Average Error: 14.4 → 0.4
Time: 4.3m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{-2}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{-2}{(x \cdot \left(x + -1\right) + \left(x + -1\right))_*}
double f(double x) {
        double r19521729 = 1.0;
        double r19521730 = x;
        double r19521731 = r19521730 + r19521729;
        double r19521732 = r19521729 / r19521731;
        double r19521733 = r19521730 - r19521729;
        double r19521734 = r19521729 / r19521733;
        double r19521735 = r19521732 - r19521734;
        return r19521735;
}
↓
double f(double x) {
        double r19521736 = -2.0;
        double r19521737 = x;
        double r19521738 = -1.0;
        double r19521739 = r19521737 + r19521738;
        double r19521740 = fma(r19521737, r19521739, r19521739);
        double r19521741 = r19521736 / r19521740;
        return r19521741;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.4

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

  3. Applied frac-sub13.8

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))