Average Error: 28.8 → 0.1
Time: 32.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13931.998939596693:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{\frac{3}{x \cdot x}}{x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 12121.984188186829:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + {x}^{3}}{\left(1 + \left(x \cdot x - x\right)\right) \cdot \left(-1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{\frac{3}{x \cdot x}}{x}\right) - \frac{3}{x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13931.998939596693:\\
\;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{\frac{3}{x \cdot x}}{x}\right) - \frac{3}{x}\\

\mathbf{elif}\;x \le 12121.984188186829:\\
\;\;\;\;\frac{x}{1 + x} - \frac{1 + {x}^{3}}{\left(1 + \left(x \cdot x - x\right)\right) \cdot \left(-1 + x\right)}\\

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

\end{array}
double f(double x) {
        double r4380514 = x;
        double r4380515 = 1.0;
        double r4380516 = r4380514 + r4380515;
        double r4380517 = r4380514 / r4380516;
        double r4380518 = r4380514 - r4380515;
        double r4380519 = r4380516 / r4380518;
        double r4380520 = r4380517 - r4380519;
        return r4380520;
}
↓
double f(double x) {
        double r4380521 = x;
        double r4380522 = -13931.998939596693;
        bool r4380523 = r4380521 <= r4380522;
        double r4380524 = -1.0;
        double r4380525 = r4380524 / r4380521;
        double r4380526 = r4380525 / r4380521;
        double r4380527 = 3.0;
        double r4380528 = r4380521 * r4380521;
        double r4380529 = r4380527 / r4380528;
        double r4380530 = r4380529 / r4380521;
        double r4380531 = r4380526 - r4380530;
        double r4380532 = r4380527 / r4380521;
        double r4380533 = r4380531 - r4380532;
        double r4380534 = 12121.984188186829;
        bool r4380535 = r4380521 <= r4380534;
        double r4380536 = 1.0;
        double r4380537 = r4380536 + r4380521;
        double r4380538 = r4380521 / r4380537;
        double r4380539 = pow(r4380521, r4380527);
        double r4380540 = r4380536 + r4380539;
        double r4380541 = r4380528 - r4380521;
        double r4380542 = r4380536 + r4380541;
        double r4380543 = r4380524 + r4380521;
        double r4380544 = r4380542 * r4380543;
        double r4380545 = r4380540 / r4380544;
        double r4380546 = r4380538 - r4380545;
        double r4380547 = r4380535 ? r4380546 : r4380533;
        double r4380548 = r4380523 ? r4380533 : r4380547;
        return r4380548;
}

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 < -13931.998939596693 or 12121.984188186829 < x

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.0

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.0

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

    if -13931.998939596693 < x < 12121.984188186829

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/l/0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13931.998939596693:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{\frac{3}{x \cdot x}}{x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 12121.984188186829:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + {x}^{3}}{\left(1 + \left(x \cdot x - x\right)\right) \cdot \left(-1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{\frac{3}{x \cdot x}}{x}\right) - \frac{3}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))