Average Error: 28.9 → 0.4
Time: 12.4s
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 r3743804 = x;
        double r3743805 = 1.0;
        double r3743806 = r3743804 + r3743805;
        double r3743807 = r3743804 / r3743806;
        double r3743808 = r3743804 - r3743805;
        double r3743809 = r3743806 / r3743808;
        double r3743810 = r3743807 - r3743809;
        return r3743810;
}
↓
double f(double x) {
        double r3743811 = x;
        double r3743812 = -0.9983730585657322;
        bool r3743813 = r3743811 <= r3743812;
        double r3743814 = 3.0;
        double r3743815 = -r3743814;
        double r3743816 = r3743811 * r3743811;
        double r3743817 = r3743816 * r3743811;
        double r3743818 = r3743815 / r3743817;
        double r3743819 = 1.0;
        double r3743820 = r3743819 / r3743816;
        double r3743821 = r3743814 / r3743811;
        double r3743822 = r3743820 + r3743821;
        double r3743823 = r3743818 - r3743822;
        double r3743824 = 1.021370211112824;
        bool r3743825 = r3743811 <= r3743824;
        double r3743826 = r3743819 * r3743811;
        double r3743827 = r3743826 + r3743814;
        double r3743828 = r3743811 * r3743827;
        double r3743829 = r3743828 + r3743819;
        double r3743830 = r3743825 ? r3743829 : r3743823;
        double r3743831 = r3743813 ? r3743823 : r3743830;
        return r3743831;
}

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))))