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The default IEEE 754 rounding mode minimizes errors, and most normal computations should not suffer any serious accumulation of errors from rounding.

Of course, you can contrive examples where that is not so. Here is one: iterate a square root, then attempt to recover the original value by repeated squaring.

#include <stdio.h> #include <math.h> int main (void) { double x = 100.0; double y; for (n = 10; n <= 100; n += 10) { y = x; for (k = 0; k < n; ++k) y = sqrt (y); for (k = 0; k < n; ++k) y *= y; printf ("n = %3d; x = %.0f\ty = %.6f\n", n, x, y); } return 0; }

Here is the output:

n = 10; x = 100 y = 100.000000 n = 20; x = 100 y = 100.000000 n = 30; x = 100 y = 99.999977 n = 40; x = 100 y = 99.981025 n = 50; x = 100 y = 90.017127 n = 60; x = 100 y = 1.000000 n = 70; x = 100 y = 1.000000 n = 80; x = 100 y = 1.000000 n = 90; x = 100 y = 1.000000 n = 100; x = 100 y = 1.000000

After 50 iterations, `y`

has barely one correct digit, and
soon after, there are no correct digits.