Average Error: 29.2 → 0.0
Time: 14.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6041717527243.163:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 180005.41283506868:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6041717527243.163:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 180005.41283506868:\\
\;\;\;\;\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 r4583636 = x;
        double r4583637 = 1.0;
        double r4583638 = r4583636 + r4583637;
        double r4583639 = r4583636 / r4583638;
        double r4583640 = r4583636 - r4583637;
        double r4583641 = r4583638 / r4583640;
        double r4583642 = r4583639 - r4583641;
        return r4583642;
}
↓
double f(double x) {
        double r4583643 = x;
        double r4583644 = -6041717527243.163;
        bool r4583645 = r4583643 <= r4583644;
        double r4583646 = -1.0;
        double r4583647 = r4583643 * r4583643;
        double r4583648 = r4583646 / r4583647;
        double r4583649 = 3.0;
        double r4583650 = r4583649 / r4583643;
        double r4583651 = r4583648 - r4583650;
        double r4583652 = r4583650 / r4583647;
        double r4583653 = r4583651 - r4583652;
        double r4583654 = 180005.41283506868;
        bool r4583655 = r4583643 <= r4583654;
        double r4583656 = -3.0;
        double r4583657 = r4583656 * r4583643;
        double r4583658 = r4583657 + r4583646;
        double r4583659 = 1.0;
        double r4583660 = r4583643 - r4583659;
        double r4583661 = r4583659 + r4583643;
        double r4583662 = r4583660 * r4583661;
        double r4583663 = r4583658 / r4583662;
        double r4583664 = r4583655 ? r4583663 : r4583653;
        double r4583665 = r4583645 ? r4583653 : r4583664;
        return r4583665;
}

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 < -6041717527243.163 or 180005.41283506868 < x

    1. Initial program 59.9

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -6041717527243.163 < x < 180005.41283506868

    1. Initial program 0.5

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

    3. Applied frac-sub0.5

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

    4. Taylor expanded around inf 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6041717527243.163:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 180005.41283506868:\\ \;\;\;\;\frac{-3 \cdot x + -1}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))