Average Error: 15.2 → 0.3
Time: 1.5m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{(x \cdot x + x)_*}\]
double f(double x) {
        double r3707859 = 1.0;
        double r3707860 = x;
        double r3707861 = r3707860 + r3707859;
        double r3707862 = r3707859 / r3707861;
        double r3707863 = r3707859 / r3707860;
        double r3707864 = r3707862 - r3707863;
        return r3707864;
}
↓
double f(double x) {
        double r3707865 = -1.0;
        double r3707866 = x;
        double r3707867 = fma(r3707866, r3707866, r3707866);
        double r3707868 = r3707865 / r3707867;
        return r3707868;
}


\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-1}{(x \cdot x + x)_*}

Error

Bits error versus x

Derivation

  1. Initial program 15.2

    \[\frac{1}{x + 1} - \frac{1}{x}\]
  2. Using strategy rm

  3. Applied frac-sub14.6

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]

  4. Simplified0.3

    \[\leadsto \frac{\color{blue}{-1}}{\left(x + 1\right) \cdot x}\]

  5. Simplified0.3

    \[\leadsto \frac{-1}{\color{blue}{(x \cdot x + x)_*}}\]

  6. Final simplification0.3

    \[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))