Average Error: 29.7 → 0.1
Time: 9.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9516.22025087149:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 11259.474582447798:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9516.22025087149:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 11259.474582447798:\\
\;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\

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

\end{array}
double f(double x) {
        double r2058514 = x;
        double r2058515 = 1.0;
        double r2058516 = r2058514 + r2058515;
        double r2058517 = r2058514 / r2058516;
        double r2058518 = r2058514 - r2058515;
        double r2058519 = r2058516 / r2058518;
        double r2058520 = r2058517 - r2058519;
        return r2058520;
}
↓
double f(double x) {
        double r2058521 = x;
        double r2058522 = -9516.22025087149;
        bool r2058523 = r2058521 <= r2058522;
        double r2058524 = -3.0;
        double r2058525 = r2058521 * r2058521;
        double r2058526 = r2058525 * r2058521;
        double r2058527 = r2058524 / r2058526;
        double r2058528 = 1.0;
        double r2058529 = r2058528 / r2058525;
        double r2058530 = 3.0;
        double r2058531 = r2058530 / r2058521;
        double r2058532 = r2058529 + r2058531;
        double r2058533 = r2058527 - r2058532;
        double r2058534 = 11259.474582447798;
        bool r2058535 = r2058521 <= r2058534;
        double r2058536 = r2058521 + r2058528;
        double r2058537 = r2058521 / r2058536;
        double r2058538 = r2058521 - r2058528;
        double r2058539 = r2058536 / r2058538;
        double r2058540 = r2058537 - r2058539;
        double r2058541 = exp(r2058540);
        double r2058542 = log(r2058541);
        double r2058543 = r2058535 ? r2058542 : r2058533;
        double r2058544 = r2058523 ? r2058533 : r2058543;
        return r2058544;
}

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 < -9516.22025087149 or 11259.474582447798 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -9516.22025087149 < x < 11259.474582447798

    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. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9516.22025087149:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 11259.474582447798:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))