Average Error: 9.6 → 0.2
Time: 14.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -104.3426691909725434470601612702012062073:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{1}{x \cdot x} \cdot \frac{2}{x}\right) + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \le 117.9969754864155078166731982491910457611:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{3}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -104.3426691909725434470601612702012062073:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{1}{x \cdot x} \cdot \frac{2}{x}\right) + \frac{2}{{x}^{5}}\\

\mathbf{elif}\;x \le 117.9969754864155078166731982491910457611:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{3}}\right) + \frac{2}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r88716 = 1.0;
        double r88717 = x;
        double r88718 = r88717 + r88716;
        double r88719 = r88716 / r88718;
        double r88720 = 2.0;
        double r88721 = r88720 / r88717;
        double r88722 = r88719 - r88721;
        double r88723 = r88717 - r88716;
        double r88724 = r88716 / r88723;
        double r88725 = r88722 + r88724;
        return r88725;
}
↓
double f(double x) {
        double r88726 = x;
        double r88727 = -104.34266919097254;
        bool r88728 = r88726 <= r88727;
        double r88729 = 2.0;
        double r88730 = 7.0;
        double r88731 = pow(r88726, r88730);
        double r88732 = r88729 / r88731;
        double r88733 = 1.0;
        double r88734 = r88726 * r88726;
        double r88735 = r88733 / r88734;
        double r88736 = r88729 / r88726;
        double r88737 = r88735 * r88736;
        double r88738 = r88732 + r88737;
        double r88739 = 5.0;
        double r88740 = pow(r88726, r88739);
        double r88741 = r88729 / r88740;
        double r88742 = r88738 + r88741;
        double r88743 = 117.99697548641551;
        bool r88744 = r88726 <= r88743;
        double r88745 = 1.0;
        double r88746 = r88726 + r88745;
        double r88747 = r88745 / r88746;
        double r88748 = r88747 - r88736;
        double r88749 = r88726 - r88745;
        double r88750 = r88745 / r88749;
        double r88751 = r88748 + r88750;
        double r88752 = 3.0;
        double r88753 = pow(r88726, r88752);
        double r88754 = r88729 / r88753;
        double r88755 = r88732 + r88754;
        double r88756 = r88755 + r88741;
        double r88757 = r88744 ? r88751 : r88756;
        double r88758 = r88728 ? r88742 : r88757;
        return r88758;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -104.34266919097254

    1. Initial program 19.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    5. Applied add-cube-cbrt1.1

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

    6. Applied unpow-prod-down1.1

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

    7. Applied *-un-lft-identity1.1

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

    8. Applied times-frac0.8

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

    9. Simplified0.4

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

    10. Simplified0.1

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

    if -104.34266919097254 < x < 117.99697548641551

    1. Initial program 0.0

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

    if 117.99697548641551 < x

    1. Initial program 19.0

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -104.3426691909725434470601612702012062073:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{1}{x \cdot x} \cdot \frac{2}{x}\right) + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \le 117.9969754864155078166731982491910457611:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{3}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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))))