Average Error: 29.4 → 0.1
Time: 7.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x} \cdot \left(\frac{1}{x \cdot x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\
\;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x} \cdot \left(\frac{1}{x \cdot x} + 1\right)\\

\mathbf{else}:\\
\;\;\;\;\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)}\\

\end{array}
double f(double x) {
        double r133472 = x;
        double r133473 = 1.0;
        double r133474 = r133472 + r133473;
        double r133475 = r133472 / r133474;
        double r133476 = r133472 - r133473;
        double r133477 = r133474 / r133476;
        double r133478 = r133475 - r133477;
        return r133478;
}
↓
double f(double x) {
        double r133479 = x;
        double r133480 = -12787.016553276117;
        bool r133481 = r133479 <= r133480;
        double r133482 = 9971.293289135534;
        bool r133483 = r133479 <= r133482;
        double r133484 = !r133483;
        bool r133485 = r133481 || r133484;
        double r133486 = 1.0;
        double r133487 = r133479 * r133479;
        double r133488 = r133486 / r133487;
        double r133489 = -r133488;
        double r133490 = 3.0;
        double r133491 = r133490 / r133479;
        double r133492 = 1.0;
        double r133493 = r133492 / r133487;
        double r133494 = r133493 + r133492;
        double r133495 = r133491 * r133494;
        double r133496 = r133489 - r133495;
        double r133497 = r133479 - r133486;
        double r133498 = r133479 * r133497;
        double r133499 = r133479 + r133486;
        double r133500 = r133499 * r133499;
        double r133501 = r133498 - r133500;
        double r133502 = r133497 * r133499;
        double r133503 = r133501 / r133502;
        double r133504 = r133485 ? r133496 : r133503;
        return r133504;
}

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 < -12787.016553276117 or 9971.293289135534 < x

    1. Initial program 59.3

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

    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

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

    5. Applied *-un-lft-identity0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied times-frac0.0

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

    8. Applied unpow30.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied times-frac0.0

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

    11. Applied distribute-rgt-out0.0

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

    12. Simplified0.0

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

    if -12787.016553276117 < x < 9971.293289135534

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \frac{3}{x} \cdot \left(\frac{1}{x \cdot x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))