Average Error: 29.3 → 0.1
Time: 17.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9689.783168792597280116751790046691894531:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 12586.31945398888638010248541831970214844:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9689.783168792597280116751790046691894531:\\
\;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\

\mathbf{elif}\;x \le 12586.31945398888638010248541831970214844:\\
\;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\

\end{array}
double f(double x) {
        double r4979624 = x;
        double r4979625 = 1.0;
        double r4979626 = r4979624 + r4979625;
        double r4979627 = r4979624 / r4979626;
        double r4979628 = r4979624 - r4979625;
        double r4979629 = r4979626 / r4979628;
        double r4979630 = r4979627 - r4979629;
        return r4979630;
}
↓
double f(double x) {
        double r4979631 = x;
        double r4979632 = -9689.783168792597;
        bool r4979633 = r4979631 <= r4979632;
        double r4979634 = 3.0;
        double r4979635 = r4979631 * r4979631;
        double r4979636 = r4979635 * r4979631;
        double r4979637 = r4979634 / r4979636;
        double r4979638 = -r4979637;
        double r4979639 = r4979634 / r4979631;
        double r4979640 = 1.0;
        double r4979641 = r4979640 / r4979631;
        double r4979642 = r4979641 / r4979631;
        double r4979643 = r4979639 + r4979642;
        double r4979644 = r4979638 - r4979643;
        double r4979645 = 12586.319453988886;
        bool r4979646 = r4979631 <= r4979645;
        double r4979647 = r4979631 - r4979640;
        double r4979648 = r4979631 + r4979640;
        double r4979649 = r4979648 * r4979647;
        double r4979650 = r4979631 / r4979649;
        double r4979651 = r4979647 * r4979650;
        double r4979652 = r4979648 / r4979647;
        double r4979653 = r4979651 - r4979652;
        double r4979654 = r4979646 ? r4979653 : r4979644;
        double r4979655 = r4979633 ? r4979644 : r4979654;
        return r4979655;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -9689.783168792597 or 12586.319453988886 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.0

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

    if -9689.783168792597 < x < 12586.319453988886

    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. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9689.783168792597280116751790046691894531:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 12586.31945398888638010248541831970214844:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))