Average Error: 29.4 → 0.2
Time: 10.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 2.0571526871 \cdot 10^{-7}:\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 2.0571526871 \cdot 10^{-7}:\\
\;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r142803 = x;
        double r142804 = 1.0;
        double r142805 = r142803 + r142804;
        double r142806 = r142803 / r142805;
        double r142807 = r142803 - r142804;
        double r142808 = r142805 / r142807;
        double r142809 = r142806 - r142808;
        return r142809;
}
↓
double f(double x) {
        double r142810 = x;
        double r142811 = 1.0;
        double r142812 = r142810 + r142811;
        double r142813 = r142810 / r142812;
        double r142814 = r142810 - r142811;
        double r142815 = r142812 / r142814;
        double r142816 = r142813 - r142815;
        double r142817 = 2.057152687084951e-07;
        bool r142818 = r142816 <= r142817;
        double r142819 = r142810 * r142810;
        double r142820 = r142811 / r142819;
        double r142821 = -r142820;
        double r142822 = 3.0;
        double r142823 = r142822 / r142810;
        double r142824 = 3.0;
        double r142825 = pow(r142810, r142824);
        double r142826 = r142822 / r142825;
        double r142827 = r142823 + r142826;
        double r142828 = r142821 - r142827;
        double r142829 = pow(r142816, r142824);
        double r142830 = cbrt(r142829);
        double r142831 = r142818 ? r142828 : r142830;
        return r142831;
}

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 (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 2.057152687084951e-07

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.3

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

    if 2.057152687084951e-07 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 2.0571526871 \cdot 10^{-7}:\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}^{3}}\\ \end{array}\]

Reproduce

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