Average Error: 29.5 → 0.5
Time: 6.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.327620599565707992795182690315414220095 \lor \neg \left(x \le 1.014919163926425227373329107649624347687\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(-\left(2 \cdot \left({x}^{2} + x\right) + 1\right)\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.327620599565707992795182690315414220095 \lor \neg \left(x \le 1.014919163926425227373329107649624347687\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \left(-\left(2 \cdot \left({x}^{2} + x\right) + 1\right)\right)\\

\end{array}
double f(double x) {
        double r164699 = x;
        double r164700 = 1.0;
        double r164701 = r164699 + r164700;
        double r164702 = r164699 / r164701;
        double r164703 = r164699 - r164700;
        double r164704 = r164701 / r164703;
        double r164705 = r164702 - r164704;
        return r164705;
}
↓
double f(double x) {
        double r164706 = x;
        double r164707 = -1.327620599565708;
        bool r164708 = r164706 <= r164707;
        double r164709 = 1.0149191639264252;
        bool r164710 = r164706 <= r164709;
        double r164711 = !r164710;
        bool r164712 = r164708 || r164711;
        double r164713 = 1.0;
        double r164714 = -r164713;
        double r164715 = 2.0;
        double r164716 = pow(r164706, r164715);
        double r164717 = r164714 / r164716;
        double r164718 = 3.0;
        double r164719 = r164718 / r164706;
        double r164720 = r164717 - r164719;
        double r164721 = 3.0;
        double r164722 = pow(r164706, r164721);
        double r164723 = r164718 / r164722;
        double r164724 = r164720 - r164723;
        double r164725 = r164706 + r164713;
        double r164726 = r164706 / r164725;
        double r164727 = 2.0;
        double r164728 = r164716 + r164706;
        double r164729 = r164727 * r164728;
        double r164730 = r164729 + r164713;
        double r164731 = -r164730;
        double r164732 = r164726 - r164731;
        double r164733 = r164712 ? r164724 : r164732;
        return r164733;
}

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 < -1.327620599565708 or 1.0149191639264252 < x

    1. Initial program 58.4

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.4

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

    if -1.327620599565708 < x < 1.0149191639264252

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.327620599565707992795182690315414220095 \lor \neg \left(x \le 1.014919163926425227373329107649624347687\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(-\left(2 \cdot \left({x}^{2} + x\right) + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))