Double-precision floating-point format

In computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point (in which case it is often referred to as FP64).

Modern computers with 32-bit storage locations use two memory locations to store a 64-bit double-precision number (a single storage location can hold a single-precision number). Double-precision floating-point is an IEEE 754 standard for encoding binary or decimal floating-point numbers in 64 bits (8 bytes).

Floating-point precisions
IEEE 754
16-bit: Half (binary16) 32-bit: Single (binary32), decimal32 64-bit: Double (binary64), decimal64 128-bit: Quadruple (binary128), decimal128 Extended precision formats
Other
Minifloat 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 width: 11 bits
• Significand precision: 53 bits (52 explicitly stored)

This gives from 15–17 significant decimal digits precision. If a decimal string with at most 15 significant decimal is converted to IEEE 754 double precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 double precision is converted to a decimal string with at least 17 significant decimal and then converted back to double, then the final number must match the original.

The format is written with the significand having an implicit integer bit of value 1, unless the written exponent is all zeros. 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 data with a given biased exponent e and a 52-bit fraction is $= (-1)^{\text{sign}}(1.b_{-1}b_{-2}...b_{-52})_2 \times 2^{e-1023}$ or more precisely: $\text{value} = (-1)^{\text{sign}}(1 + \sum_{i=1}^{52} b_{52-i} 2^{-i} )\times 2^{(e-1023)}$

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.

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_min (1) = −1022
• E (50) = −973
• E_max (2046) = 1023

Thus, as defined by the offset-binary representation, in order to get the true exponent the exponent bias of 1023 has to be subtracted from the written exponent.

The exponents 00016 and 7ff16 have a special meaning:

• 00016 is used to represent zero (if M=0) and subnormals (if M≠0); and
• 7ff16 is used to represent ∞ (if M=0) and NaNs (if M≠0),

where M is the fraction mantissa. All bit patterns are valid encoding.

Except for the above exceptions, the entire double-precision number is described by:

$(-1)^{\text{sign}} \times 2^{\text{exponent} - \text{exponent bias}} \times 1.\text{mantissa}$

Double-precision examples

3ff0 0000 0000 000016   = 1
3ff0 0000 0000 000116   ≈ 1.0000000000000002, the smallest number > 1
3ff0 0000 0000 000216   ≈ 1.0000000000000004
4000 0000 0000 000016   = 2
c000 0000 0000 000016   = –2

0000 0000 0000 000116   = 2-1022-52 
                       ≈ 5 x 10−324 (Min subnormal positive double, has a precision of only one bit, i.e. ±2−1075)
000f ffff ffff ffff16   = 2−1022 - 2-1022-52 
                       ≈ 2.2250738585072009 x 10−308 (Max subnormal double)
0010 0000 0000 000016   = 2−1022 
                       ≈ 2.2250738585072014 x 10−308 (Min normal positive double)
7fef ffff ffff ffff16   = (1 + (1 - 2−52)) x 21023 
                       ≈ 1.7976931348623157 x 10308 (Max Double)

0000 0000 0000 000016   = 0
8000 0000 0000 000016   = –0

7ff0 0000 0000 000016   = ∞
fff0 0000 0000 000016   = −∞

3fd5 5555 5555 555516   ≈ 1/3

(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)
  Significand = 5 5555 5555 555516
  Value = 2(Exponent − Exponent Bias) × 1.Significand – Note the Significand 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 floating-point variables and mathematical functions (sin(), cos(), atan2(), log(), exp(), sqrt() are the most popular ones) of double precision as opposed to single precision comes at execution cost: the operations with double precision are usually slower. On average, on a PC of year 2012 build, calculations with double precision are 1.1–1.6 times slower than with single precision.^{[citation needed]}

See also

• IEEE Standard for Floating-Point Arithmetic (IEEE 754)
• Extended precision (80-bit)