Average Error: 14.0 → 0.4
Time: 25.7s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -215.53143675154297:\\ \;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\ \mathbf{elif}\;x \le 206.8284768138118:\\ \;\;\;\;\frac{1}{1 + x} - \sqrt[3]{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -215.53143675154297:\\
\;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\

\mathbf{elif}\;x \le 206.8284768138118:\\
\;\;\;\;\frac{1}{1 + x} - \sqrt[3]{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\\

\mathbf{else}:\\
\;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\

\end{array}
double f(double x) {
        double r3554790 = 1.0;
        double r3554791 = x;
        double r3554792 = r3554791 + r3554790;
        double r3554793 = r3554790 / r3554792;
        double r3554794 = r3554791 - r3554790;
        double r3554795 = r3554790 / r3554794;
        double r3554796 = r3554793 - r3554795;
        return r3554796;
}
↓
double f(double x) {
        double r3554797 = x;
        double r3554798 = -215.53143675154297;
        bool r3554799 = r3554797 <= r3554798;
        double r3554800 = 2.0;
        double r3554801 = r3554797 * r3554797;
        double r3554802 = r3554800 / r3554801;
        double r3554803 = 4.0;
        double r3554804 = pow(r3554797, r3554803);
        double r3554805 = r3554800 / r3554804;
        double r3554806 = 6.0;
        double r3554807 = pow(r3554797, r3554806);
        double r3554808 = r3554800 / r3554807;
        double r3554809 = r3554805 + r3554808;
        double r3554810 = r3554802 + r3554809;
        double r3554811 = -r3554810;
        double r3554812 = 206.8284768138118;
        bool r3554813 = r3554797 <= r3554812;
        double r3554814 = 1.0;
        double r3554815 = r3554814 + r3554797;
        double r3554816 = r3554814 / r3554815;
        double r3554817 = r3554797 - r3554814;
        double r3554818 = r3554814 / r3554817;
        double r3554819 = r3554818 * r3554818;
        double r3554820 = r3554818 * r3554819;
        double r3554821 = cbrt(r3554820);
        double r3554822 = r3554816 - r3554821;
        double r3554823 = r3554813 ? r3554822 : r3554811;
        double r3554824 = r3554799 ? r3554811 : r3554823;
        return r3554824;
}

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 < -215.53143675154297 or 206.8284768138118 < x

    1. Initial program 28.7

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

    3. Applied add-cbrt-cube58.7

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

    4. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    5. Simplified0.8

      \[\leadsto \color{blue}{-\left(\left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right) + \frac{2}{x \cdot x}\right)}\]

    if -215.53143675154297 < x < 206.8284768138118

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -215.53143675154297:\\ \;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\ \mathbf{elif}\;x \le 206.8284768138118:\\ \;\;\;\;\frac{1}{1 + x} - \sqrt[3]{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{2}{x \cdot x} + \left(\frac{2}{{x}^{4}} + \frac{2}{{x}^{6}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))