Average Error: 29.3 → 0.0
Time: 4.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4423751.3748676982 \lor \neg \left(x \le 134430.98924424738\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4423751.3748676982 \lor \neg \left(x \le 134430.98924424738\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r135162 = x;
        double r135163 = 1.0;
        double r135164 = r135162 + r135163;
        double r135165 = r135162 / r135164;
        double r135166 = r135162 - r135163;
        double r135167 = r135164 / r135166;
        double r135168 = r135165 - r135167;
        return r135168;
}
↓
double f(double x) {
        double r135169 = x;
        double r135170 = -4423751.374867698;
        bool r135171 = r135169 <= r135170;
        double r135172 = 134430.98924424738;
        bool r135173 = r135169 <= r135172;
        double r135174 = !r135173;
        bool r135175 = r135171 || r135174;
        double r135176 = 1.0;
        double r135177 = -r135176;
        double r135178 = 2.0;
        double r135179 = pow(r135169, r135178);
        double r135180 = r135177 / r135179;
        double r135181 = 3.0;
        double r135182 = r135181 / r135169;
        double r135183 = r135180 - r135182;
        double r135184 = 3.0;
        double r135185 = pow(r135169, r135184);
        double r135186 = r135181 / r135185;
        double r135187 = r135183 - r135186;
        double r135188 = r135181 * r135169;
        double r135189 = r135188 + r135176;
        double r135190 = -r135189;
        double r135191 = r135169 * r135169;
        double r135192 = r135176 * r135176;
        double r135193 = r135191 - r135192;
        double r135194 = r135190 / r135193;
        double r135195 = r135175 ? r135187 : r135194;
        return r135195;
}

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 < -4423751.374867698 or 134430.98924424738 < x

    1. Initial program 59.7

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -4423751.374867698 < x < 134430.98924424738

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Simplified0.1

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

    5. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4423751.3748676982 \lor \neg \left(x \le 134430.98924424738\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))