Average Error: 29.9 → 0.1
Time: 13.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15613.93143224409:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x \cdot x}}{x}\\ \mathbf{elif}\;x \le 12671.507747924104:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x \cdot x}}{x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15613.93143224409:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x \cdot x}}{x}\\

\mathbf{elif}\;x \le 12671.507747924104:\\
\;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right)}}\right)\\

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

\end{array}
double f(double x) {
        double r5450042 = x;
        double r5450043 = 1.0;
        double r5450044 = r5450042 + r5450043;
        double r5450045 = r5450042 / r5450044;
        double r5450046 = r5450042 - r5450043;
        double r5450047 = r5450044 / r5450046;
        double r5450048 = r5450045 - r5450047;
        return r5450048;
}
↓
double f(double x) {
        double r5450049 = x;
        double r5450050 = -15613.93143224409;
        bool r5450051 = r5450049 <= r5450050;
        double r5450052 = -1.0;
        double r5450053 = r5450049 * r5450049;
        double r5450054 = r5450052 / r5450053;
        double r5450055 = 3.0;
        double r5450056 = r5450055 / r5450049;
        double r5450057 = r5450054 - r5450056;
        double r5450058 = r5450055 / r5450053;
        double r5450059 = r5450058 / r5450049;
        double r5450060 = r5450057 - r5450059;
        double r5450061 = 12671.507747924104;
        bool r5450062 = r5450049 <= r5450061;
        double r5450063 = 1.0;
        double r5450064 = r5450063 + r5450049;
        double r5450065 = r5450049 / r5450064;
        double r5450066 = r5450049 - r5450063;
        double r5450067 = r5450064 / r5450066;
        double r5450068 = r5450065 - r5450067;
        double r5450069 = r5450068 * r5450068;
        double r5450070 = r5450068 * r5450069;
        double r5450071 = cbrt(r5450070);
        double r5450072 = exp(r5450071);
        double r5450073 = log(r5450072);
        double r5450074 = r5450062 ? r5450073 : r5450060;
        double r5450075 = r5450051 ? r5450060 : r5450074;
        return r5450075;
}

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

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

    3. Applied add-log-exp59.3

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

    5. Applied add-cbrt-cube59.3

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

    6. Taylor expanded around inf 0.3

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

    7. Simplified0.0

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

    if -15613.93143224409 < x < 12671.507747924104

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    5. Applied add-cbrt-cube0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15613.93143224409:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x \cdot x}}{x}\\ \mathbf{elif}\;x \le 12671.507747924104:\\ \;\;\;\;\log \left(e^{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))