Average Error: 29.2 → 0.0
Time: 36.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12548923.376107145:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 4977255.094115384:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12548923.376107145:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 4977255.094115384:\\
\;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(1 + x\right)}\\

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

\end{array}
double f(double x) {
        double r4474622 = x;
        double r4474623 = 1.0;
        double r4474624 = r4474622 + r4474623;
        double r4474625 = r4474622 / r4474624;
        double r4474626 = r4474622 - r4474623;
        double r4474627 = r4474624 / r4474626;
        double r4474628 = r4474625 - r4474627;
        return r4474628;
}
↓
double f(double x) {
        double r4474629 = x;
        double r4474630 = -12548923.376107145;
        bool r4474631 = r4474629 <= r4474630;
        double r4474632 = -1.0;
        double r4474633 = r4474629 * r4474629;
        double r4474634 = r4474632 / r4474633;
        double r4474635 = 3.0;
        double r4474636 = r4474635 / r4474629;
        double r4474637 = r4474634 - r4474636;
        double r4474638 = r4474636 / r4474633;
        double r4474639 = r4474637 - r4474638;
        double r4474640 = 4977255.094115384;
        bool r4474641 = r4474629 <= r4474640;
        double r4474642 = -3.0;
        double r4474643 = r4474642 * r4474629;
        double r4474644 = r4474643 + r4474632;
        double r4474645 = 1.0;
        double r4474646 = r4474629 - r4474645;
        double r4474647 = r4474645 + r4474629;
        double r4474648 = r4474646 * r4474647;
        double r4474649 = r4474644 / r4474648;
        double r4474650 = r4474641 ? r4474649 : r4474639;
        double r4474651 = r4474631 ? r4474639 : r4474650;
        return r4474651;
}

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 < -12548923.376107145 or 4977255.094115384 < x

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -12548923.376107145 < x < 4977255.094115384

    1. Initial program 0.2

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

    3. Applied frac-sub0.2

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12548923.376107145:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 4977255.094115384:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))