Average Error: 14.5 → 0.4
Time: 3.8m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{1}{(x \cdot x + -1)_*} \cdot -2\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{1}{(x \cdot x + -1)_*} \cdot -2
double f(double x) {
        double r21702532 = 1.0;
        double r21702533 = x;
        double r21702534 = r21702533 + r21702532;
        double r21702535 = r21702532 / r21702534;
        double r21702536 = r21702533 - r21702532;
        double r21702537 = r21702532 / r21702536;
        double r21702538 = r21702535 - r21702537;
        return r21702538;
}
↓
double f(double x) {
        double r21702539 = 1.0;
        double r21702540 = x;
        double r21702541 = -1.0;
        double r21702542 = fma(r21702540, r21702540, r21702541);
        double r21702543 = r21702539 / r21702542;
        double r21702544 = -2.0;
        double r21702545 = r21702543 * r21702544;
        return r21702545;
}

Error

Bits error versus x

Derivation

  1. Initial program 14.5

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

  3. Applied flip--29.3

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

  4. Applied associate-/r/29.3

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

  5. Applied flip-+14.6

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

  6. Applied associate-/r/14.5

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

  7. Applied distribute-lft-out--14.0

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

  8. Simplified14.0

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))