Average Error: 29.4 → 0.2
Time: 8.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12366.7094165989129 \lor \neg \left(x \le 0.99621864459541976\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \left(\frac{3}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12366.7094165989129 \lor \neg \left(x \le 0.99621864459541976\right):\\
\;\;\;\;\frac{-1}{{x}^{2}} - \left(\frac{3}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\\

\end{array}
double f(double x) {
        double r110870 = x;
        double r110871 = 1.0;
        double r110872 = r110870 + r110871;
        double r110873 = r110870 / r110872;
        double r110874 = r110870 - r110871;
        double r110875 = r110872 / r110874;
        double r110876 = r110873 - r110875;
        return r110876;
}
↓
double f(double x) {
        double r110877 = x;
        double r110878 = -12366.709416598913;
        bool r110879 = r110877 <= r110878;
        double r110880 = 0.9962186445954198;
        bool r110881 = r110877 <= r110880;
        double r110882 = !r110881;
        bool r110883 = r110879 || r110882;
        double r110884 = 1.0;
        double r110885 = -r110884;
        double r110886 = 2.0;
        double r110887 = pow(r110877, r110886);
        double r110888 = r110885 / r110887;
        double r110889 = 3.0;
        double r110890 = r110889 / r110877;
        double r110891 = 1.0;
        double r110892 = 3.0;
        double r110893 = pow(r110877, r110892);
        double r110894 = r110891 / r110893;
        double r110895 = r110889 * r110894;
        double r110896 = r110890 + r110895;
        double r110897 = r110888 - r110896;
        double r110898 = r110877 + r110884;
        double r110899 = r110877 / r110898;
        double r110900 = r110877 - r110884;
        double r110901 = r110898 / r110900;
        double r110902 = r110899 - r110901;
        double r110903 = sqrt(r110902);
        double r110904 = r110877 * r110900;
        double r110905 = r110898 * r110898;
        double r110906 = r110904 - r110905;
        double r110907 = r110877 * r110877;
        double r110908 = r110884 * r110884;
        double r110909 = r110907 - r110908;
        double r110910 = r110906 / r110909;
        double r110911 = sqrt(r110910);
        double r110912 = r110903 * r110911;
        double r110913 = r110883 ? r110897 : r110912;
        return r110913;
}

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 < -12366.709416598913 or 0.9962186445954198 < x

    1. Initial program 59.0

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.5

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

    6. Simplified0.2

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

    if -12366.709416598913 < x < 0.9962186445954198

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    5. Applied frac-sub0.1

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

    6. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12366.7094165989129 \lor \neg \left(x \le 0.99621864459541976\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \left(\frac{3}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))