Average Error: 29.3 → 0.1
Time: 37.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8404.501425772927:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 6607.223075937772:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8404.501425772927:\\
\;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 6607.223075937772:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}} - \frac{1 + x}{x - 1}\\

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

\end{array}
double f(double x) {
        double r4644531 = x;
        double r4644532 = 1.0;
        double r4644533 = r4644531 + r4644532;
        double r4644534 = r4644531 / r4644533;
        double r4644535 = r4644531 - r4644532;
        double r4644536 = r4644533 / r4644535;
        double r4644537 = r4644534 - r4644536;
        return r4644537;
}
↓
double f(double x) {
        double r4644538 = x;
        double r4644539 = -8404.501425772927;
        bool r4644540 = r4644538 <= r4644539;
        double r4644541 = -3.0;
        double r4644542 = r4644541 / r4644538;
        double r4644543 = -1.0;
        double r4644544 = r4644538 * r4644538;
        double r4644545 = r4644543 / r4644544;
        double r4644546 = r4644542 / r4644544;
        double r4644547 = r4644545 + r4644546;
        double r4644548 = r4644542 + r4644547;
        double r4644549 = 6607.223075937772;
        bool r4644550 = r4644538 <= r4644549;
        double r4644551 = 1.0;
        double r4644552 = r4644551 + r4644538;
        double r4644553 = cbrt(r4644552);
        double r4644554 = r4644553 * r4644553;
        double r4644555 = r4644538 / r4644554;
        double r4644556 = r4644555 / r4644553;
        double r4644557 = r4644538 - r4644551;
        double r4644558 = r4644552 / r4644557;
        double r4644559 = r4644556 - r4644558;
        double r4644560 = r4644550 ? r4644559 : r4644548;
        double r4644561 = r4644540 ? r4644548 : r4644560;
        return r4644561;
}

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 < -8404.501425772927 or 6607.223075937772 < x

    1. Initial program 59.1

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

    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

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

    4. Taylor expanded around -inf 0.3

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

    5. Simplified0.0

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

    if -8404.501425772927 < x < 6607.223075937772

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8404.501425772927:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 6607.223075937772:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))