Average Error: 28.9 → 0.4
Time: 12.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\ \;\;\;\;x \cdot \left(1 \cdot x + 3\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\
\;\;\;\;x \cdot \left(1 \cdot x + 3\right) + 1\\

\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 r5211737 = x;
        double r5211738 = 1.0;
        double r5211739 = r5211737 + r5211738;
        double r5211740 = r5211737 / r5211739;
        double r5211741 = r5211737 - r5211738;
        double r5211742 = r5211739 / r5211741;
        double r5211743 = r5211740 - r5211742;
        return r5211743;
}
↓
double f(double x) {
        double r5211744 = x;
        double r5211745 = -0.9983730585657322;
        bool r5211746 = r5211744 <= r5211745;
        double r5211747 = 3.0;
        double r5211748 = -r5211747;
        double r5211749 = r5211744 * r5211744;
        double r5211750 = r5211749 * r5211744;
        double r5211751 = r5211748 / r5211750;
        double r5211752 = 1.0;
        double r5211753 = r5211752 / r5211749;
        double r5211754 = r5211747 / r5211744;
        double r5211755 = r5211753 + r5211754;
        double r5211756 = r5211751 - r5211755;
        double r5211757 = 1.021370211112824;
        bool r5211758 = r5211744 <= r5211757;
        double r5211759 = r5211752 * r5211744;
        double r5211760 = r5211759 + r5211747;
        double r5211761 = r5211744 * r5211760;
        double r5211762 = r5211761 + r5211752;
        double r5211763 = r5211758 ? r5211762 : r5211756;
        double r5211764 = r5211746 ? r5211756 : r5211763;
        return r5211764;
}

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 < -0.9983730585657322 or 1.021370211112824 < 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

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

    if -0.9983730585657322 < x < 1.021370211112824

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\ \;\;\;\;x \cdot \left(1 \cdot x + 3\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))