Double-precision floating-point format
|Floating point precisions|
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.
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 and a 52-bit fraction is
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.
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.
7ff16 have a special meaning:
00016is used to represent a signed zero (if M=0) and subnormals (if M≠0); and
7ff16is 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:
In the case of subnormals (E=0) the double-precision number is described by:
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. 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. 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.)
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 = Infinity fff0 0000 0000 000016 = −Infinity 7fff ffff ffff ffff16 = NaN
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) 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, on video cards designed for gaming, calculations with double precision take 3 to 24 times longer to complete than calculations using single precision.
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.
All arithmetic in Lua is done using double-precision floating-point arithmetic. Also, automatic type conversions between doubles and strings are provided.
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 issues.
- 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).
- "Floating point formats".
- "pack – convert a list into a binary representation".
- ECMA-262 ECMAScript Language Specification (PDF) (5th ed.). Ecma International. p. 29, §8.5 The Number Type.
- GCC Bug 323 - optimized code gives strange floating point results