Average Error: 29.3 → 0.1
Time: 5.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11287.389275273414 \lor \neg \left(x \le 10812.5003292367\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11287.389275273414 \lor \neg \left(x \le 10812.5003292367\right):\\
\;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\

\end{array}
double f(double x) {
        double r146877 = x;
        double r146878 = 1.0;
        double r146879 = r146877 + r146878;
        double r146880 = r146877 / r146879;
        double r146881 = r146877 - r146878;
        double r146882 = r146879 / r146881;
        double r146883 = r146880 - r146882;
        return r146883;
}
↓
double f(double x) {
        double r146884 = x;
        double r146885 = -11287.389275273414;
        bool r146886 = r146884 <= r146885;
        double r146887 = 10812.5003292367;
        bool r146888 = r146884 <= r146887;
        double r146889 = !r146888;
        bool r146890 = r146886 || r146889;
        double r146891 = 1.0;
        double r146892 = 2.0;
        double r146893 = pow(r146884, r146892);
        double r146894 = r146891 / r146893;
        double r146895 = 3.0;
        double r146896 = r146895 / r146884;
        double r146897 = r146894 + r146896;
        double r146898 = -r146897;
        double r146899 = 1.0;
        double r146900 = 3.0;
        double r146901 = pow(r146884, r146900);
        double r146902 = r146899 / r146901;
        double r146903 = r146895 * r146902;
        double r146904 = r146898 - r146903;
        double r146905 = r146884 + r146891;
        double r146906 = r146884 / r146905;
        double r146907 = r146906 * r146906;
        double r146908 = r146884 - r146891;
        double r146909 = r146905 / r146908;
        double r146910 = r146909 * r146909;
        double r146911 = r146907 - r146910;
        double r146912 = r146906 + r146909;
        double r146913 = r146911 / r146912;
        double r146914 = r146890 ? r146904 : r146913;
        return r146914;
}

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 < -11287.389275273414 or 10812.5003292367 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    5. Applied fma-udef0.3

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

    6. Applied associate--r+0.3

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

    7. Simplified0.0

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

    if -11287.389275273414 < x < 10812.5003292367

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11287.389275273414 \lor \neg \left(x \le 10812.5003292367\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))