Double-precision floating-point format

Double-precision floating-point format is a computer number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.

Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Double precision may be chosen when the range or precision of single precision would be insufficient.

In the IEEE 754-2008 standard, the 64-bit base-2 format is officially referred to as binary64; it was called double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 single precision and, more recently, base-10 representations.

One of the first programming languages to provide single- and double-precision floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision data type was the 64-bit MBF floating-point format.

Floating-point formats
IEEE 754
16-bit: Half (binary16) 32-bit: Single (binary32), decimal32 64-bit: Double (binary64), decimal64 128-bit: Quadruple (binary128), decimal128 256-bit: Octuple (binary256) Extended-precision formats (40-bit or 80-bit)
Other
Minifloat bfloat16 Microsoft Binary Format IBM floating-point architecture Arbitrary precision

IEEE 754 double-precision binary floating-point format: binary64

Double-precision binary floating-point is a commonly used format on PCs, due to its wider range over single-precision floating point, in spite of its performance and bandwidth cost. As with single-precision floating-point format, it lacks precision on integer numbers when compared with an integer format of the same size. 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 can be interpreted as either an 11-bit signed integer from −1024 to 1023 (2's complement) or an 11-bit unsigned integer from 0 to 2047, which is the accepted biased form in the IEEE 754 binary64 definition. If the unsigned integer format is used, the exponent value used in the arithmetic is the exponent shifted by a bias – for the IEEE 754 binary64 case, an exponent value of 1023 represents the actual zero (i.e. for 2^{e − 1023} to be one, e must be 1023). Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved for special numbers.

The 53-bit 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 IEEE 754 double-precision representation, 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 double-precision number is converted to a decimal string with at least 17 significant digits, and then converted back to double-precision representation, the final result must match the original number.

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 significand appearing in the memory format, the total precision is therefore 53 bits (approximately 16 decimal digits, 53 log₁₀(2) ≈ 15.955). The bits are laid out as follows:

The real value assumed by a given 64-bit double-precision datum with a given biased exponent $e$ and a 52-bit fraction is

$(-1)^{\text{sign}}(1.b_{51}b_{50}...b_{0})_{2}\times 2^{e-1023}$

or

$(-1)^{\text{sign}}\left(1+\sum _{i=1}^{52}b_{52-i}2^{-i}\right)\times 2^{e-1023}$

Between 2⁵²=4,503,599,627,370,496 and 2⁵³=9,007,199,254,740,992 the representable numbers are exactly the integers. For the next range, from 2⁵³ to 2⁵⁴, everything is multiplied by 2, so the representable numbers are the even ones, etc. Conversely, for the previous range from 2⁵¹ to 2⁵², the spacing is 0.5, etc.

The spacing as a fraction of the numbers in the range from 2ⁿ to 2ⁿ⁺¹ is 2ⁿ⁻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 10³⁰⁸, 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 double-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 1023; also known as exponent bias in the IEEE 754 standard. Examples of such representations would be:

e=000000000012=00116=1:		$2^{1-1023}=2^{-1022}$	(smallest exponent for normal numbers)
e=011111111112=3ff16=1023:		$2^{1023-1023}=2^{0}$	(zero offset)
e=100000001012=40516=1029:		$2^{1029-1023}=2^{6}$
e=111111111102=7fe16=2046:		$2^{2046-1023}=2^{1023}$	(highest exponent)

The exponents 00016 and 7ff16 have a special meaning:

• 000000000002=00016 is used to represent a signed zero (if F=0) and subnormals (if F≠0); and
• 111111111112=7ff16 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 double-precision number is described by:

$(-1)^{\text{sign}}\times 2^{e-1023}\times 1.{\text{fraction}}$

In the case of subnormals (e=0) the double-precision number is described by:

$(-1)^{\text{sign}}\times 2^{1-1023}\times 0.{\text{fraction}}=(-1)^{\text{sign}}\times 2^{-1022}\times 0.{\text{fraction}}$

Endianness

Although the ubiquitous x86 processors of today use little-endian storage for all types of data (integer, floating point, BCD), there are a number of hardware architectures where floating-point numbers are represented in big-endian form while integers are represented in little-endian form. There are ARM processors that have half little-endian, half big-endian floating-point representation for double-precision numbers: both 32-bit words are stored in little-endian like integer registers, but the most significant one first. Because there have been many floating-point formats with no "network" standard representation for them, the XDR standard uses big-endian IEEE 754 as its representation. It may therefore appear strange that the widespread IEEE 754 floating-point standard does not specify endianness. Theoretically, this means that even standard IEEE floating-point data written by one machine might not be readable by another. However, on modern standard computers (i.e., implementing IEEE 754), one may in practice safely assume that the endianness is the same for floating-point numbers as for integers, making the conversion straightforward regardless of data type. (Small embedded systems using special floating-point formats may be another matter however.)

