Average Error: 29.0 → 0.1
Time: 15.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11192.06208307448832783848047256469726562 \lor \neg \left(x \le 7503.829948731607146328315138816833496094\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\frac{1 + x}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11192.06208307448832783848047256469726562 \lor \neg \left(x \le 7503.829948731607146328315138816833496094\right):\\
\;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{\frac{1 + x}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\

\end{array}
double f(double x) {
        double r89723 = x;
        double r89724 = 1.0;
        double r89725 = r89723 + r89724;
        double r89726 = r89723 / r89725;
        double r89727 = r89723 - r89724;
        double r89728 = r89725 / r89727;
        double r89729 = r89726 - r89728;
        return r89729;
}
↓
double f(double x) {
        double r89730 = x;
        double r89731 = -11192.062083074488;
        bool r89732 = r89730 <= r89731;
        double r89733 = 7503.829948731607;
        bool r89734 = r89730 <= r89733;
        double r89735 = !r89734;
        bool r89736 = r89732 || r89735;
        double r89737 = 1.0;
        double r89738 = r89730 * r89730;
        double r89739 = r89737 / r89738;
        double r89740 = 3.0;
        double r89741 = r89740 / r89730;
        double r89742 = r89739 + r89741;
        double r89743 = 3.0;
        double r89744 = pow(r89730, r89743);
        double r89745 = r89740 / r89744;
        double r89746 = r89742 + r89745;
        double r89747 = -r89746;
        double r89748 = r89730 + r89737;
        double r89749 = r89730 / r89748;
        double r89750 = r89737 + r89730;
        double r89751 = r89730 - r89737;
        double r89752 = cbrt(r89751);
        double r89753 = r89752 * r89752;
        double r89754 = r89750 / r89753;
        double r89755 = r89754 / r89752;
        double r89756 = r89749 - r89755;
        double r89757 = r89736 ? r89747 : r89756;
        return r89757;
}

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 < -11192.062083074488 or 7503.829948731607 < x

    1. Initial program 59.4

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

    2. Simplified59.4

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{1 + x}{x - 1}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt60.3

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

    5. Applied associate-/r*60.3

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{\frac{1 + x}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}}\]

    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

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

    if -11192.062083074488 < x < 7503.829948731607

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\frac{x}{x + 1} - \frac{1 + x}{x - 1}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.1

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

    5. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11192.06208307448832783848047256469726562 \lor \neg \left(x \le 7503.829948731607146328315138816833496094\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\frac{1 + x}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))