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# Double-precision floating-point format

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### Double-precision floating-point format

Double-precision floating-point format is a computer number format that occupies 8 bytes (64 bits) in computer memory and represents a wide dynamic range of values by using floating point.

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 format usually refers to binary64, as specified by the IEEE 754 standard, not to the 64-bit decimal format decimal64.

## 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:

This gives 15–17 significant decimal digits precision. If a decimal string with at most 15 significant digits is converted to IEEE 754 double precision representation and then converted back to a string with the same number of significant digits, then the final string should match the original. If an IEEE 754 double precision is converted to a decimal string with at least 17 significant digits and then converted back to double, then the final number must match the original.[1]

The format is written with the significand having an implicit integer bit of value 1 (except for special datums, 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 with a decimal exponent between 10−308 and 10308, with full 15–17 decimal digits precision. By compromising precision, subnormal representation allows values smaller than 10−323.

### 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:

• Emin (1) = −1022
• E (50) = −973
• Emax (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 a signed 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}

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

(-1)^{\text{sign}} \times 2^{1 - \text{exponent bias}} \times 0.\text{mantissa}

### Endianness

Although the ubiquitous x86 of today use little-endian storage for all types of data (integer, floating point, BCD), there have been a few historical machines where floating point numbers were represented in big-endian form while integers were represented in little-endian form.[2] There are old 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, there is no formal standard for transferring floating point values between diverse systems. It may therefore appear strange that the widespread IEEE 754 floating point standard does not specify endianness.[3] 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

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 = 2−1074
≈ 4.9406564584124654 × 10−324 (Min subnormal positive double)
000f ffff ffff ffff16   = 2−1022 − 2−1022−52
≈ 2.2250738585072009 × 10−308 (Max subnormal double)
0010 0000 0000 000016   = 2−1022
≈ 2.2250738585072014 × 10−308 (Min normal positive double)
7fef ffff ffff ffff16   = (1 + (1 − 2−52)) × 21023
≈ 1.7976931348623157 × 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


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)
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 double precision floating-point variables and mathematical functions (e.g., sin(), cos(), atan2(), log(), exp(), 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, on gaming cards, calculations with double precision take 3 to 24 times longer to complete than calculations using single precision.[4]

## Notes and references

1. ^ William Kahan (1 October 1987). "Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic".
2. ^ "Floating point formats".
3. ^ "pack – convert a list into a binary representation".
4. ^ http://www.tomshardware.com/reviews/geforce-gtx-titan-gk110-review,3438-3.html
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