Average Error: 29.2 → 0.1
Time: 15.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13212.25074346146:\\ \;\;\;\;-3 \cdot \frac{\frac{\frac{1}{x}}{x}}{x} - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \frac{\frac{\frac{1}{x}}{x}}{x} - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13212.25074346146:\\
\;\;\;\;-3 \cdot \frac{\frac{\frac{1}{x}}{x}}{x} - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\

\mathbf{elif}\;x \le 9551.679161702108:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r2214082 = x;
        double r2214083 = 1.0;
        double r2214084 = r2214082 + r2214083;
        double r2214085 = r2214082 / r2214084;
        double r2214086 = r2214082 - r2214083;
        double r2214087 = r2214084 / r2214086;
        double r2214088 = r2214085 - r2214087;
        return r2214088;
}
↓
double f(double x) {
        double r2214089 = x;
        double r2214090 = -13212.25074346146;
        bool r2214091 = r2214089 <= r2214090;
        double r2214092 = -3.0;
        double r2214093 = 1.0;
        double r2214094 = r2214093 / r2214089;
        double r2214095 = r2214094 / r2214089;
        double r2214096 = r2214095 / r2214089;
        double r2214097 = r2214092 * r2214096;
        double r2214098 = 3.0;
        double r2214099 = r2214098 / r2214089;
        double r2214100 = r2214099 + r2214095;
        double r2214101 = r2214097 - r2214100;
        double r2214102 = 9551.679161702108;
        bool r2214103 = r2214089 <= r2214102;
        double r2214104 = r2214089 + r2214093;
        double r2214105 = cbrt(r2214104);
        double r2214106 = r2214105 * r2214105;
        double r2214107 = r2214089 / r2214106;
        double r2214108 = r2214107 / r2214105;
        double r2214109 = r2214089 - r2214093;
        double r2214110 = r2214104 / r2214109;
        double r2214111 = r2214108 - r2214110;
        double r2214112 = r2214103 ? r2214111 : r2214101;
        double r2214113 = r2214091 ? r2214101 : r2214112;
        return r2214113;
}

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 < -13212.25074346146 or 9551.679161702108 < x

    1. Initial program 59.4

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

    3. Applied add-cube-cbrt60.2

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

    4. Applied associate-/r*60.2

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}}} - \frac{x + 1}{x - 1}\]

    5. Taylor expanded around inf 0.3

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

    6. Simplified0.0

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

    if -13212.25074346146 < x < 9551.679161702108

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13212.25074346146:\\ \;\;\;\;-3 \cdot \frac{\frac{\frac{1}{x}}{x}}{x} - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \frac{\frac{\frac{1}{x}}{x}}{x} - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))