Average Error: 14.3 → 0.1
Time: 12.2s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\frac{-2}{x + 1}}{x - 1}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{\frac{-2}{x + 1}}{x - 1}
double f(double x) {
        double r4688900 = 1.0;
        double r4688901 = x;
        double r4688902 = r4688901 + r4688900;
        double r4688903 = r4688900 / r4688902;
        double r4688904 = r4688901 - r4688900;
        double r4688905 = r4688900 / r4688904;
        double r4688906 = r4688903 - r4688905;
        return r4688906;
}
↓
double f(double x) {
        double r4688907 = -2.0;
        double r4688908 = x;
        double r4688909 = 1.0;
        double r4688910 = r4688908 + r4688909;
        double r4688911 = r4688907 / r4688910;
        double r4688912 = r4688908 - r4688909;
        double r4688913 = r4688911 / r4688912;
        return r4688913;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 14.3

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

  3. Applied frac-sub13.7

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

  4. Simplified13.7

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

  5. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{-2}}{\left(x + 1\right) \cdot \left(x - 1\right)}\]
  6. Using strategy rm

  7. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{-2}{x + 1}}{x - 1}}\]

  8. Final simplification0.1

    \[\leadsto \frac{\frac{-2}{x + 1}}{x - 1}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))