Average Error: 29.3 → 0.2
Time: 5.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -92822810.013569295:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{2}{x}}{x}\\ \mathbf{elif}\;x \le 12712.5671000054735:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -92822810.013569295:\\
\;\;\;\;\frac{-3}{x} - \frac{\frac{2}{x}}{x}\\

\mathbf{elif}\;x \le 12712.5671000054735:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r128623 = x;
        double r128624 = 1.0;
        double r128625 = r128623 + r128624;
        double r128626 = r128623 / r128625;
        double r128627 = r128623 - r128624;
        double r128628 = r128625 / r128627;
        double r128629 = r128626 - r128628;
        return r128629;
}
↓
double f(double x) {
        double r128630 = x;
        double r128631 = -92822810.0135693;
        bool r128632 = r128630 <= r128631;
        double r128633 = 3.0;
        double r128634 = -r128633;
        double r128635 = r128634 / r128630;
        double r128636 = 2.0;
        double r128637 = r128636 / r128630;
        double r128638 = r128637 / r128630;
        double r128639 = r128635 - r128638;
        double r128640 = 12712.567100005474;
        bool r128641 = r128630 <= r128640;
        double r128642 = cbrt(r128630);
        double r128643 = r128642 * r128642;
        double r128644 = 1.0;
        double r128645 = r128643 / r128644;
        double r128646 = 1.0;
        double r128647 = r128630 + r128646;
        double r128648 = r128642 / r128647;
        double r128649 = r128630 - r128646;
        double r128650 = r128647 / r128649;
        double r128651 = -r128650;
        double r128652 = fma(r128645, r128648, r128651);
        double r128653 = -r128646;
        double r128654 = 2.0;
        double r128655 = pow(r128630, r128654);
        double r128656 = r128653 / r128655;
        double r128657 = r128644 / r128630;
        double r128658 = 3.0;
        double r128659 = pow(r128630, r128658);
        double r128660 = r128644 / r128659;
        double r128661 = r128633 * r128660;
        double r128662 = fma(r128633, r128657, r128661);
        double r128663 = r128656 - r128662;
        double r128664 = r128641 ? r128652 : r128663;
        double r128665 = r128632 ? r128639 : r128664;
        return r128665;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -92822810.0135693

    1. Initial program 59.7

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

    3. Applied *-un-lft-identity59.7

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

    4. Applied add-cube-cbrt60.4

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

    5. Applied times-frac60.5

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

    6. Applied fma-neg60.6

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

    7. Taylor expanded around inf 0.6

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

    8. Simplified0.3

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{2}{x}}{x}}\]

    if -92822810.0135693 < x < 12712.567100005474

    1. Initial program 0.1

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

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

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

    4. Applied add-cube-cbrt0.2

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

    5. Applied times-frac0.2

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

    6. Applied fma-neg0.2

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

    if 12712.567100005474 < x

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -92822810.013569295:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{2}{x}}{x}\\ \mathbf{elif}\;x \le 12712.5671000054735:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))