Average Error: 29.3 → 0.1
Time: 4.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12471.6665092964158 \lor \neg \left(x \le 20005.7029934287893\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x + 1\right) \cdot \left(x + 1\right)} \cdot \frac{{x}^{3}}{x + 1} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12471.6665092964158 \lor \neg \left(x \le 20005.7029934287893\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r156761 = x;
        double r156762 = 1.0;
        double r156763 = r156761 + r156762;
        double r156764 = r156761 / r156763;
        double r156765 = r156761 - r156762;
        double r156766 = r156763 / r156765;
        double r156767 = r156764 - r156766;
        return r156767;
}
↓
double f(double x) {
        double r156768 = x;
        double r156769 = -12471.666509296416;
        bool r156770 = r156768 <= r156769;
        double r156771 = 20005.70299342879;
        bool r156772 = r156768 <= r156771;
        double r156773 = !r156772;
        bool r156774 = r156770 || r156773;
        double r156775 = 1.0;
        double r156776 = -r156775;
        double r156777 = 2.0;
        double r156778 = pow(r156768, r156777);
        double r156779 = r156776 / r156778;
        double r156780 = 3.0;
        double r156781 = r156780 / r156768;
        double r156782 = r156779 - r156781;
        double r156783 = 3.0;
        double r156784 = pow(r156768, r156783);
        double r156785 = r156780 / r156784;
        double r156786 = r156782 - r156785;
        double r156787 = 1.0;
        double r156788 = r156768 + r156775;
        double r156789 = r156788 * r156788;
        double r156790 = r156787 / r156789;
        double r156791 = r156784 / r156788;
        double r156792 = r156790 * r156791;
        double r156793 = r156768 - r156775;
        double r156794 = r156788 / r156793;
        double r156795 = pow(r156794, r156783);
        double r156796 = r156792 - r156795;
        double r156797 = r156768 / r156788;
        double r156798 = r156794 + r156797;
        double r156799 = r156794 * r156798;
        double r156800 = r156797 * r156797;
        double r156801 = r156799 + r156800;
        double r156802 = r156796 / r156801;
        double r156803 = r156774 ? r156786 : r156802;
        return r156803;
}

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 < -12471.666509296416 or 20005.70299342879 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -12471.666509296416 < x < 20005.70299342879

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied times-frac0.1

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

    9. Applied unpow-prod-down0.1

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

    10. Simplified0.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12471.6665092964158 \lor \neg \left(x \le 20005.7029934287893\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x + 1\right) \cdot \left(x + 1\right)} \cdot \frac{{x}^{3}}{x + 1} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))