Average Error: 14.4 → 0.4
Time: 4.5m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot -2\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot -2
double f(double x) {
        double r20257678 = 1.0;
        double r20257679 = x;
        double r20257680 = r20257679 + r20257678;
        double r20257681 = r20257678 / r20257680;
        double r20257682 = r20257679 - r20257678;
        double r20257683 = r20257678 / r20257682;
        double r20257684 = r20257681 - r20257683;
        return r20257684;
}
↓
double f(double x) {
        double r20257685 = 1.0;
        double r20257686 = x;
        double r20257687 = -1.0;
        double r20257688 = fma(r20257686, r20257686, r20257687);
        double r20257689 = r20257685 / r20257688;
        double r20257690 = -2.0;
        double r20257691 = r20257689 * r20257690;
        return r20257691;
}

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 flip--29.2

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

  4. Applied associate-/r/29.2

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

  5. Applied flip-+14.5

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

  6. Applied associate-/r/14.4

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

  7. Applied distribute-lft-out--13.9

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

  8. Simplified13.9

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))