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Excel maintains 15 figures in its numbers, but they are not always accurate; mathematically, the bottom line should be the same as the top line, in 'fp-math' the step '1 + 1/9000' leads to a rounding up as the first bit of the 14 bit tail '10111000110010' of the mantissa falling off the table when adding 1 is a '1', this up-rounding is not undone when subtracting the 1 again, since there is no ...
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 variant of the round-to-nearest method is also called convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, [6] or bankers' rounding. [ 7 ] This is the default rounding mode used in IEEE 754 operations for results in binary floating-point formats.
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.
However, it is quite likely that it is not safe to round off the intermediate steps in the calculation to the same number of digits. Be aware that roundoff errors can accumulate. Be aware that roundoff errors can accumulate.
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.
Radial basis function; Radial basis function interpolation; Rate of convergence; Redundant CORDIC; Regge calculus; Regularized meshless method; Relative change; Remez algorithm; Mauricio Resende; Residual (numerical analysis) Riemann solver; Rounding Errors in Algebraic Processes; Round-off error; Runge–Kutta methods; Runge–Kutta–Fehlberg ...
Gal's accurate tables — table of function values with unequal spacing to reduce round-off error; Spigot algorithm — algorithms that can compute individual digits of a real number; Approximations of π: Liu Hui's π algorithm — first algorithm that can compute π to arbitrary precision