Average Error: 29.4 → 0.3
Time: 7.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11573.705281839704:\\ \;\;\;\;\log \left(e^{\frac{-1}{x \cdot x}}\right) + \left(\frac{-3}{x} + \frac{-3}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 10678.074230429429:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1}{x - 1} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{-1}{x \cdot x}}\right) + \left(\frac{-3}{x} + \frac{-3}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11573.705281839704:\\
\;\;\;\;\log \left(e^{\frac{-1}{x \cdot x}}\right) + \left(\frac{-3}{x} + \frac{-3}{\left(x \cdot x\right) \cdot x}\right)\\

\mathbf{elif}\;x \le 10678.074230429429:\\
\;\;\;\;\frac{x}{1 + x} - \frac{1}{x - 1} \cdot \left(1 + x\right)\\

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

\end{array}
double f(double x) {
        double r1449889 = x;
        double r1449890 = 1.0;
        double r1449891 = r1449889 + r1449890;
        double r1449892 = r1449889 / r1449891;
        double r1449893 = r1449889 - r1449890;
        double r1449894 = r1449891 / r1449893;
        double r1449895 = r1449892 - r1449894;
        return r1449895;
}
↓
double f(double x) {
        double r1449896 = x;
        double r1449897 = -11573.705281839704;
        bool r1449898 = r1449896 <= r1449897;
        double r1449899 = -1.0;
        double r1449900 = r1449896 * r1449896;
        double r1449901 = r1449899 / r1449900;
        double r1449902 = exp(r1449901);
        double r1449903 = log(r1449902);
        double r1449904 = -3.0;
        double r1449905 = r1449904 / r1449896;
        double r1449906 = r1449900 * r1449896;
        double r1449907 = r1449904 / r1449906;
        double r1449908 = r1449905 + r1449907;
        double r1449909 = r1449903 + r1449908;
        double r1449910 = 10678.074230429429;
        bool r1449911 = r1449896 <= r1449910;
        double r1449912 = 1.0;
        double r1449913 = r1449912 + r1449896;
        double r1449914 = r1449896 / r1449913;
        double r1449915 = r1449896 - r1449912;
        double r1449916 = r1449912 / r1449915;
        double r1449917 = r1449916 * r1449913;
        double r1449918 = r1449914 - r1449917;
        double r1449919 = r1449911 ? r1449918 : r1449909;
        double r1449920 = r1449898 ? r1449909 : r1449919;
        return r1449920;
}

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 < -11573.705281839704 or 10678.074230429429 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    5. Applied add-log-exp0.5

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

    if -11573.705281839704 < x < 10678.074230429429

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Applied associate-/r/0.1

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

    6. Applied difference-of-squares0.1

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

    7. Applied associate-/r*0.1

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

    8. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11573.705281839704:\\ \;\;\;\;\log \left(e^{\frac{-1}{x \cdot x}}\right) + \left(\frac{-3}{x} + \frac{-3}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 10678.074230429429:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1}{x - 1} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{-1}{x \cdot x}}\right) + \left(\frac{-3}{x} + \frac{-3}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))