Average Error: 29.9 → 0.0
Time: 3.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\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 r137565 = x;
        double r137566 = 1.0;
        double r137567 = r137565 + r137566;
        double r137568 = r137565 / r137567;
        double r137569 = r137565 - r137566;
        double r137570 = r137567 / r137569;
        double r137571 = r137568 - r137570;
        return r137571;
}
↓
double f(double x) {
        double r137572 = x;
        double r137573 = -3547768106768.249;
        bool r137574 = r137572 <= r137573;
        double r137575 = 7926639408581.21;
        bool r137576 = r137572 <= r137575;
        double r137577 = !r137576;
        bool r137578 = r137574 || r137577;
        double r137579 = 1.0;
        double r137580 = -r137579;
        double r137581 = 2.0;
        double r137582 = pow(r137572, r137581);
        double r137583 = r137580 / r137582;
        double r137584 = 3.0;
        double r137585 = r137584 / r137572;
        double r137586 = r137583 - r137585;
        double r137587 = 3.0;
        double r137588 = pow(r137572, r137587);
        double r137589 = r137584 / r137588;
        double r137590 = r137586 - r137589;
        double r137591 = r137584 * r137572;
        double r137592 = r137591 + r137579;
        double r137593 = -r137592;
        double r137594 = r137572 * r137572;
        double r137595 = r137579 * r137579;
        double r137596 = r137594 - r137595;
        double r137597 = r137593 / r137596;
        double r137598 = r137578 ? r137590 : r137597;
        return r137598;
}

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 < -3547768106768.249 or 7926639408581.21 < x

    1. Initial program 60.3

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

    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 -3547768106768.249 < x < 7926639408581.21

    1. Initial program 0.8

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

    3. Applied frac-sub0.8

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

    4. Simplified0.8

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))