Average Error: 29.7 → 0.1
Time: 15.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7452.829832440166683227289468050003051758 \lor \neg \left(x \le 8087.115897509891510708257555961608886719\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - \frac{1}{{x}^{3}} \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(\sqrt[3]{\frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7452.829832440166683227289468050003051758 \lor \neg \left(x \le 8087.115897509891510708257555961608886719\right):\\
\;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - \frac{1}{{x}^{3}} \cdot 3\\

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

\end{array}
double f(double x) {
        double r87068 = x;
        double r87069 = 1.0;
        double r87070 = r87068 + r87069;
        double r87071 = r87068 / r87070;
        double r87072 = r87068 - r87069;
        double r87073 = r87070 / r87072;
        double r87074 = r87071 - r87073;
        return r87074;
}
↓
double f(double x) {
        double r87075 = x;
        double r87076 = -7452.829832440167;
        bool r87077 = r87075 <= r87076;
        double r87078 = 8087.1158975098915;
        bool r87079 = r87075 <= r87078;
        double r87080 = !r87079;
        bool r87081 = r87077 || r87080;
        double r87082 = 1.0;
        double r87083 = 2.0;
        double r87084 = pow(r87075, r87083);
        double r87085 = r87082 / r87084;
        double r87086 = 3.0;
        double r87087 = r87086 / r87075;
        double r87088 = r87085 + r87087;
        double r87089 = -r87088;
        double r87090 = 1.0;
        double r87091 = 3.0;
        double r87092 = pow(r87075, r87091);
        double r87093 = r87090 / r87092;
        double r87094 = r87093 * r87086;
        double r87095 = r87089 - r87094;
        double r87096 = r87075 + r87082;
        double r87097 = r87075 / r87096;
        double r87098 = r87075 - r87082;
        double r87099 = r87096 / r87098;
        double r87100 = cbrt(r87099);
        double r87101 = r87100 * r87100;
        double r87102 = r87101 * r87100;
        double r87103 = r87097 - r87102;
        double r87104 = r87081 ? r87095 : r87103;
        return r87104;
}

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 < -7452.829832440167 or 8087.1158975098915 < x

    1. Initial program 59.1

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

    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

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

    5. Applied distribute-rgt-in0.3

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

    6. Applied associate--r+0.3

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

    7. Simplified0.0

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

    if -7452.829832440167 < x < 8087.1158975098915

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7452.829832440166683227289468050003051758 \lor \neg \left(x \le 8087.115897509891510708257555961608886719\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - \frac{1}{{x}^{3}} \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(\sqrt[3]{\frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))