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The design of floating-point format allows various optimisations, resulting from the easy generation of a base-2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics.
Like the binary floating-point formats, the number is divided into a sign, an exponent, and a significand. Unlike binary floating-point, numbers are not necessarily normalized; values with few significant digits have multiple possible representations: 1×10 2 =0.1×10 3 =0.01×10 4, etc. When the significand is zero, the exponent can be any ...
In the floating-point case, a variable exponent would represent the power of ten to which the mantissa of the number is multiplied. Languages that support a rational data type usually allow the construction of such a value from two integers, instead of a base-2 floating-point number, due to the loss of exactness the latter would cause.
Julia: the built-in BigFloat and BigInt types provide arbitrary-precision floating point and integer arithmetic respectively. newRPL: integers and floats can be of arbitrary precision (up to at least 2000 digits); maximum number of digits configurable (default 32 digits) Nim: bigints and multiple GMP bindings.
Bfloat16 is designed to maintain the number range from the 32-bit IEEE 754 single-precision floating-point format (binary32), while reducing the precision from 24 bits to 8 bits. This means that the precision is between two and three decimal digits, and bfloat16 can represent finite values up to about 3.4 × 10 38 .
The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it.
The resulting 'raw' exponent is a 8 bit binary integer where the leading bits are not '11', thus values 0 ... 1011 1111 b = 0 ... 191 d , appr. bias is to be subtracted. The resulting significand could be a positive binary integer of 24 bits up to 1001 1111111111 1111111111 b = 10485759 d , but values above 10 7 − 1 = 9 999 999 = 98967F 16 ...
The binary interchange formats have the "half precision" (16-bit storage format) and "quad precision" (128-bit format) added, together with generalized formulae for some wider formats; the basic formats have 32-bit, 64-bit, and 128-bit encodings. Three new decimal formats are described, matching the lengths of the 32–128-bit binary formats.