Average Error: 29.3 → 0.1
Time: 7.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10645.547231578275 \lor \neg \left(x \le 9789.5565685403017\right):\\ \;\;\;\;-\left(\left(\frac{1}{{x}^{2}} + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) - \frac{x + 1}{x - 1}\right)}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10645.547231578275 \lor \neg \left(x \le 9789.5565685403017\right):\\
\;\;\;\;-\left(\left(\frac{1}{{x}^{2}} + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\\

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

\end{array}
double f(double x) {
        double r208686 = x;
        double r208687 = 1.0;
        double r208688 = r208686 + r208687;
        double r208689 = r208686 / r208688;
        double r208690 = r208686 - r208687;
        double r208691 = r208688 / r208690;
        double r208692 = r208689 - r208691;
        return r208692;
}
↓
double f(double x) {
        double r208693 = x;
        double r208694 = -10645.547231578275;
        bool r208695 = r208693 <= r208694;
        double r208696 = 9789.556568540302;
        bool r208697 = r208693 <= r208696;
        double r208698 = !r208697;
        bool r208699 = r208695 || r208698;
        double r208700 = 1.0;
        double r208701 = 2.0;
        double r208702 = pow(r208693, r208701);
        double r208703 = r208700 / r208702;
        double r208704 = 3.0;
        double r208705 = 3.0;
        double r208706 = pow(r208693, r208705);
        double r208707 = r208704 / r208706;
        double r208708 = r208703 + r208707;
        double r208709 = r208704 / r208693;
        double r208710 = r208708 + r208709;
        double r208711 = -r208710;
        double r208712 = pow(r208700, r208705);
        double r208713 = r208706 + r208712;
        double r208714 = r208693 / r208713;
        double r208715 = r208693 * r208693;
        double r208716 = r208700 * r208700;
        double r208717 = r208693 * r208700;
        double r208718 = r208716 - r208717;
        double r208719 = r208715 + r208718;
        double r208720 = r208714 * r208719;
        double r208721 = r208693 + r208700;
        double r208722 = r208693 - r208700;
        double r208723 = r208721 / r208722;
        double r208724 = r208720 - r208723;
        double r208725 = pow(r208724, r208705);
        double r208726 = cbrt(r208725);
        double r208727 = r208699 ? r208711 : r208726;
        return r208727;
}

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 < -10645.547231578275 or 9789.556568540302 < x

    1. Initial program 59.2

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

    3. Applied add-cbrt-cube59.2

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

    4. Simplified59.2

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

    6. Applied add-cube-cbrt59.2

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

    7. Simplified59.2

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

    8. Taylor expanded around inf 0.3

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

    9. Simplified0.0

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

    if -10645.547231578275 < x < 9789.556568540302

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    6. Applied flip3-+0.1

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

    7. Applied associate-/r/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10645.547231578275 \lor \neg \left(x \le 9789.5565685403017\right):\\ \;\;\;\;-\left(\left(\frac{1}{{x}^{2}} + \frac{3}{{x}^{3}}\right) + \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) - \frac{x + 1}{x - 1}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))