Average Error: 29.5 → 0.5
Time: 28.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\left(\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 1.02083405594772891333832376403734087944:\\ \;\;\;\;\left(1 \cdot x + 3\right) \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\
\;\;\;\;\left(\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 1.02083405594772891333832376403734087944:\\
\;\;\;\;\left(1 \cdot x + 3\right) \cdot x + 1\\

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

\end{array}
double f(double x) {
        double r6023586 = x;
        double r6023587 = 1.0;
        double r6023588 = r6023586 + r6023587;
        double r6023589 = r6023586 / r6023588;
        double r6023590 = r6023586 - r6023587;
        double r6023591 = r6023588 / r6023590;
        double r6023592 = r6023589 - r6023591;
        return r6023592;
}
↓
double f(double x) {
        double r6023593 = x;
        double r6023594 = -1.0026376323582022;
        bool r6023595 = r6023593 <= r6023594;
        double r6023596 = 1.0;
        double r6023597 = r6023593 * r6023593;
        double r6023598 = r6023596 / r6023597;
        double r6023599 = -r6023598;
        double r6023600 = 3.0;
        double r6023601 = r6023600 / r6023593;
        double r6023602 = r6023599 - r6023601;
        double r6023603 = r6023601 / r6023597;
        double r6023604 = r6023602 - r6023603;
        double r6023605 = 1.020834055947729;
        bool r6023606 = r6023593 <= r6023605;
        double r6023607 = r6023596 * r6023593;
        double r6023608 = r6023607 + r6023600;
        double r6023609 = r6023608 * r6023593;
        double r6023610 = r6023609 + r6023596;
        double r6023611 = r6023606 ? r6023610 : r6023604;
        double r6023612 = r6023595 ? r6023604 : r6023611;
        return r6023612;
}

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 < -1.0026376323582022 or 1.020834055947729 < x

    1. Initial program 58.6

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.4

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

    if -1.0026376323582022 < x < 1.020834055947729

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Taylor expanded around 0 0.6

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

    5. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002637632358202246507516974816098809242:\\ \;\;\;\;\left(\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 1.02083405594772891333832376403734087944:\\ \;\;\;\;\left(1 \cdot x + 3\right) \cdot x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))