# IEEE-754 Functions

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IEEE-754 is a standard for the representation of and computations with floating point numbers in binary systems. It is widely used by floating point implementations in CPUs. These functions implement encoding and decoding binary representations of floating point numbers according to IEEE-754.

An IEEE-754 binary float consists of three parts: a sign bit, the exponent and the significand (sometimes called the mantissa). From these parts, the value is then calculated using the following formula: `-1 ^ sign * 2 ^ (exponent - bias) * 1.significand`. The standard defines multiple binary formats of different sizes that all follow these rules, but differ in the number of bits allocated for the exponent and significand. The bias for the default formats is defined as `bias = (2 ^ (exponent_bits - 1)) - 1`. See this article for a more detailed introduction into the subject.

The following binary float formats are defined by the standard:

Name | Also known as | Exponent bits | Significand bits
--- | --- | --- | ---
`binary16` | Half precision | 5 | 10
`binary32` | Single precision | 8 | 23
`binary64` | Double precision | 11 | 52
`binary128` | Quad precision | 15 | 112

In many programming languages, the `binary32` format is available as `float` and `binary64` is available as `double`.

`ieee754_encode`(x; exponent_bits; significand_bits[; exponent_bias])

  Encode a floating point number into a IEEE-754 binary representation.

  Parameters: | x – The floating point value to encode. exponent_bits – The length of the exponent part, in bits. significand_bits – The length of the significand part, in bits. exponent_bias – The exponent bias to use. Derived from the length of the exponent if not specified.

`ieee754_decode`(x; exponent_bits; significand_bits[; exponent_bias])

  Calculate the value of an IEEE-754 binary float.

  Parameters: | x – The binary float to decode. exponent_bits – The length of the exponent part, in bits. significand_bits – The length of the significand part, in bits. exponent_bias – The exponent bias to use. Derived from the length of the exponent if not specified.

`ieee754_half_encode`(x)

  Encode `x` in the half-precision binary format.

`ieee754_half_decode`(x)

  Decode the half-precision binary float `x`.

`ieee754_single_encode`(x)

  Encode `x` in the single-precision binary format.

`ieee754_single_decode`(x)

  Decode the single-precision binary float `x`.

`ieee754_double_encode`(x)

  Encode `x` in the double-precision binary format.

`ieee754_double_decode`(x)

  Decode the double-precision binary float `x`.

`ieee754_quad_encode`(x)

  Encode `x` in the quad-precision binary format.

`ieee754_quad_decode`(x)

  Decode the quad-precision binary float `x`.