Average Error: 29.6 → 0.1
Time: 5.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8016.6118392120898 \lor \neg \left(x \le 7951.093111795664\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8016.6118392120898 \lor \neg \left(x \le 7951.093111795664\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\

\end{array}
double f(double x) {
        double r148820 = x;
        double r148821 = 1.0;
        double r148822 = r148820 + r148821;
        double r148823 = r148820 / r148822;
        double r148824 = r148820 - r148821;
        double r148825 = r148822 / r148824;
        double r148826 = r148823 - r148825;
        return r148826;
}
↓
double f(double x) {
        double r148827 = x;
        double r148828 = -8016.61183921209;
        bool r148829 = r148827 <= r148828;
        double r148830 = 7951.093111795664;
        bool r148831 = r148827 <= r148830;
        double r148832 = !r148831;
        bool r148833 = r148829 || r148832;
        double r148834 = 1.0;
        double r148835 = -r148834;
        double r148836 = 2.0;
        double r148837 = pow(r148827, r148836);
        double r148838 = r148835 / r148837;
        double r148839 = 3.0;
        double r148840 = r148839 / r148827;
        double r148841 = r148838 - r148840;
        double r148842 = 3.0;
        double r148843 = pow(r148827, r148842);
        double r148844 = r148839 / r148843;
        double r148845 = r148841 - r148844;
        double r148846 = cbrt(r148827);
        double r148847 = r148846 * r148846;
        double r148848 = r148827 + r148834;
        double r148849 = r148846 / r148848;
        double r148850 = r148847 * r148849;
        double r148851 = r148827 - r148834;
        double r148852 = r148848 / r148851;
        double r148853 = exp(r148852);
        double r148854 = log(r148853);
        double r148855 = r148850 - r148854;
        double r148856 = r148833 ? r148845 : r148855;
        return r148856;
}

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 < -8016.61183921209 or 7951.093111795664 < x

    1. Initial program 59.3

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

    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 -8016.61183921209 < x < 7951.093111795664

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\log \left(e^{\frac{x + 1}{x - 1}}\right)}\]
    4. Using strategy rm

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

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied times-frac0.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8016.6118392120898 \lor \neg \left(x \le 7951.093111795664\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))