Double-precision examples

0 01111111111 00000000000000000000000000000000000000000000000000002 ≙ +20 × 1 = 1

0 01111111111 00000000000000000000000000000000000000000000000000012 ≙ +20 × (1 + 2−52) ≈ 1.0000000000000002, the smallest number > 1

0 01111111111 00000000000000000000000000000000000000000000000000102 ≙ +20 × (1 + 2−51) ≈ 1.0000000000000004

0 10000000000 00000000000000000000000000000000000000000000000000002 ≙ +21 × 1 = 2

1 10000000000 00000000000000000000000000000000000000000000000000002 ≙ −21 × 1 = −2

0 10000000000 10000000000000000000000000000000000000000000000000002 ≙ +21 × 1.12	= 112 = 3
0 10000000001 00000000000000000000000000000000000000000000000000002 ≙ +22 × 1	= 1002 = 4
0 10000000001 01000000000000000000000000000000000000000000000000002 ≙ +22 × 1.012	= 1012 = 5
0 10000000001 10000000000000000000000000000000000000000000000000002 ≙ +22 × 1.12	= 1102 = 6
0 10000000011 01110000000000000000000000000000000000000000000000002 ≙ +24 × 1.01112	= 101112 = 23

0 00000000000 00000000000000000000000000000000000000000000000000012	≙ +2−1022 × 2−52 = 2−1074 ≈ 4.9406564584124654 × 10−324	(Min. subnormal positive double)
0 00000000000 11111111111111111111111111111111111111111111111111112	≙ +2−1022 × (1 − 2−52) ≈ 2.2250738585072009 × 10−308	(Max. subnormal double)
0 00000000001 00000000000000000000000000000000000000000000000000002	≙ +2−1022 × 1 ≈ 2.2250738585072014 × 10−308	(Min. normal positive double)
0 11111111110 11111111111111111111111111111111111111111111111111112	≙ +21023 × (1 + (1 − 2−52)) ≈ 1.7976931348623157 × 10308	(Max. Double)

0 00000000000 00000000000000000000000000000000000000000000000000002 ≙ +0
1 00000000000 00000000000000000000000000000000000000000000000000002 ≙ −0
0 11111111111 00000000000000000000000000000000000000000000000000002 ≙ +∞	(positive infinity)
1 11111111111 00000000000000000000000000000000000000000000000000002 ≙ −∞	(negative infinity)
0 11111111111 00000000000000000000000000000000000000000000000000012 ≙ NaN	(sNaN on most processors, such as x86 and ARM)
0 11111111111 10000000000000000000000000000000000000000000000000012 ≙ NaN	(qNaN on most processors, such as x86 and ARM)
0 11111111111 11111111111111111111111111111111111111111111111111112 ≙ NaN	(an alternative encoding)

0 01111111101 01010101010101010101010101010101010101010101010101012 = 3fd5 5555 5555 555516

≙ +2−2 × (1 + 2−2 + 2−4 + ... + 2−52) ≈ 1/3

0 10000000000 10010010000111111011010101000100010000101101000110002 = 4009 21fb 5444 2d1816

≈ 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 PA-RISC 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 555516,
  Sign = 0
  Exponent = 3FD16 = 1021
  Exponent Bias = 1023 (constant value; see above)
  Fraction = 5 5555 5555 555516
  Value = 2(Exponent − Exponent Bias) × 1.Fraction – Note that Fraction must not be converted to decimal here
        = 2−2 × (15 5555 5555 555516 × 2−52)
        = 2−54 × 15 5555 5555 555516
        = 0.333333333333333314829616256247390992939472198486328125
        ≈ 1/3

Execution speed with double-precision arithmetic

Using double-precision floating-point variables and mathematical functions (e.g., sin, cos, atan2, log, exp and sqrt) are slower than working with their single precision counterparts. One area of computing where this is a particular issue is for parallel code running on GPUs. For example, when using NVIDIA's CUDA platform, calculations with double precision take approximately 2 times longer to complete compared to those done using single precision.

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 32-bit x86 with extended precision by default, some compilers may not conform to the C standard and/or the arithmetic may suffer from double rounding.

Common Lisp

Common Lisp provides the types SHORT-FLOAT, SINGLE-FLOAT, DOUBLE-FLOAT and LONG-FLOAT. Most implementations provide SINGLE-FLOATs and DOUBLE-FLOATs with the other types appropriate synonyms. Common Lisp provides exceptions for catching floating-point underflows and overflows, and the inexact floating-point 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.

JavaScript

As specified by the ECMAScript standard, all arithmetic in JavaScript shall be done using double-precision floating-point arithmetic.

See also

• IEEE 754, IEEE standard for floating-point arithmetic