Average Error: 29.3 → 0.0
Time: 15.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -350706.4836362626:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right) + \frac{-3}{x}\\ \mathbf{elif}\;x \le 176552.52851234598:\\ \;\;\;\;\left(\left(x + 1\right) \cdot \left(1 - x\right)\right) \cdot \frac{\left(x \cdot 2 + 1\right) - 3 \cdot \left(x \cdot x\right)}{\left(-1 + x \cdot x\right) \cdot \left(\left(-1 + x\right) \cdot \left(1 - x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right) + \frac{-3}{x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -350706.4836362626:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right) + \frac{-3}{x}\\

\mathbf{elif}\;x \le 176552.52851234598:\\
\;\;\;\;\left(\left(x + 1\right) \cdot \left(1 - x\right)\right) \cdot \frac{\left(x \cdot 2 + 1\right) - 3 \cdot \left(x \cdot x\right)}{\left(-1 + x \cdot x\right) \cdot \left(\left(-1 + x\right) \cdot \left(1 - x \cdot x\right)\right)}\\

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

\end{array}
double f(double x) {
        double r5794537 = x;
        double r5794538 = 1.0;
        double r5794539 = r5794537 + r5794538;
        double r5794540 = r5794537 / r5794539;
        double r5794541 = r5794537 - r5794538;
        double r5794542 = r5794539 / r5794541;
        double r5794543 = r5794540 - r5794542;
        return r5794543;
}
↓
double f(double x) {
        double r5794544 = x;
        double r5794545 = -350706.4836362626;
        bool r5794546 = r5794544 <= r5794545;
        double r5794547 = -1.0;
        double r5794548 = r5794544 * r5794544;
        double r5794549 = r5794547 / r5794548;
        double r5794550 = -3.0;
        double r5794551 = r5794550 / r5794544;
        double r5794552 = r5794551 / r5794548;
        double r5794553 = r5794549 + r5794552;
        double r5794554 = r5794553 + r5794551;
        double r5794555 = 176552.52851234598;
        bool r5794556 = r5794544 <= r5794555;
        double r5794557 = 1.0;
        double r5794558 = r5794544 + r5794557;
        double r5794559 = r5794557 - r5794544;
        double r5794560 = r5794558 * r5794559;
        double r5794561 = 2.0;
        double r5794562 = r5794544 * r5794561;
        double r5794563 = r5794562 + r5794557;
        double r5794564 = 3.0;
        double r5794565 = r5794564 * r5794548;
        double r5794566 = r5794563 - r5794565;
        double r5794567 = r5794547 + r5794548;
        double r5794568 = r5794547 + r5794544;
        double r5794569 = r5794557 - r5794548;
        double r5794570 = r5794568 * r5794569;
        double r5794571 = r5794567 * r5794570;
        double r5794572 = r5794566 / r5794571;
        double r5794573 = r5794560 * r5794572;
        double r5794574 = r5794556 ? r5794573 : r5794554;
        double r5794575 = r5794546 ? r5794554 : r5794574;
        return r5794575;
}

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 < -350706.4836362626 or 176552.52851234598 < x

    1. Initial program 59.6

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

    3. Applied div-inv59.7

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

    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

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

    if -350706.4836362626 < x < 176552.52851234598

    1. Initial program 0.2

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

    3. Applied div-inv0.2

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

    5. Applied flip-+0.2

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}} \cdot \frac{1}{x - 1}\]

    6. Applied frac-times0.2

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

    7. Applied frac-sub0.2

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

    8. Simplified0.2

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

    9. Simplified0.2

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

    10. Taylor expanded around 0 0.1

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

    11. Simplified0.1

      \[\leadsto \frac{\color{blue}{x \cdot 2 + \left(1 - \left(x \cdot x\right) \cdot 3\right)}}{\left(1 + x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}\]
    12. Using strategy rm

    13. Applied flip--0.0

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

    14. Applied associate-*r/0.0

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

    15. Applied flip-+0.0

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

    16. Applied frac-times0.0

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

    17. Applied associate-/r/0.0

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

    18. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -350706.4836362626:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right) + \frac{-3}{x}\\ \mathbf{elif}\;x \le 176552.52851234598:\\ \;\;\;\;\left(\left(x + 1\right) \cdot \left(1 - x\right)\right) \cdot \frac{\left(x \cdot 2 + 1\right) - 3 \cdot \left(x \cdot x\right)}{\left(-1 + x \cdot x\right) \cdot \left(\left(-1 + x\right) \cdot \left(1 - x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{\frac{-3}{x}}{x \cdot x}\right) + \frac{-3}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))