
C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard). Just as decimal numbers can be represented in exponential notation, so
Hexadecimal
Loss of precision in numerical analysis
2^{-52}} . Although the radix conversion from decimal floating-point to binary floating-point only incurs a small relative error, catastrophic cancellation
Catastrophic_cancellation
Floating-point number format used in computer processors
floating-point format (binary32), while reducing the precision from 24 bits to 8 bits. This means that the precision is between two and three decimal
Bfloat16 floating-point format
Bfloat16_floating-point_format
Algorithm in numerical analysis
example easier to read, it will be given in decimal. Suppose we are using six-digit decimal floating-point arithmetic, sum has attained the value 10000
Kahan_summation_algorithm
32-bit microprocessor
Savard, John J. G. (2018) [2007]. "The Decimal Floating-Point Standard". quadibloc. Other Recent Decimal Floating-Point Formats. Archived from the original
Motorola_68040
Base memory unit handled by a computer
use binary-coded decimal rather than plain binary, typically having a word size of 10 or 12 decimal digits, and some early decimal computers have no
Word_(computer_architecture)
2008 64-bit mainframe microprocessor by IBM
software/hardware cache optimizations support for 1 MB page frames decimal floating point fully implemented in hardware. Error detection and recovery is emphasized
IBM_z10
Computer architecture bit width
two's complement integers; 64-bit and 128-bit floating point; 32-bit, 64-bit, and 128-bit packed decimal and a 128-bit accumulator register. The architecture
64-bit_computing
Second edition of the IEEE 754 floating-point standard
(previously known as 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
IEEE_754-2008_revision
Floating-point values coded as few bits
In computing, minifloats are floating-point values represented with very few bits. This reduced precision makes them ill-suited for general-purpose numerical
Minifloat
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
26-bit_computing
Data types supported by the C programming language
DECIMAL_DIG (C99) – minimum number of decimal digits such that any number of the widest supported floating-point type can be represented in decimal with
C_data_types
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
1-bit_computing
Computer architecture bit width
was a convenient size. This also made it useful for storing a single decimal digit along with a sign. Possibly the best-known 12-bit CPUs are the PDP-8
12-bit_computing
Replacing a number with a simpler value
The Toronto Star. "java.math.RoundingMode". Oracle. "decimal – Decimal fixed point and floating point arithmetic". Python Software Foundation. Engineering
Rounding
Computer architecture bit width
addressing. It is binary compatible with the 8-bit 6502. Several fixed-point digital signal processors have a 24-bit data bus, selected as the basic
24-bit_computing
Data type approximating a real number
although both ten and sixteen have been used. The decimal type is similar to fixed-point or floating-point data type, but with a denominator that is a power
Real_data_type
2010 family of multi-core microprocessors by IBM
core: 2 fixed-point units 2 load/store units 4 double-precision floating-point units 1 vector unit supporting VSX 1 decimal floating-point unit 1 branch
POWER7
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
45-bit_computing
Computer designed in the early 1960s
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
UNIVAC_490
Arithmetic operation
contained the representation of the exponent. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer
Exponentiation
Floating-point number format by Microsoft
binary-coded decimal (BCD) floating point. Even so, for a while MBF became the de facto floating-point format on home computers, to the point where people
Microsoft_Binary_Format
Number type in floating-point arithmetic
number in a floating-point representation which is within the balanced range supported by a given floating-point format: it is a floating point number that
Normal_number_(computing)
Internal representation of numeric values in a digital computer
integers and fixed-point numbers and go to a "floating-point" format. In the decimal system, we are familiar with floating-point numbers of the form
Computer_number_format
2010 64-bit mainframe microprocessor by IBM
two integer units, two load–store units, one binary floating point unit and one decimal floating point unit. The z196 chip can decode three instructions
IBM_z196
Details of data storage in a spreadsheet application
Standard floating-point error mitigation techniques apply. In the top figure the fraction 1/9000 in Excel is displayed. Although this number has a decimal representation
Numeric precision in Microsoft Excel
Numeric_precision_in_Microsoft_Excel
First edition of the IEEE 754 floating-point standard
IEEE 754-1985 is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by
IEEE_754-1985
SIMD instruction set extension for the PowerPC ISA
to support up to 64 registers, with support for regular floating point, decimal floating point and vector execution. POWER7 is the first Power ISA processor
AltiVec
Integrated development environment
and support for visualizing memory in several numeric formats (decimal, floating-point).[citation needed] In 2015, a cloud computing version of CCS was
Code_Composer_Studio
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
48-bit_computing
Computer architecture bit width
Application 8 16 32 64 Binary floating-point precision 16 (×½) 24 32 (×1) 40 64 (×2) 80 128 (×4) 256 (×8) Decimal floating-point precision 32 64 128 v t e
18-bit_computing
Series of microprocessors from IBM
inception of POWER and PowerPC processors. POWER6 also introduced the decimal floating point unit to the Power ISA, which it shares with z/Architecture. With
IBM_Power_microprocessors
Model independent architecture for the S/360 line of mainframe computers
instruction set, adding decimal arithmetic instructions, is optional on some models, as is a scientific instruction set, which adds floating-point instructions.
IBM_System/360_architecture
Programming language
applicable), additional modifiers (e.g. the number of implied decimals for a packed decimal field) and optionally an initial value: * Primitive types: DATA:
ABAP
Computer architecture bit width
buses of that size. A 60-bit word is typically used for high-precision floating-point calculations; it can also store 10 6-bit characters. The only widely-used
60-bit_computing
I (Luxor ABC 80 in ROM) – Semicompiled to byte code for speed, decimal floating point, support for bignum arithmetic BASIC II (Luxor ABC 800/Facit DTC
List_of_BASIC_dialects
Method of constructing a computer processor
addresses, in which 9 modules of type 6203 held the 9-bit shift count and floating point exponent registers, 36 modules of type 6205 held the several 36-bit
Bit_slicing
Measure of time intervals using the metric system
since January 1, 1900. Microsoft Excel uses number of days (with decimals, floating point) since January 1, 1900. All these systems present time for the
Metric_time
DECIMAL FLOATING-POINT
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