Average Error: 29.4 → 0.4
Time: 18.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.397680696053489324270913130021654069424:\\ \;\;\;\;1 \cdot \left(\left(\left(x \cdot x\right) \cdot x + x\right) - x \cdot x\right) - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\
\;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 1.397680696053489324270913130021654069424:\\
\;\;\;\;1 \cdot \left(\left(\left(x \cdot x\right) \cdot x + x\right) - x \cdot x\right) - \frac{x + 1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r4026641 = x;
        double r4026642 = 1.0;
        double r4026643 = r4026641 + r4026642;
        double r4026644 = r4026641 / r4026643;
        double r4026645 = r4026641 - r4026642;
        double r4026646 = r4026643 / r4026645;
        double r4026647 = r4026644 - r4026646;
        return r4026647;
}
↓
double f(double x) {
        double r4026648 = x;
        double r4026649 = -1.0098163723001696;
        bool r4026650 = r4026648 <= r4026649;
        double r4026651 = 3.0;
        double r4026652 = r4026648 * r4026648;
        double r4026653 = r4026652 * r4026648;
        double r4026654 = r4026651 / r4026653;
        double r4026655 = -r4026654;
        double r4026656 = 1.0;
        double r4026657 = r4026656 / r4026652;
        double r4026658 = r4026651 / r4026648;
        double r4026659 = r4026657 + r4026658;
        double r4026660 = r4026655 - r4026659;
        double r4026661 = 1.3976806960534893;
        bool r4026662 = r4026648 <= r4026661;
        double r4026663 = r4026653 + r4026648;
        double r4026664 = r4026663 - r4026652;
        double r4026665 = r4026656 * r4026664;
        double r4026666 = r4026648 + r4026656;
        double r4026667 = r4026648 - r4026656;
        double r4026668 = r4026666 / r4026667;
        double r4026669 = r4026665 - r4026668;
        double r4026670 = r4026662 ? r4026669 : r4026660;
        double r4026671 = r4026650 ? r4026660 : r4026670;
        return r4026671;
}

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 < -1.0098163723001696 or 1.3976806960534893 < x

    1. Initial program 58.6

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.4

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

    if -1.0098163723001696 < x < 1.3976806960534893

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.009816372300169629028232520795427262783:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.397680696053489324270913130021654069424:\\ \;\;\;\;1 \cdot \left(\left(\left(x \cdot x\right) \cdot x + x\right) - x \cdot x\right) - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{1}{x \cdot x} + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))