Average Error: 9.6 → 0.1
Time: 9.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.331636163847744521973196434576919413339 \cdot 10^{154} \lor \neg \left(x \le 249669183499528.375\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left({x}^{2} \cdot \sqrt{1} + 1 \cdot \left(\frac{x}{\sqrt{1}} - \left(x \cdot \sqrt{1} + \frac{{x}^{2}}{\sqrt{1}}\right)\right)\right) + 2 \cdot \frac{1}{\sqrt{1}}}{x}}{\frac{x + 1}{\sqrt{1}}}}{x - 1}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.331636163847744521973196434576919413339 \cdot 10^{154} \lor \neg \left(x \le 249669183499528.375\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\left({x}^{2} \cdot \sqrt{1} + 1 \cdot \left(\frac{x}{\sqrt{1}} - \left(x \cdot \sqrt{1} + \frac{{x}^{2}}{\sqrt{1}}\right)\right)\right) + 2 \cdot \frac{1}{\sqrt{1}}}{x}}{\frac{x + 1}{\sqrt{1}}}}{x - 1}\\

\end{array}
double f(double x) {
        double r140724 = 1.0;
        double r140725 = x;
        double r140726 = r140725 + r140724;
        double r140727 = r140724 / r140726;
        double r140728 = 2.0;
        double r140729 = r140728 / r140725;
        double r140730 = r140727 - r140729;
        double r140731 = r140725 - r140724;
        double r140732 = r140724 / r140731;
        double r140733 = r140730 + r140732;
        return r140733;
}
↓
double f(double x) {
        double r140734 = x;
        double r140735 = -1.3316361638477445e+154;
        bool r140736 = r140734 <= r140735;
        double r140737 = 249669183499528.38;
        bool r140738 = r140734 <= r140737;
        double r140739 = !r140738;
        bool r140740 = r140736 || r140739;
        double r140741 = 2.0;
        double r140742 = 1.0;
        double r140743 = 7.0;
        double r140744 = pow(r140734, r140743);
        double r140745 = r140742 / r140744;
        double r140746 = 5.0;
        double r140747 = pow(r140734, r140746);
        double r140748 = r140742 / r140747;
        double r140749 = 3.0;
        double r140750 = pow(r140734, r140749);
        double r140751 = r140742 / r140750;
        double r140752 = r140748 + r140751;
        double r140753 = r140745 + r140752;
        double r140754 = r140741 * r140753;
        double r140755 = 2.0;
        double r140756 = pow(r140734, r140755);
        double r140757 = 1.0;
        double r140758 = sqrt(r140757);
        double r140759 = r140756 * r140758;
        double r140760 = r140734 / r140758;
        double r140761 = r140734 * r140758;
        double r140762 = r140756 / r140758;
        double r140763 = r140761 + r140762;
        double r140764 = r140760 - r140763;
        double r140765 = r140757 * r140764;
        double r140766 = r140759 + r140765;
        double r140767 = r140742 / r140758;
        double r140768 = r140741 * r140767;
        double r140769 = r140766 + r140768;
        double r140770 = r140769 / r140734;
        double r140771 = r140734 + r140757;
        double r140772 = r140771 / r140758;
        double r140773 = r140770 / r140772;
        double r140774 = r140734 - r140757;
        double r140775 = r140773 / r140774;
        double r140776 = r140740 ? r140754 : r140775;
        return r140776;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3316361638477445e+154 or 249669183499528.38 < x

    1. Initial program 12.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -1.3316361638477445e+154 < x < 249669183499528.38

    1. Initial program 8.1

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

    3. Applied add-sqr-sqrt8.1

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

    4. Applied associate-/l*8.1

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

    6. Applied frac-sub9.0

      \[\leadsto \color{blue}{\frac{\sqrt{1} \cdot x - \frac{x + 1}{\sqrt{1}} \cdot 2}{\frac{x + 1}{\sqrt{1}} \cdot x}} + \frac{1}{x - 1}\]

    7. Applied frac-add7.6

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

    8. Taylor expanded around 0 7.6

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

    9. Simplified0.2

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

    11. Applied associate-/r*0.1

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

    12. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.331636163847744521973196434576919413339 \cdot 10^{154} \lor \neg \left(x \le 249669183499528.375\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left({x}^{2} \cdot \sqrt{1} + 1 \cdot \left(\frac{x}{\sqrt{1}} - \left(x \cdot \sqrt{1} + \frac{{x}^{2}}{\sqrt{1}}\right)\right)\right) + 2 \cdot \frac{1}{\sqrt{1}}}{x}}{\frac{x + 1}{\sqrt{1}}}}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))