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The 24-bit significand will stop at position 23, shown as the underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit. It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values, which is not the case here).
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This type of rounding, which is also named rounding to a logarithmic scale, is a variant of rounding to a specified power. Rounding on a logarithmic scale is accomplished by taking the log of the amount and doing normal rounding to the nearest value on the log scale. For example, resistors are supplied with preferred numbers on a logarithmic scale.
In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]
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 existing 64- and 128-bit formats follow this rule, but the 16- and 32-bit formats have more exponent bits (5 and 8 respectively) than this formula would provide (3 and 7 respectively). As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits to indicate either infinity (trailing significand field = 0) or a NaN (trailing ...
IEEE 754-1985 [1] is a historic industry standard for representing floating-point numbers in computers, officially adopted in 1985 and superseded in 2008 by IEEE 754-2008, and then again in 2019 by minor revision IEEE 754-2019. [2]
In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: petītiō principiī) is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion.