Average Error: 9.8 → 0.1
Time: 36.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -109.157505507267487 \lor \neg \left(x \le 120.214345156184848\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} - \frac{2 \cdot \left(x + 1\right) - x \cdot 1}{x \cdot \left(x + 1\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -109.157505507267487 \lor \neg \left(x \le 120.214345156184848\right):\\
\;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\

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

\end{array}
double f(double x) {
        double r116638 = 1.0;
        double r116639 = x;
        double r116640 = r116639 + r116638;
        double r116641 = r116638 / r116640;
        double r116642 = 2.0;
        double r116643 = r116642 / r116639;
        double r116644 = r116641 - r116643;
        double r116645 = r116639 - r116638;
        double r116646 = r116638 / r116645;
        double r116647 = r116644 + r116646;
        return r116647;
}
↓
double f(double x) {
        double r116648 = x;
        double r116649 = -109.15750550726749;
        bool r116650 = r116648 <= r116649;
        double r116651 = 120.21434515618485;
        bool r116652 = r116648 <= r116651;
        double r116653 = !r116652;
        bool r116654 = r116650 || r116653;
        double r116655 = 2.0;
        double r116656 = 5.0;
        double r116657 = pow(r116648, r116656);
        double r116658 = r116655 / r116657;
        double r116659 = r116648 * r116648;
        double r116660 = r116655 / r116659;
        double r116661 = r116660 / r116648;
        double r116662 = 7.0;
        double r116663 = pow(r116648, r116662);
        double r116664 = r116655 / r116663;
        double r116665 = r116661 + r116664;
        double r116666 = r116658 + r116665;
        double r116667 = 1.0;
        double r116668 = r116648 - r116667;
        double r116669 = r116667 / r116668;
        double r116670 = r116648 + r116667;
        double r116671 = r116655 * r116670;
        double r116672 = r116648 * r116667;
        double r116673 = r116671 - r116672;
        double r116674 = r116648 * r116670;
        double r116675 = r116673 / r116674;
        double r116676 = r116669 - r116675;
        double r116677 = r116654 ? r116666 : r116676;
        return r116677;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -109.15750550726749 or 120.21434515618485 < x

    1. Initial program 19.4

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

    2. Simplified19.4

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

    3. Taylor expanded around inf 0.5

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

    4. Simplified0.5

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

    6. Applied unpow30.5

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

    7. Applied associate-/r*0.1

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

    if -109.15750550726749 < x < 120.21434515618485

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied *-un-lft-identity0.0

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

    6. Applied times-frac0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied *-un-lft-identity0.0

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

    9. Applied times-frac0.0

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

    10. Applied distribute-lft-out--0.0

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

    12. Applied frac-sub0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -109.157505507267487 \lor \neg \left(x \le 120.214345156184848\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} - \frac{2 \cdot \left(x + 1\right) - x \cdot 1}{x \cdot \left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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))))