Average Error: 29.1 → 0.1
Time: 14.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11235.7395424927436 \lor \neg \left(x \le 8706.16731299591083\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11235.7395424927436 \lor \neg \left(x \le 8706.16731299591083\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{x + 1}{x - 1}\right)\\

\end{array}
double f(double x) {
        double r158597 = x;
        double r158598 = 1.0;
        double r158599 = r158597 + r158598;
        double r158600 = r158597 / r158599;
        double r158601 = r158597 - r158598;
        double r158602 = r158599 / r158601;
        double r158603 = r158600 - r158602;
        return r158603;
}
↓
double f(double x) {
        double r158604 = x;
        double r158605 = -11235.739542492744;
        bool r158606 = r158604 <= r158605;
        double r158607 = 8706.16731299591;
        bool r158608 = r158604 <= r158607;
        double r158609 = !r158608;
        bool r158610 = r158606 || r158609;
        double r158611 = 1.0;
        double r158612 = -r158611;
        double r158613 = 2.0;
        double r158614 = pow(r158604, r158613);
        double r158615 = r158612 / r158614;
        double r158616 = 3.0;
        double r158617 = r158616 / r158604;
        double r158618 = r158615 - r158617;
        double r158619 = 3.0;
        double r158620 = pow(r158604, r158619);
        double r158621 = r158616 / r158620;
        double r158622 = r158618 - r158621;
        double r158623 = r158604 * r158604;
        double r158624 = r158611 * r158611;
        double r158625 = r158623 - r158624;
        double r158626 = r158604 / r158625;
        double r158627 = r158604 - r158611;
        double r158628 = r158604 + r158611;
        double r158629 = r158628 / r158627;
        double r158630 = -r158629;
        double r158631 = fma(r158626, r158627, r158630);
        double r158632 = r158610 ? r158622 : r158631;
        return r158632;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -11235.739542492744 or 8706.16731299591 < x

    1. Initial program 59.2

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

    3. Applied flip-+60.4

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

    4. Applied associate-/r/60.4

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

    5. Applied fma-neg60.2

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

    6. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}}\]

    if -11235.739542492744 < x < 8706.16731299591

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied fma-neg0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{x + 1}{x - 1}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11235.7395424927436 \lor \neg \left(x \le 8706.16731299591083\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))