Average Error: 9.8 → 0.3
Time: 18.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1440571.9585081101395189762115478515625:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 168550.15379253146238625049591064453125:\\ \;\;\;\;\frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) - \left(x + 1\right) \cdot \left(2 \cdot \left(x - 1\right) - x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1440571.9585081101395189762115478515625:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 168550.15379253146238625049591064453125:\\
\;\;\;\;\frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) - \left(x + 1\right) \cdot \left(2 \cdot \left(x - 1\right) - x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\\

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

\end{array}
double f(double x) {
        double r9164602 = 1.0;
        double r9164603 = x;
        double r9164604 = r9164603 + r9164602;
        double r9164605 = r9164602 / r9164604;
        double r9164606 = 2.0;
        double r9164607 = r9164606 / r9164603;
        double r9164608 = r9164605 - r9164607;
        double r9164609 = r9164603 - r9164602;
        double r9164610 = r9164602 / r9164609;
        double r9164611 = r9164608 + r9164610;
        return r9164611;
}
↓
double f(double x) {
        double r9164612 = x;
        double r9164613 = -1440571.9585081101;
        bool r9164614 = r9164612 <= r9164613;
        double r9164615 = 2.0;
        double r9164616 = 7.0;
        double r9164617 = pow(r9164612, r9164616);
        double r9164618 = r9164615 / r9164617;
        double r9164619 = r9164612 * r9164612;
        double r9164620 = r9164612 * r9164619;
        double r9164621 = r9164615 / r9164620;
        double r9164622 = 5.0;
        double r9164623 = pow(r9164612, r9164622);
        double r9164624 = r9164615 / r9164623;
        double r9164625 = r9164621 + r9164624;
        double r9164626 = r9164618 + r9164625;
        double r9164627 = 168550.15379253146;
        bool r9164628 = r9164612 <= r9164627;
        double r9164629 = 1.0;
        double r9164630 = r9164612 - r9164629;
        double r9164631 = r9164612 * r9164630;
        double r9164632 = r9164629 * r9164631;
        double r9164633 = r9164612 + r9164629;
        double r9164634 = r9164615 * r9164630;
        double r9164635 = r9164612 * r9164629;
        double r9164636 = r9164634 - r9164635;
        double r9164637 = r9164633 * r9164636;
        double r9164638 = r9164632 - r9164637;
        double r9164639 = r9164633 * r9164631;
        double r9164640 = r9164638 / r9164639;
        double r9164641 = r9164628 ? r9164640 : r9164626;
        double r9164642 = r9164614 ? r9164626 : r9164641;
        return r9164642;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1440571.9585081101 or 168550.15379253146 < x

    1. Initial program 19.5

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.6

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

    if -1440571.9585081101 < x < 168550.15379253146

    1. Initial program 0.3

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    2. Using strategy rm

    3. Applied associate-+l-0.3

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

    5. Applied frac-sub0.3

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

    6. Applied frac-sub0.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1440571.9585081101395189762115478515625:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 168550.15379253146238625049591064453125:\\ \;\;\;\;\frac{1 \cdot \left(x \cdot \left(x - 1\right)\right) - \left(x + 1\right) \cdot \left(2 \cdot \left(x - 1\right) - x \cdot 1\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))))