In computing, double precision is a computer numbering 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.

Modern computers with 32-bit stores (single precision) provide 64-bit double precision. Double precision floating point is an IEEE 754 standard for encoding binary or decimal floating point numbers in 8 bytes.

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Double precision binary floating-point format

The format is written with an implicit integer bit with value 1 unless the written exponent is all zeros. With the 52 bits of the fraction mantissa appearing in the memory format the total precision is therefore 53 bits (approximately 16 decimal digits, \log_{10}(2^{53}) \approx 15.955). The bits are laid out as follows:

Image:IEEE_754_Double_Floating_Point_Format.svg

Exponent encoding

  • Emin (0x001) = -1022
  • Emax (0x7fe) = 1023
  • Exponent bias (0x3ff) = 1023

The true exponent = written exponent - exponent bias

The exponents 0x000 and 0x7ff have a special meaning:

All bit patterns are valid encoding.

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

0x3ff0 0000 0000 0000   = 1

0xc000 0000 0000 0000   = –2

0x7fef ffff ffff ffff   ≈ 1.7976931348623157 x 10308 (Max Double)

0x0000 0000 0000 0000   = 0
0x8000 0000 0000 0000   = –0

0x7ff0 0000 0000 0000   = Infinity
0xfff0 0000 0000 0000   = -Infinity

0x3fd5 5555 5555 5555   ≈ 1/3

(1/3 rounds down instead of up like single precision, because of the odd number of bits in the significant.)

In more detail:

Given the binary representation 0x3fd5 5555 5555 5555,
  Sign = 0x0
  Exponent = 0x3fd = 1021
  Exponent Bias = 1023 (above)
  Mantissa = 0x5 5555 5555 5555
  Value = 2(Exponent − Exponent Bias) × 1.Mantissa – Note the Mantissa must not be converted to decimal here
        = 2–2 × (0x15 5555 5555 5555 × 2–52)
        = 2–54 × 0x15 5555 5555 5555
        = 0.333333333333333314829616256247390992939472198486328125
        ≈ 1/3

Double precision decimal floating-point format

to be added

See also

Retrieved from "http://en.wikipedia.org/wiki/Double_precision"

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