Average Error: 29.4 → 0.1
Time: 6.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\

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

\end{array}
double f(double x) {
        double r97731 = x;
        double r97732 = 1.0;
        double r97733 = r97731 + r97732;
        double r97734 = r97731 / r97733;
        double r97735 = r97731 - r97732;
        double r97736 = r97733 / r97735;
        double r97737 = r97734 - r97736;
        return r97737;
}
↓
double f(double x) {
        double r97738 = x;
        double r97739 = -11931.41196902016;
        bool r97740 = r97738 <= r97739;
        double r97741 = 12679.85899306799;
        bool r97742 = r97738 <= r97741;
        double r97743 = !r97742;
        bool r97744 = r97740 || r97743;
        double r97745 = 1.0;
        double r97746 = -r97745;
        double r97747 = r97738 * r97738;
        double r97748 = r97746 / r97747;
        double r97749 = 3.0;
        double r97750 = 3.0;
        double r97751 = pow(r97738, r97750);
        double r97752 = r97749 / r97751;
        double r97753 = r97749 / r97738;
        double r97754 = r97752 + r97753;
        double r97755 = r97748 - r97754;
        double r97756 = r97738 - r97745;
        double r97757 = r97756 * r97756;
        double r97758 = r97747 * r97757;
        double r97759 = r97738 + r97745;
        double r97760 = r97759 * r97759;
        double r97761 = r97760 * r97760;
        double r97762 = r97758 - r97761;
        double r97763 = r97738 / r97759;
        double r97764 = r97759 / r97756;
        double r97765 = r97763 + r97764;
        double r97766 = r97760 * r97757;
        double r97767 = r97765 * r97766;
        double r97768 = r97762 / r97767;
        double r97769 = r97744 ? r97755 : r97768;
        return r97769;
}

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 < -11931.41196902016 or 12679.85899306799 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -11931.41196902016 < x < 12679.85899306799

    1. Initial program 0.1

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

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied frac-times0.1

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

    6. Applied frac-times0.1

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

    7. Applied frac-sub0.1

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

    8. Applied associate-/l/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{{x}^{3}} + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))