Average Error: 29.1 → 0.1
Time: 14.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10757.1582330256224 \lor \neg \left(x \le 11108.9645686635795\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10757.1582330256224 \lor \neg \left(x \le 11108.9645686635795\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\

\end{array}
double f(double x) {
        double r209004 = x;
        double r209005 = 1.0;
        double r209006 = r209004 + r209005;
        double r209007 = r209004 / r209006;
        double r209008 = r209004 - r209005;
        double r209009 = r209006 / r209008;
        double r209010 = r209007 - r209009;
        return r209010;
}
↓
double f(double x) {
        double r209011 = x;
        double r209012 = -10757.158233025622;
        bool r209013 = r209011 <= r209012;
        double r209014 = 11108.96456866358;
        bool r209015 = r209011 <= r209014;
        double r209016 = !r209015;
        bool r209017 = r209013 || r209016;
        double r209018 = 1.0;
        double r209019 = -r209018;
        double r209020 = 2.0;
        double r209021 = pow(r209011, r209020);
        double r209022 = r209019 / r209021;
        double r209023 = 3.0;
        double r209024 = r209023 / r209011;
        double r209025 = r209022 - r209024;
        double r209026 = 3.0;
        double r209027 = pow(r209011, r209026);
        double r209028 = r209023 / r209027;
        double r209029 = r209025 - r209028;
        double r209030 = r209011 + r209018;
        double r209031 = r209011 / r209030;
        double r209032 = r209031 * r209031;
        double r209033 = r209011 - r209018;
        double r209034 = r209030 / r209033;
        double r209035 = r209034 * r209034;
        double r209036 = r209032 - r209035;
        double r209037 = r209031 + r209034;
        double r209038 = r209036 / r209037;
        double r209039 = r209017 ? r209029 : r209038;
        return r209039;
}

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 < -10757.158233025622 or 11108.96456866358 < x

    1. Initial program 59.2

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

    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 -10757.158233025622 < x < 11108.96456866358

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10757.1582330256224 \lor \neg \left(x \le 11108.9645686635795\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))