# 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 a floating point.

Computers with 32-bit storage locations use two memory locations to store a 64-bit double-precision number; each storage location holds 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**.

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## Contents

## 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 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 log_{10}(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 and a 52-bit fraction is

or

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 with a decimal exponent between 10^{−308} and 10^{308}, 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:

- 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 `000`

and _{16}`7ff`

have a special meaning:_{16}

`000`

is used to represent a signed zero (if M=0) and subnormals (if M≠0); and_{16}`7ff`

is used to represent ∞ (if M=0) and NaNs (if M≠0),_{16}

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:

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

### Endianness

Although the ubiquitous x86 processors 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 0000_{16}= 1 3ff0 0000 0000 0001_{16}≈ 1.0000000000000002, the smallest number > 1 3ff0 0000 0000 0002_{16}≈ 1.0000000000000004 4000 0000 0000 0000_{16}= 2 c000 0000 0000 0000_{16}= –2

0000 0000 0000 0001_{16}= 2^{−1022−52}= 2^{−1074}≈ 4.9406564584124654 × 10^{−324}(Min subnormal positive double) 000f ffff ffff ffff_{16}= 2^{−1022}− 2^{−1022−52}≈ 2.2250738585072009 × 10^{−308}(Max subnormal double) 0010 0000 0000 0000_{16}= 2^{−1022}≈ 2.2250738585072014 × 10^{−308}(Min normal positive double) 7fef ffff ffff ffff_{16}= (1 + (1 − 2^{−52})) × 2^{1023}≈ 1.7976931348623157 × 10^{308}(Max Double)

0000 0000 0000 0000_{16}= 0 8000 0000 0000 0000_{16}= –0

7ff0 0000 0000 0000_{16}= Infinity fff0 0000 0000 0000_{16}= −Infinity 7fff ffff ffff ffff_{16}= NaN

3fd5 5555 5555 5555_{16}≈ 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 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 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 video cards designed for gaming, calculations with double precision take 3 to 24 times longer to complete than calculations using single precision.^{[4]}

## Implementations

Doubles are implemented in many programming languages in different ways such as the following.

### Lua

All arithmetic in Lua is done using double-precision floating-point arithmetic. Also, automatic type conversions between doubles and strings are provided.

### JavaScript

All arithmetic in JavaScript is done using double-precision floating-point arithmetic.

### C and C++

C and C++ offer a wide variety of arithmetic types. Double precision is not required by the standards, but on most systems, the `double`

type corresponds to double precision.

## See also

- IEEE floating point: IEEE standard for floating-point arithmetic (IEEE 754)

## Notes and references

- ↑ William Kahan (1 October 1997). "Lecture Notes on the Status of IEEE Standard 754 for Binary Floating-Point Arithmetic" (PDF).<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Floating point formats".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "pack – convert a list into a binary representation".<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ http://www.tomshardware.com/reviews/geforce-gtx-titan-gk110-review,3438-3.html