Average Error: 29.9 → 0.1
Time: 28.2s
Precision: 64
\[\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15613.93143224409:\\ \;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 12671.507747924104:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15613.93143224409:\\
\;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 12671.507747924104:\\
\;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right)\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\

\end{array}
double f(double x) {
        double r5071915 = x;
        double r5071916 = 1.0;
        double r5071917 = r5071915 + r5071916;
        double r5071918 = r5071915 / r5071917;
        double r5071919 = r5071915 - r5071916;
        double r5071920 = r5071917 / r5071919;
        double r5071921 = r5071918 - r5071920;
        return r5071921;
}
↓
double f(double x) {
        double r5071922 = x;
        double r5071923 = -15613.93143224409;
        bool r5071924 = r5071922 <= r5071923;
        double r5071925 = 1.0;
        double r5071926 = -r5071925;
        double r5071927 = r5071922 * r5071922;
        double r5071928 = r5071926 / r5071927;
        double r5071929 = 3.0;
        double r5071930 = r5071929 / r5071922;
        double r5071931 = r5071928 - r5071930;
        double r5071932 = r5071930 / r5071927;
        double r5071933 = r5071931 - r5071932;
        double r5071934 = 12671.507747924104;
        bool r5071935 = r5071922 <= r5071934;
        double r5071936 = r5071925 + r5071922;
        double r5071937 = r5071922 / r5071936;
        double r5071938 = r5071922 - r5071925;
        double r5071939 = r5071936 / r5071938;
        double r5071940 = r5071937 - r5071939;
        double r5071941 = r5071940 * r5071940;
        double r5071942 = r5071940 * r5071941;
        double r5071943 = cbrt(r5071942);
        double r5071944 = exp(r5071943);
        double r5071945 = log(r5071944);
        double r5071946 = r5071935 ? r5071945 : r5071933;
        double r5071947 = r5071924 ? r5071933 : r5071946;
        return r5071947;
}

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 < -15613.93143224409 or 12671.507747924104 < x

    1. Initial program 59.4

      \[\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\]

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(3.0 \cdot \frac{1}{{x}^{3}} + \left(1.0 \cdot \frac{1}{{x}^{2}} + 3.0 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\left(-\frac{1.0}{x \cdot x}\right) - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}}\]

    if -15613.93143224409 < x < 12671.507747924104

    1. Initial program 0.1

      \[\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \frac{x}{x + 1.0} - \color{blue}{\log \left(e^{\frac{x + 1.0}{x - 1.0}}\right)}\]

    4. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x + 1.0}}\right)} - \log \left(e^{\frac{x + 1.0}{x - 1.0}}\right)\]

    5. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{x}{x + 1.0}}}{e^{\frac{x + 1.0}{x - 1.0}}}\right)}\]

    6. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}}\right)}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube0.1

      \[\leadsto \log \left(e^{\color{blue}{\sqrt[3]{\left(\left(\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\right) \cdot \left(\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\right)\right) \cdot \left(\frac{x}{x + 1.0} - \frac{x + 1.0}{x - 1.0}\right)}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15613.93143224409:\\ \;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 12671.507747924104:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right) \cdot \left(\frac{x}{1.0 + x} - \frac{1.0 + x}{x - 1.0}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1.0}{x \cdot x} - \frac{3.0}{x}\right) - \frac{\frac{3.0}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))