Doubleprecision floatingpoint format
Doubleprecision floatingpoint format (sometimes called FP64 or float64) is a floatingpoint number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
Double precision may be chosen when the range or precision of single precision would be insufficient.
In the IEEE 7542008 standard, the 64bit base2 format is officially referred to as binary64; it was called double in IEEE 7541985. IEEE 754 specifies additional floatingpoint formats, including 32bit base2 single precision and, more recently, base10 representations (decimal floating point).
One of the first programming languages to provide floatingpoint data types was Fortran.^{[citation needed]} Before the widespread adoption of IEEE 7541985, the representation and properties of floatingpoint data types depended on the computer manufacturer and computer model, and upon decisions made by programminglanguage implementers. E.g., GWBASIC's doubleprecision data type was the 64bit MBF floatingpoint format.
Floatingpoint formats 

IEEE 754 
Other 
IEEE 754 doubleprecision binary floatingpoint format: binary64
Doubleprecision binary floatingpoint is a commonly used format on PCs, due to its wider range over singleprecision floating point, in spite of its performance and bandwidth cost. It is commonly known simply as double. The IEEE 754 standard specifies a binary64 as having:
 Sign bit: 1 bit
 Exponent: 11 bits
 Significand precision: 53 bits (52 explicitly stored)
The sign bit determines the sign of the number (including when this number is zero, which is signed).
The exponent field is an 11bit unsigned integer from 0 to 2047, in biased form: an exponent value of 1023 represents the actual zero. Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved for special numbers.
The 53bit significand precision gives from 15 to 17 significant decimal digits precision (2^{−53} ≈ 1.11 × 10^{−16}). If a decimal string with at most 15 significant digits is converted to the IEEE 754 doubleprecision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 doubleprecision number is converted to a decimal string with at least 17 significant digits, and then converted back to doubleprecision representation, the final result must match the original number.^{[1]}
The format is written with the significand having an implicit integer bit of value 1 (except for special data, see the exponent encoding below). With the 52 bits of the fraction (F) significand appearing in the memory format, the total precision is therefore 53 bits (approximately 16 decimal digits, 53 log_{10}(2) ≈ 15.955). The bits are laid out as follows:
The real value assumed by a given 64bit doubleprecision datum with a given biased exponent [math]\displaystyle{ e }[/math] and a 52bit fraction is
 [math]\displaystyle{ (1)^{\text{sign}}(1.b_{51}b_{50}...b_{0})_2 \times 2^{e1023} }[/math]
or
 [math]\displaystyle{ (1)^{\text{sign}}\left(1 + \sum_{i=1}^{52} b_{52i} 2^{i} \right)\times 2^{e1023} }[/math]
Between 2^{52}=4,503,599,627,370,496 and 2^{53}=9,007,199,254,740,992 the representable numbers are exactly the integers. For the next range, from 2^{53} to 2^{54}, everything is multiplied by 2, so the representable numbers are the even ones, etc. Conversely, for the previous range from 2^{51} to 2^{52}, the spacing is 0.5, etc.
The spacing as a fraction of the numbers in the range from 2^{n} to 2^{n+1} is 2^{n−52}. The maximum relative rounding error when rounding a number to the nearest representable one (the machine epsilon) is therefore 2^{−53}.
The 11 bit width of the exponent allows the representation of numbers between 10^{−308} and 10^{308}, with full 15–17 decimal digits precision. By compromising precision, the subnormal representation allows even smaller values up to about 5 × 10^{−324}.
