Average Error: 14.5 → 0.4
Time: 4.9m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{-2}{\mathsf{fma}\left(x, \left(x + -1\right), \left(x + -1\right)\right)}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{-2}{\mathsf{fma}\left(x, \left(x + -1\right), \left(x + -1\right)\right)}
double f(double x) {
        double r30738433 = 1.0;
        double r30738434 = x;
        double r30738435 = r30738434 + r30738433;
        double r30738436 = r30738433 / r30738435;
        double r30738437 = r30738434 - r30738433;
        double r30738438 = r30738433 / r30738437;
        double r30738439 = r30738436 - r30738438;
        return r30738439;
}
↓
double f(double x) {
        double r30738440 = -2.0;
        double r30738441 = x;
        double r30738442 = -1.0;
        double r30738443 = r30738441 + r30738442;
        double r30738444 = fma(r30738441, r30738443, r30738443);
        double r30738445 = r30738440 / r30738444;
        return r30738445;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.5

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

  3. Applied frac-sub14.0

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

  4. Simplified0.4

    \[\leadsto \frac{\color{blue}{-2}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]

  5. Simplified0.4

    \[\leadsto \frac{-2}{\color{blue}{\mathsf{fma}\left(x, \left(x + -1\right), \left(x + -1\right)\right)}}\]

  6. Final simplification0.4

    \[\leadsto \frac{-2}{\mathsf{fma}\left(x, \left(x + -1\right), \left(x + -1\right)\right)}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))