Average Error: 9.7 → 0.1
Time: 22.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -137.4367895734916373839951120316982269287:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 108.745999215554448369402962271124124527:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -137.4367895734916373839951120316982269287:\\
\;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\

\mathbf{elif}\;x \le 108.745999215554448369402962271124124527:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r3560744 = 1.0;
        double r3560745 = x;
        double r3560746 = r3560745 + r3560744;
        double r3560747 = r3560744 / r3560746;
        double r3560748 = 2.0;
        double r3560749 = r3560748 / r3560745;
        double r3560750 = r3560747 - r3560749;
        double r3560751 = r3560745 - r3560744;
        double r3560752 = r3560744 / r3560751;
        double r3560753 = r3560750 + r3560752;
        return r3560753;
}
↓
double f(double x) {
        double r3560754 = x;
        double r3560755 = -137.43678957349164;
        bool r3560756 = r3560754 <= r3560755;
        double r3560757 = 2.0;
        double r3560758 = 5.0;
        double r3560759 = pow(r3560754, r3560758);
        double r3560760 = r3560757 / r3560759;
        double r3560761 = r3560757 / r3560754;
        double r3560762 = r3560754 * r3560754;
        double r3560763 = r3560761 / r3560762;
        double r3560764 = 7.0;
        double r3560765 = pow(r3560754, r3560764);
        double r3560766 = r3560757 / r3560765;
        double r3560767 = r3560763 + r3560766;
        double r3560768 = r3560760 + r3560767;
        double r3560769 = 108.74599921555445;
        bool r3560770 = r3560754 <= r3560769;
        double r3560771 = 1.0;
        double r3560772 = r3560771 + r3560754;
        double r3560773 = r3560771 / r3560772;
        double r3560774 = r3560773 - r3560761;
        double r3560775 = r3560754 - r3560771;
        double r3560776 = r3560771 / r3560775;
        double r3560777 = r3560774 + r3560776;
        double r3560778 = r3560757 / r3560762;
        double r3560779 = r3560778 / r3560754;
        double r3560780 = r3560779 + r3560766;
        double r3560781 = r3560780 + r3560760;
        double r3560782 = r3560770 ? r3560777 : r3560781;
        double r3560783 = r3560756 ? r3560768 : r3560782;
        return r3560783;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -137.43678957349164

    1. Initial program 20.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]

    4. Taylor expanded around 0 0.4

      \[\leadsto \left(\color{blue}{\frac{2}{{x}^{3}}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

    5. Simplified0.1

      \[\leadsto \left(\color{blue}{\frac{\frac{2}{x}}{x \cdot x}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

    if -137.43678957349164 < x < 108.74599921555445

    1. Initial program 0.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    if 108.74599921555445 < x

    1. Initial program 19.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -137.4367895734916373839951120316982269287:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 108.745999215554448369402962271124124527:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))