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Single precision is termed REAL in Fortran; [1] SINGLE-FLOAT in Common Lisp; [2] float in C, C++, C# and Java; [3] Float in Haskell [4] and Swift; [5] and Single in Object Pascal , Visual Basic, and MATLAB. However, float in Python, Ruby, PHP, and OCaml and single in versions of Octave before 3.2 refer to double-precision numbers.
This can express values in the range ±65,504, with the minimum value above 1 being 1 + 1/1024. Depending on the computer, half-precision can be over an order of magnitude faster than double precision, e.g. 550 PFLOPS for half-precision vs 37 PFLOPS for double precision on one cloud provider. [1]
A 2-bit float with 1-bit exponent and 1-bit mantissa would only have 0, 1, Inf, NaN values. If the mantissa is allowed to be 0-bit, a 1-bit float format would have a 1-bit exponent, and the only two values would be 0 and Inf. The exponent must be at least 1 bit or else it no longer makes sense as a float (it would just be a signed number).
The three fields in a 64bit IEEE 754 float. Floating-point numbers in IEEE 754 format consist of three fields: a sign bit, a biased exponent, and a fraction. The following example illustrates the meaning of each. The decimal number 0.15625 10 represented in binary is 0.00101 2 (that is, 1/8 + 1/32). (Subscripts indicate the number base.)
The TensorFloat-32 [31] format combines the 8 bits of exponent of the Bfloat16 with the 10 bits of trailing significand field of half-precision formats, resulting in a size of 19 bits. This format was introduced by Nvidia , which provides hardware support for it in the Tensor Cores of its GPUs based on the Nvidia Ampere architecture.
The minimum strictly positive (subnormal) value is 2 −16494 ≈ 10 −4965 and has a precision of only one bit. The minimum positive normal value is 2 −16382 ≈ 3.3621 × 10 −4932 and has a precision of 113 bits, i.e. ±2 −16494 as well. The maximum representable value is 2 16384 − 2 16271 ≈ 1.1897 × 10 4932.
This alternative definition is significantly more widespread: machine epsilon is the difference between 1 and the next larger floating point number.This definition is used in language constants in Ada, C, C++, Fortran, MATLAB, Mathematica, Octave, Pascal, Python and Rust etc., and defined in textbooks like «Numerical Recipes» by Press et al.
The bfloat16 format, being a shortened IEEE 754 single-precision 32-bit float, allows for fast conversion to and from an IEEE 754 single-precision 32-bit float; in conversion to the bfloat16 format, the exponent bits are preserved while the significand field can be reduced by truncation (thus corresponding to round toward 0) or other rounding ...