Exponent encoding
The doubleprecision binary floatingpoint exponent is encoded using an offsetbinary representation, with the zero offset being 1023; also known as exponent bias in the IEEE 754 standard. Examples of such representations would be:
e =00000000001_{2} =001_{16} =1:

[math]\displaystyle{ 2^{11023}=2^{1022} }[/math]  (smallest exponent for normal numbers)  
e =01111111111_{2} =3ff_{16} =1023:

[math]\displaystyle{ 2^{10231023}=2^0 }[/math]  (zero offset)  
e =10000000101_{2} =405_{16} =1029:

[math]\displaystyle{ 2^{10291023}=2^6 }[/math]  
e =11111111110_{2} =7fe_{16} =2046:

[math]\displaystyle{ 2^{20461023}=2^{1023} }[/math]  (highest exponent) 
The exponents 000_{16}
and 7ff_{16}
have a special meaning:
00000000000_{2}
=000_{16}
is used to represent a signed zero (if F = 0) and subnormal numbers (if F ≠ 0); and11111111111_{2}
=7ff_{16}
is used to represent ∞ (if F = 0) and NaNs (if F ≠ 0),
where F is the fractional part of the significand. All bit patterns are valid encoding.
Except for the above exceptions, the entire doubleprecision number is described by:
 [math]\displaystyle{ (1)^{\text{sign}} \times 2^{e  1023} \times 1.\text{fraction} }[/math]
In the case of subnormal numbers (e = 0) the doubleprecision number is described by:
 [math]\displaystyle{ (1)^{\text{sign}} \times 2^{11023} \times 0.\text{fraction} = (1)^{\text{sign}} \times 2^{1022} \times 0.\text{fraction} }[/math]
Endianness
Doubleprecision examples
0 01111111111 0000000000000000000000000000000000000000000000000000_{2} ≙ 3FF0 0000 0000 0000_{16} ≙ +2^{0} × 1 = 1 
0 01111111111 0000000000000000000000000000000000000000000000000001_{2} ≙ 3FF0 0000 0000 0001_{16} ≙ +2^{0} × (1 + 2^{−52}) ≈ 1.0000000000000002, the smallest number > 1 
0 01111111111 0000000000000000000000000000000000000000000000000010_{2} ≙ 3FF0 0000 0000 0002_{16} ≙ +2^{0} × (1 + 2^{−51}) ≈ 1.0000000000000004 
0 10000000000 0000000000000000000000000000000000000000000000000000_{2} ≙ 4000 0000 0000 0000_{16} ≙ +2^{1} × 1 = 2 
1 10000000000 0000000000000000000000000000000000000000000000000000_{2} ≙ C000 0000 0000 0000_{16} ≙ −2^{1} × 1 = −2 
0 10000000000 1000000000000000000000000000000000000000000000000000_{2} ≙ 4008 0000 0000 0000_{16} ≙ +2^{1} × 1.1_{2} = 11_{2} = 3 
0 10000000001 0000000000000000000000000000000000000000000000000000_{2} ≙ 4010 0000 0000 0000_{16} ≙ +2^{2} × 1 = 100_{2} = 4 
0 10000000001 0100000000000000000000000000000000000000000000000000_{2} ≙ 4014 0000 0000 0000_{16} ≙ +2^{2} × 1.01_{2} = 101_{2} = 5 
0 10000000001 1000000000000000000000000000000000000000000000000000_{2} ≙ 4018 0000 0000 0000_{16} ≙ +2^{2} × 1.1_{2} = 110_{2} = 6 
0 10000000011 0111000000000000000000000000000000000000000000000000_{2} ≙ 4037 0000 0000 0000_{16} ≙ +2^{4} × 1.0111_{2} = 10111_{2} = 23 
0 01111111000 1000000000000000000000000000000000000000000000000000_{2} ≙ 3F88 0000 0000 0000_{16} ≙ +2^{−7} × 1.1_{2} = 0.00000011_{2} = 0.01171875 (3/256) 
0 00000000000 0000000000000000000000000000000000000000000000000001_{2} ≙ 0000 0000 0000 0001_{16} ≙ +2^{−1022} × 2^{−52} = 2^{−1074} ≈ 4.9406564584124654 × 10^{−324} (Min. subnormal positive double) 
0 00000000000 1111111111111111111111111111111111111111111111111111_{2} ≙ 000F FFFF FFFF FFFF_{16} ≙ +2^{−1022} × (1 − 2^{−52}) ≈ 2.2250738585072009 × 10^{−308} (Max. subnormal double) 
