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The bfloat16 (brain floating point) [1] [2] floating-point format is a computer number format occupying 16 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. This format is a shortened (16-bit) version of the 32-bit IEEE 754 single-precision floating-point format (binary32) with the ...
Swift introduced half-precision floating point numbers in Swift 5.3 with the Float16 type. [20] OpenCL also supports half-precision floating point numbers with the half datatype on IEEE 754-2008 half-precision storage format. [21] As of 2024, Rust is currently working on adding a new f16 type for IEEE half-precision 16-bit floats. [22]
Unums (universal numbers [1]) are a family of number formats and arithmetic for implementing real numbers on a computer, proposed by John L. Gustafson in 2015. [2] They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version is known as posits. [3]
On a typical computer system, a double-precision (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 10 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2 −1022 ≈ 2 × 10 −308 to approximately 2 1024 ≈ ...
For the exchange of binary floating-point numbers, interchange formats of length 16 bits, 32 bits, 64 bits, and any multiple of 32 bits ≥ 128 [e] are defined. The 16-bit format is intended for the exchange or storage of small numbers (e.g., for graphics).
A common usage of mixed-precision arithmetic is for operating on inaccurate numbers with a small width and expanding them to a larger, more accurate representation. For example, two half-precision or bfloat16 (16-bit) floating-point numbers may be multiplied together to result in a more accurate single-precision (32-bit) float. [1]
In addition to the assumption about bit-representation of floating-point numbers, the above floating-point type-punning example also violates the C language's constraints on how objects are accessed: [3] the declared type of x is float but it is read through an expression of type unsigned int.
Even floating-point numbers are soon outranged, so it may help to recast the calculations in terms of the logarithm of the number. But if exact values for large factorials are desired, then special software is required, as in the pseudocode that follows, which implements the classic algorithm to calculate 1, 1×2, 1×2×3, 1×2×3×4, etc. the ...