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A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23) × 2 127 ≈ 3.4028235 ...
It is intended for storage of floating-point values in applications where higher precision is not essential, in particular image processing and neural networks. Almost all modern uses follow the IEEE 754-2008 standard, where the 16-bit base-2 format is referred to as binary16, and the exponent uses 5 bits. This can express values in the range ...
The minimum strictly positive (subnormal) value is 2 −262378 ≈ 10 −78984 and has a precision of only one bit. The minimum positive normal value is 2 −262142 ≈ 2.4824 × 10 −78913. The maximum representable value is 2 262144 − 2 261907 ≈ 1.6113 × 10 78913.
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient.
E max = FE H −7F H = 127; Exponent bias = 7F H = 127; Thus, in order to get the true exponent as defined by the offset-binary representation, the offset of 127 has to be subtracted from the value of the exponent field. The minimum and maximum values of the exponent field (00 H and FF H) are interpreted specially, like in the IEEE 754 standard ...
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.
decimal128 supports 'normal' values that can have 34 digit precision from ±1.000 000 000 000 000 000 000 000 000 000 000 × 10 ^ −6143 to ±9.999 999 999 999 999 999 999 999 999 999 999 × 10 ^ +6144, plus 'denormal' values with ramp-down relative precision down to ±1 × 10 −6176, signed zeros, signed infinities and NaN (Not a Number).
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.