0 00000000001 0000000000000000000000000000000000000000000000000000_{2} ≙ 0010 0000 0000 0000_{16} ≙ +2^{−1022} × 1 ≈ 2.2250738585072014 × 10^{−308} (Min. normal positive double) 
0 11111111110 1111111111111111111111111111111111111111111111111111_{2} ≙ 7FEF FFFF FFFF FFFF_{16} ≙ +2^{1023} × (1 + (1 − 2^{−52})) ≈ 1.7976931348623157 × 10^{308} (Max. double) 
0 00000000000 0000000000000000000000000000000000000000000000000000_{2} ≙ 0000 0000 0000 0000_{16} ≙ +0 
1 00000000000 0000000000000000000000000000000000000000000000000000_{2} ≙ 8000 0000 0000 0000_{16} ≙ −0 
0 11111111111 0000000000000000000000000000000000000000000000000000_{2} ≙ 7FF0 0000 0000 0000_{16} ≙ +∞ (positive infinity) 
1 11111111111 0000000000000000000000000000000000000000000000000000_{2} ≙ FFF0 0000 0000 0000_{16} ≙ −∞ (negative infinity) 
0 11111111111 0000000000000000000000000000000000000000000000000001_{2} ≙ 7FF0 0000 0000 0001_{16} ≙ NaN (sNaN on most processors, such as x86 and ARM) 
0 11111111111 1000000000000000000000000000000000000000000000000001_{2} ≙ 7FF8 0000 0000 0001_{16} ≙ NaN (qNaN on most processors, such as x86 and ARM) 
0 11111111111 1111111111111111111111111111111111111111111111111111_{2} ≙ 7FFF FFFF FFFF FFFF_{16} ≙ NaN (an alternative encoding of NaN) 
0 01111111101 0101010101010101010101010101010101010101010101010101_{2} = 3FD5 5555 5555 5555_{16} ≙ +2^{−2} × (1 + 2^{−2} + 2^{−4} + ... + 2^{−52}) ≈ ^{1}/_{3} 
0 10000000000 1001001000011111101101010100010001000010110100011000_{2} = 4009 21FB 5444 2D18_{16} ≈ pi 
Encodings of qNaN and sNaN are not completely specified in IEEE 754 and depend on the processor. Most processors, such as the x86 family and the ARM family processors, use the most significant bit of the significand field to indicate a quiet NaN; this is what is recommended by IEEE 754. The PARISC processors use the bit to indicate a signaling NaN.
By default, ^{1}/_{3} rounds down, instead of up like single precision, because of the odd number of bits in the significand.
In more detail:
Given the hexadecimal representation 3FD5 5555 5555 5555_{16}, Sign = 0 Exponent = 3FD_{16} = 1021 Exponent Bias = 1023 (constant value; see above) Fraction = 5 5555 5555 5555_{16} Value = 2^{(Exponent − Exponent Bias)} × 1.Fraction – Note that Fraction must not be converted to decimal here = 2^{−2} × (15 5555 5555 5555_{16} × 2^{−52}) = 2^{−54} × 15 5555 5555 5555_{16} = 0.333333333333333314829616256247390992939472198486328125 ≈ 1/3
Execution speed with doubleprecision arithmetic
Using doubleprecision floatingpoint variables is usually slower than working with their single precision counterparts. One area of computing where this is a particular issue is parallel code running on GPUs. For example, when using NVIDIA's CUDA platform, calculations with double precision can take, depending on hardware, from 2 to 32 times as long to complete compared to those done using single precision.^{[4]}
Additionally, many mathematical functions (e.g., sin, cos, atan2, log, exp and sqrt) need more computations to give accurate doubleprecision results, and are therefore slower.
Precision limitations on integer values
 Integers from −2^{53} to 2^{53} (−9,007,199,254,740,992 to 9,007,199,254,740,992) can be exactly represented.
