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Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
Here, the product notation indicates a binary floating point representation with the exponent of the representation given as a power of two and with the significand given with three bits after the binary point. To compute the subtraction it is necessary to change the forms of these numbers so that they have the same exponent, and so that when ...
The IEEE 754 specification—followed by all modern floating-point hardware—requires that the result of an elementary arithmetic operation (addition, subtraction, multiplication, division, and square root since 1985, and FMA since 2008) be correctly rounded, which implies that in rounding to nearest, the rounded result is within 0.5 ulp of ...
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions.
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 ...
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.
returns next representable floating-point value towards the given value copysign: copies the sign of a floating-point value Classification fpclassify: categorizes the given floating-point value isfinite: checks if the argument has finite value isinf: checks if the argument is infinite isnan: checks if the argument is NaN isnormal
The GNU Multiple Precision Floating-Point Reliable Library (GNU MPFR) is a GNU portable C library for arbitrary-precision binary floating-point computation with correct rounding, based on GNU Multi-Precision Library.