Average Error: 29.3 → 0.1
Time: 18.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10645.547231578275 \lor \neg \left(x \le 9789.5565685403017\right):\\ \;\;\;\;-\left(\frac{3}{x} + \left(\frac{3}{{x}^{3}} + \frac{1}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) - \frac{x + 1}{x - 1}\right)}^{3}}\\ \end{array}\]
\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(\frac{3}{x} + \left(\frac{3}{{x}^{3}} + \frac{1}{x \cdot x}\right)\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 r169505 = x;
        double r169506 = 1.0;
        double r169507 = r169505 + r169506;
        double r169508 = r169505 / r169507;
        double r169509 = r169505 - r169506;
        double r169510 = r169507 / r169509;
        double r169511 = r169508 - r169510;
        return r169511;
}
↓
double f(double x) {
        double r169512 = x;
        double r169513 = -10645.547231578275;
        bool r169514 = r169512 <= r169513;
        double r169515 = 9789.556568540302;
        bool r169516 = r169512 <= r169515;
        double r169517 = !r169516;
        bool r169518 = r169514 || r169517;
        double r169519 = 3.0;
        double r169520 = r169519 / r169512;
        double r169521 = 3.0;
        double r169522 = pow(r169512, r169521);
        double r169523 = r169519 / r169522;
        double r169524 = 1.0;
        double r169525 = r169512 * r169512;
        double r169526 = r169524 / r169525;
        double r169527 = r169523 + r169526;
        double r169528 = r169520 + r169527;
        double r169529 = -r169528;
        double r169530 = pow(r169524, r169521);
        double r169531 = r169522 + r169530;
        double r169532 = r169512 / r169531;
        double r169533 = r169524 * r169524;
        double r169534 = r169512 * r169524;
        double r169535 = r169533 - r169534;
        double r169536 = r169525 + r169535;
        double r169537 = r169532 * r169536;
        double r169538 = r169512 + r169524;
        double r169539 = r169512 - r169524;
        double r169540 = r169538 / r169539;
        double r169541 = r169537 - r169540;
        double r169542 = pow(r169541, r169521);
        double r169543 = cbrt(r169542);
        double r169544 = r169518 ? r169529 : r169543;
        return r169544;
}

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

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

    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

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

Reproduce

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