Average Error: 29.7 → 0.1
Time: 10.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11361.07922411017534614074975252151489258 \lor \neg \left(x \le 10814.79793970629907562397420406341552734\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \sqrt[3]{{\left(\frac{1}{x - 1}\right)}^{3}} \cdot \left(1 + x\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11361.07922411017534614074975252151489258 \lor \neg \left(x \le 10814.79793970629907562397420406341552734\right):\\
\;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r73869 = x;
        double r73870 = 1.0;
        double r73871 = r73869 + r73870;
        double r73872 = r73869 / r73871;
        double r73873 = r73869 - r73870;
        double r73874 = r73871 / r73873;
        double r73875 = r73872 - r73874;
        return r73875;
}
↓
double f(double x) {
        double r73876 = x;
        double r73877 = -11361.079224110175;
        bool r73878 = r73876 <= r73877;
        double r73879 = 10814.797939706299;
        bool r73880 = r73876 <= r73879;
        double r73881 = !r73880;
        bool r73882 = r73878 || r73881;
        double r73883 = 1.0;
        double r73884 = r73876 * r73876;
        double r73885 = r73883 / r73884;
        double r73886 = 3.0;
        double r73887 = r73886 / r73876;
        double r73888 = r73885 + r73887;
        double r73889 = 3.0;
        double r73890 = pow(r73876, r73889);
        double r73891 = r73886 / r73890;
        double r73892 = r73888 + r73891;
        double r73893 = -r73892;
        double r73894 = r73883 + r73876;
        double r73895 = r73876 / r73894;
        double r73896 = 1.0;
        double r73897 = r73876 - r73883;
        double r73898 = r73896 / r73897;
        double r73899 = pow(r73898, r73889);
        double r73900 = cbrt(r73899);
        double r73901 = r73900 * r73894;
        double r73902 = r73895 - r73901;
        double r73903 = r73882 ? r73893 : r73902;
        return r73903;
}

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 < -11361.079224110175 or 10814.797939706299 < x

    1. Initial program 59.4

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

    3. Applied div-inv59.6

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

    5. Applied add-cbrt-cube60.6

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

    6. Simplified60.6

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

    7. Taylor expanded around inf 0.3

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

    8. Simplified0.0

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

    if -11361.079224110175 < x < 10814.797939706299

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    5. Applied add-cbrt-cube0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11361.07922411017534614074975252151489258 \lor \neg \left(x \le 10814.79793970629907562397420406341552734\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \sqrt[3]{{\left(\frac{1}{x - 1}\right)}^{3}} \cdot \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))