Average Error: 14.7 → 0.4
Time: 4.2m
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 r14988405 = 1.0;
        double r14988406 = x;
        double r14988407 = r14988406 + r14988405;
        double r14988408 = r14988405 / r14988407;
        double r14988409 = r14988406 - r14988405;
        double r14988410 = r14988405 / r14988409;
        double r14988411 = r14988408 - r14988410;
        return r14988411;
}
↓
double f(double x) {
        double r14988412 = -2.0;
        double r14988413 = x;
        double r14988414 = -1.0;
        double r14988415 = r14988413 + r14988414;
        double r14988416 = fma(r14988413, r14988415, r14988415);
        double r14988417 = r14988412 / r14988416;
        return r14988417;
}

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 2019124 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))