Average Error: 14.3 → 0.3
Time: 12.9s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -247.4892321533008:\\ \;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{-2}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 240.61147870412702:\\ \;\;\;\;\frac{-1}{-1 + x} + \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x}}{x} + \frac{\frac{-2}{x \cdot x}}{x \cdot x}\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -247.4892321533008:\\
\;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{-2}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 240.61147870412702:\\
\;\;\;\;\frac{-1}{-1 + x} + \frac{1}{1 + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x}}{x} + \frac{\frac{-2}{x \cdot x}}{x \cdot x}\right)\\

\end{array}
double f(double x) {
        double r3754581 = 1.0;
        double r3754582 = x;
        double r3754583 = r3754582 + r3754581;
        double r3754584 = r3754581 / r3754583;
        double r3754585 = r3754582 - r3754581;
        double r3754586 = r3754581 / r3754585;
        double r3754587 = r3754584 - r3754586;
        return r3754587;
}
↓
double f(double x) {
        double r3754588 = x;
        double r3754589 = -247.4892321533008;
        bool r3754590 = r3754588 <= r3754589;
        double r3754591 = -2.0;
        double r3754592 = r3754588 * r3754588;
        double r3754593 = r3754591 / r3754592;
        double r3754594 = r3754593 / r3754592;
        double r3754595 = r3754594 / r3754592;
        double r3754596 = r3754594 + r3754593;
        double r3754597 = r3754595 + r3754596;
        double r3754598 = 240.61147870412702;
        bool r3754599 = r3754588 <= r3754598;
        double r3754600 = -1.0;
        double r3754601 = r3754600 + r3754588;
        double r3754602 = r3754600 / r3754601;
        double r3754603 = 1.0;
        double r3754604 = r3754603 + r3754588;
        double r3754605 = r3754603 / r3754604;
        double r3754606 = r3754602 + r3754605;
        double r3754607 = r3754591 / r3754588;
        double r3754608 = r3754607 / r3754588;
        double r3754609 = r3754608 + r3754594;
        double r3754610 = r3754595 + r3754609;
        double r3754611 = r3754599 ? r3754606 : r3754610;
        double r3754612 = r3754590 ? r3754597 : r3754611;
        return r3754612;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -247.4892321533008

    1. Initial program 28.5

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

    3. Applied add-cube-cbrt50.5

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

    4. Applied *-un-lft-identity50.5

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

    5. Applied times-frac52.1

      \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}}\]
    6. Using strategy rm

    7. Applied sub-neg52.1

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

    8. Simplified28.5

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

    9. Taylor expanded around inf 1.0

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    10. Simplified1.0

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

    if -247.4892321533008 < x < 240.61147870412702

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied times-frac0.1

      \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}}\]
    6. Using strategy rm

    7. Applied sub-neg0.1

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

    8. Simplified0.0

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

    if 240.61147870412702 < x

    1. Initial program 29.6

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

    3. Applied add-cube-cbrt50.7

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

    4. Applied *-un-lft-identity50.7

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

    5. Applied times-frac51.9

      \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}}\]
    6. Using strategy rm

    7. Applied sub-neg51.9

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

    8. Simplified29.6

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

    9. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    10. Simplified0.8

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

    11. Taylor expanded around inf 0.8

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

    12. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -247.4892321533008:\\ \;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{-2}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 240.61147870412702:\\ \;\;\;\;\frac{-1}{-1 + x} + \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{x \cdot x}}{x \cdot x}}{x \cdot x} + \left(\frac{\frac{-2}{x}}{x} + \frac{\frac{-2}{x \cdot x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))