 Integers between 2^{53} and 2^{54} = 18,014,398,509,481,984 round to a multiple of 2 (even number).
 Integers between 2^{54} and 2^{55} = 36,028,797,018,963,968 round to a multiple of 4.
 Integers between 2^{n} and 2^{n+1} round to a multiple of 2^{n−52}.
Implementations
Doubles are implemented in many programming languages in different ways such as the following. On processors with only dynamic precision, such as x86 without SSE2 (or when SSE2 is not used, for compatibility purpose) and with extended precision used by default, software may have difficulties to fulfill some requirements.
C and C++
C and C++ offer a wide variety of arithmetic types. Double precision is not required by the standards (except by the optional annex F of C99, covering IEEE 754 arithmetic), but on most systems, the double
type corresponds to double precision. However, on 32bit x86 with extended precision by default, some compilers may not conform to the C standard or the arithmetic may suffer from double rounding.^{[5]}
Fortran
Fortran provides several integer and real types, and the 64bit type real64
, accessible via Fortran's intrinsic module iso_fortran_env
, corresponds to double precision.
Common Lisp
Common Lisp provides the types SHORTFLOAT, SINGLEFLOAT, DOUBLEFLOAT and LONGFLOAT. Most implementations provide SINGLEFLOATs and DOUBLEFLOATs with the other types appropriate synonyms. Common Lisp provides exceptions for catching floatingpoint underflows and overflows, and the inexact floatingpoint exception, as per IEEE 754. No infinities and NaNs are described in the ANSI standard, however, several implementations do provide these as extensions.
Java
On Java before version 1.2, every implementation had to be IEEE 754 compliant. Version 1.2 allowed implementations to bring extra precision in intermediate computations for platforms like x87. Thus a modifier strictfp was introduced to enforce strict IEEE 754 computations. Strict floating point has been restored in Java 17.^{[6]}
JavaScript
As specified by the ECMAScript standard, all arithmetic in JavaScript shall be done using doubleprecision floatingpoint arithmetic.^{[7]}
JSON
The JSON data encoding format supports numeric values, and the grammar to which numeric expressions must conform has no limits on the precision or range of the numbers so encoded. However, RFC 8259 advises that, since IEEE 754 binary64 numbers are widely implemented, good interoperability can be achieved by implementations processing JSON if they expect no more precision or range than binary64 offers.^{[8]}
See also
 IEEE 754, IEEE standard for floatingpoint arithmetic
 D notation (scientific notation)
Notes and references
 ↑ William Kahan (1 October 1997). "Lecture Notes on the Status of IEEE Standard 754 for Binary FloatingPoint Arithmetic". http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF.
 ↑ "FloatingPoint Formats", quadibloc, 2018, http://www.quadibloc.com/comp/cp0201.htm, retrieved 20180716
 ↑ "pack – convert a list into a binary representation". http://www.perl.com/doc/manual/html/pod/perlfunc/pack.html.
 ↑ "Nvidia’s New Titan V Pushes 110 Teraflops From A Single Chip" (in en). Tom's Hardware. 20171208. https://www.tomshardware.com/news/nvidiatitanv110teraflops,36085.html.
 ↑ "Bug 323 – optimized code gives strange floating point results". https://gcc.gnu.org/bugzilla/show_bug.cgi?id=323.
 ↑ Darcy, Joseph D.. "JEP 306: Restore AlwaysStrict FloatingPoint Semantics". http://openjdk.java.net/jeps/306.
 ↑ ECMA262 ECMAScript Language Specification (5th ed.). Ecma International. p. 29, §8.5 The Number Type. http://www.ecmainternational.org/publications/files/ECMASTARCH/ECMA262%205th%20edition%20December%202009.pdf.
 ↑ "The JavaScript Object Notation (JSON) Data Interchange Format". Internet Engineering Task Force. December 2017. https://datatracker.ietf.org/doc/html/rfc8259.
Original source: https://en.wikipedia.org/wiki/Doubleprecision floatingpoint format.
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