Average Error: 29.2 → 0.1
Time: 51.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13212.25074346146:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13212.25074346146:\\
\;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 9551.679161702108:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r13659313 = x;
        double r13659314 = 1.0;
        double r13659315 = r13659313 + r13659314;
        double r13659316 = r13659313 / r13659315;
        double r13659317 = r13659313 - r13659314;
        double r13659318 = r13659315 / r13659317;
        double r13659319 = r13659316 - r13659318;
        return r13659319;
}
↓
double f(double x) {
        double r13659320 = x;
        double r13659321 = -13212.25074346146;
        bool r13659322 = r13659320 <= r13659321;
        double r13659323 = -3.0;
        double r13659324 = r13659323 / r13659320;
        double r13659325 = r13659320 * r13659320;
        double r13659326 = r13659324 / r13659325;
        double r13659327 = 1.0;
        double r13659328 = r13659327 / r13659325;
        double r13659329 = r13659328 - r13659324;
        double r13659330 = r13659326 - r13659329;
        double r13659331 = 9551.679161702108;
        bool r13659332 = r13659320 <= r13659331;
        double r13659333 = r13659320 + r13659327;
        double r13659334 = cbrt(r13659333);
        double r13659335 = r13659334 * r13659334;
        double r13659336 = r13659320 / r13659335;
        double r13659337 = r13659336 / r13659334;
        double r13659338 = r13659320 - r13659327;
        double r13659339 = r13659333 / r13659338;
        double r13659340 = r13659337 - r13659339;
        double r13659341 = r13659332 ? r13659340 : r13659330;
        double r13659342 = r13659322 ? r13659330 : r13659341;
        return r13659342;
}

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 < -13212.25074346146 or 9551.679161702108 < x

    1. Initial program 59.4

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

    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

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

    if -13212.25074346146 < x < 9551.679161702108

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13212.25074346146:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))