Average Error: 29.2 → 0.0
Time: 11.6s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1816146799751588 \lor \neg \left(x \le 299480.83182101510465145111083984375\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}{x \cdot x - 1 \cdot 1} - 1 \cdot \frac{1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1816146799751588 \lor \neg \left(x \le 299480.83182101510465145111083984375\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r103746 = x;
        double r103747 = 1.0;
        double r103748 = r103746 + r103747;
        double r103749 = r103746 / r103748;
        double r103750 = r103746 - r103747;
        double r103751 = r103748 / r103750;
        double r103752 = r103749 - r103751;
        return r103752;
}
↓
double f(double x) {
        double r103753 = x;
        double r103754 = -1816146799751588.0;
        bool r103755 = r103753 <= r103754;
        double r103756 = 299480.8318210151;
        bool r103757 = r103753 <= r103756;
        double r103758 = !r103757;
        bool r103759 = r103755 || r103758;
        double r103760 = 1.0;
        double r103761 = -r103760;
        double r103762 = 2.0;
        double r103763 = pow(r103753, r103762);
        double r103764 = r103761 / r103763;
        double r103765 = 3.0;
        double r103766 = r103765 / r103753;
        double r103767 = r103764 - r103766;
        double r103768 = 3.0;
        double r103769 = pow(r103753, r103768);
        double r103770 = r103765 / r103769;
        double r103771 = r103767 - r103770;
        double r103772 = r103753 - r103760;
        double r103773 = r103753 + r103760;
        double r103774 = r103772 - r103773;
        double r103775 = r103753 * r103774;
        double r103776 = r103753 * r103753;
        double r103777 = r103760 * r103760;
        double r103778 = r103776 - r103777;
        double r103779 = r103775 / r103778;
        double r103780 = 1.0;
        double r103781 = r103780 / r103772;
        double r103782 = r103760 * r103781;
        double r103783 = r103779 - r103782;
        double r103784 = r103759 ? r103771 : r103783;
        return r103784;
}

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 < -1816146799751588.0 or 299480.8318210151 < x

    1. Initial program 59.9

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

    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

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

    if -1816146799751588.0 < x < 299480.8318210151

    1. Initial program 0.6

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

    3. Applied flip--0.6

      \[\leadsto \frac{x}{x + 1} - \frac{x + 1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

    4. Applied associate-/r/0.6

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

    5. Simplified0.6

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{x - 1}} \cdot \left(x + 1\right)\]
    6. Using strategy rm

    7. Applied distribute-rgt-in0.6

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

    8. Applied associate--r+0.6

      \[\leadsto \color{blue}{\left(\frac{x}{x + 1} - x \cdot \frac{1}{x - 1}\right) - 1 \cdot \frac{1}{x - 1}}\]
    9. Using strategy rm

    10. Applied un-div-inv0.6

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

    11. Applied frac-sub0.6

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

    12. Simplified0.0

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

    13. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1816146799751588 \lor \neg \left(x \le 299480.83182101510465145111083984375\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}{x \cdot x - 1 \cdot 1} - 1 \cdot \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))