Average Error: 29.2 → 0.1
Time: 34.6s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15147.582990964242:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 10532.01597020836:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{x \cdot x - 1} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15147.582990964242:\\
\;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 10532.01597020836:\\
\;\;\;\;\left(x - 1\right) \cdot \frac{x}{x \cdot x - 1} - \frac{1 + x}{x - 1}\\

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

\end{array}
double f(double x) {
        double r15249602 = x;
        double r15249603 = 1.0;
        double r15249604 = r15249602 + r15249603;
        double r15249605 = r15249602 / r15249604;
        double r15249606 = r15249602 - r15249603;
        double r15249607 = r15249604 / r15249606;
        double r15249608 = r15249605 - r15249607;
        return r15249608;
}
↓
double f(double x) {
        double r15249609 = x;
        double r15249610 = -15147.582990964242;
        bool r15249611 = r15249609 <= r15249610;
        double r15249612 = -1.0;
        double r15249613 = r15249609 * r15249609;
        double r15249614 = r15249612 / r15249613;
        double r15249615 = -3.0;
        double r15249616 = r15249613 * r15249609;
        double r15249617 = r15249615 / r15249616;
        double r15249618 = r15249615 / r15249609;
        double r15249619 = r15249617 + r15249618;
        double r15249620 = r15249614 + r15249619;
        double r15249621 = 10532.01597020836;
        bool r15249622 = r15249609 <= r15249621;
        double r15249623 = 1.0;
        double r15249624 = r15249609 - r15249623;
        double r15249625 = r15249613 - r15249623;
        double r15249626 = r15249609 / r15249625;
        double r15249627 = r15249624 * r15249626;
        double r15249628 = r15249623 + r15249609;
        double r15249629 = r15249628 / r15249624;
        double r15249630 = r15249627 - r15249629;
        double r15249631 = r15249622 ? r15249630 : r15249620;
        double r15249632 = r15249611 ? r15249620 : r15249631;
        return r15249632;
}

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 < -15147.582990964242 or 10532.01597020836 < x

    1. Initial program 59.2

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

    3. Applied add-log-exp59.2

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.3

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

    6. Taylor expanded around 0 0.3

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

    7. Simplified0.0

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

    if -15147.582990964242 < x < 10532.01597020836

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15147.582990964242:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 10532.01597020836:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{x \cdot x - 1} - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))