Average Error: 28.6 → 0.1
Time: 20.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11906.396240974655:\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \mathsf{fma}\left(\frac{3}{x}, \frac{1}{x \cdot x}, \frac{3}{x}\right)\right)\\ \mathbf{elif}\;x \le 10135.341772312217:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{-1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \mathsf{fma}\left(\frac{3}{x}, \frac{1}{x \cdot x}, \frac{3}{x}\right)\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11906.396240974655:\\
\;\;\;\;-\left(\frac{1}{x \cdot x} + \mathsf{fma}\left(\frac{3}{x}, \frac{1}{x \cdot x}, \frac{3}{x}\right)\right)\\

\mathbf{elif}\;x \le 10135.341772312217:\\
\;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{-1 + x}}\right)\\

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

\end{array}
double f(double x) {
        double r5319732 = x;
        double r5319733 = 1.0;
        double r5319734 = r5319732 + r5319733;
        double r5319735 = r5319732 / r5319734;
        double r5319736 = r5319732 - r5319733;
        double r5319737 = r5319734 / r5319736;
        double r5319738 = r5319735 - r5319737;
        return r5319738;
}
↓
double f(double x) {
        double r5319739 = x;
        double r5319740 = -11906.396240974655;
        bool r5319741 = r5319739 <= r5319740;
        double r5319742 = 1.0;
        double r5319743 = r5319739 * r5319739;
        double r5319744 = r5319742 / r5319743;
        double r5319745 = 3.0;
        double r5319746 = r5319745 / r5319739;
        double r5319747 = fma(r5319746, r5319744, r5319746);
        double r5319748 = r5319744 + r5319747;
        double r5319749 = -r5319748;
        double r5319750 = 10135.341772312217;
        bool r5319751 = r5319739 <= r5319750;
        double r5319752 = r5319742 + r5319739;
        double r5319753 = r5319739 / r5319752;
        double r5319754 = -1.0;
        double r5319755 = r5319754 + r5319739;
        double r5319756 = r5319752 / r5319755;
        double r5319757 = r5319753 - r5319756;
        double r5319758 = exp(r5319757);
        double r5319759 = log(r5319758);
        double r5319760 = r5319751 ? r5319759 : r5319749;
        double r5319761 = r5319741 ? r5319749 : r5319760;
        return r5319761;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -11906.396240974655 or 10135.341772312217 < x

    1. Initial program 59.2

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

    3. Applied div-inv59.4

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

    4. Applied fma-neg60.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\]
    5. Using strategy rm

    6. Applied add-log-exp59.4

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\right)}\]

    7. Simplified59.2

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

    8. Taylor expanded around inf 0.3

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

    9. Simplified0.0

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

    if -11906.396240974655 < x < 10135.341772312217

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied fma-neg0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\]
    5. Using strategy rm

    6. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\right)}\]

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11906.396240974655:\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \mathsf{fma}\left(\frac{3}{x}, \frac{1}{x \cdot x}, \frac{3}{x}\right)\right)\\ \mathbf{elif}\;x \le 10135.341772312217:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{-1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \mathsf{fma}\left(\frac{3}{x}, \frac{1}{x \cdot x}, \frac{3}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))