Average Error: 14.7 → 0.4
Time: 4.0m
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 r19270544 = 1.0;
        double r19270545 = x;
        double r19270546 = r19270545 + r19270544;
        double r19270547 = r19270544 / r19270546;
        double r19270548 = r19270545 - r19270544;
        double r19270549 = r19270544 / r19270548;
        double r19270550 = r19270547 - r19270549;
        return r19270550;
}
↓
double f(double x) {
        double r19270551 = -2.0;
        double r19270552 = x;
        double r19270553 = -1.0;
        double r19270554 = r19270552 + r19270553;
        double r19270555 = fma(r19270552, r19270554, r19270554);
        double r19270556 = r19270551 / r19270555;
        return r19270556;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.7

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

  3. Applied frac-sub14.1

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

  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 2019120 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))