Average Error: 9.6 → 0.1
Time: 23.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -105.6928587779547683567216154187917709351:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 119.6124184692429395227009081281721591949:\\ \;\;\;\;\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -105.6928587779547683567216154187917709351:\\
\;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\

\mathbf{elif}\;x \le 119.6124184692429395227009081281721591949:\\
\;\;\;\;\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\\

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

\end{array}
double f(double x) {
        double r4670544 = 1.0;
        double r4670545 = x;
        double r4670546 = r4670545 + r4670544;
        double r4670547 = r4670544 / r4670546;
        double r4670548 = 2.0;
        double r4670549 = r4670548 / r4670545;
        double r4670550 = r4670547 - r4670549;
        double r4670551 = r4670545 - r4670544;
        double r4670552 = r4670544 / r4670551;
        double r4670553 = r4670550 + r4670552;
        return r4670553;
}
↓
double f(double x) {
        double r4670554 = x;
        double r4670555 = -105.69285877795477;
        bool r4670556 = r4670554 <= r4670555;
        double r4670557 = 2.0;
        double r4670558 = 5.0;
        double r4670559 = pow(r4670554, r4670558);
        double r4670560 = r4670557 / r4670559;
        double r4670561 = r4670554 * r4670554;
        double r4670562 = r4670557 / r4670561;
        double r4670563 = r4670562 / r4670554;
        double r4670564 = 7.0;
        double r4670565 = pow(r4670554, r4670564);
        double r4670566 = r4670557 / r4670565;
        double r4670567 = r4670563 + r4670566;
        double r4670568 = r4670560 + r4670567;
        double r4670569 = 119.61241846924294;
        bool r4670570 = r4670554 <= r4670569;
        double r4670571 = 1.0;
        double r4670572 = r4670554 - r4670571;
        double r4670573 = r4670571 / r4670572;
        double r4670574 = r4670571 + r4670554;
        double r4670575 = r4670571 / r4670574;
        double r4670576 = r4670573 + r4670575;
        double r4670577 = r4670557 / r4670554;
        double r4670578 = r4670576 - r4670577;
        double r4670579 = r4670570 ? r4670578 : r4670568;
        double r4670580 = r4670556 ? r4670568 : r4670579;
        return r4670580;
}

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 < -105.69285877795477 or 119.61241846924294 < x

    1. Initial program 19.8

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

    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. Simplified0.1

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

    if -105.69285877795477 < x < 119.61241846924294

    1. Initial program 0.0

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

    3. Applied sub-neg0.0

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied associate-/r*0.2

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

    8. Applied *-un-lft-identity0.2

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

    9. Applied *-un-lft-identity0.2

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

    10. Applied distribute-lft-out0.2

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

    11. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -105.6928587779547683567216154187917709351:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 119.6124184692429395227009081281721591949:\\ \;\;\;\;\left(\frac{1}{x - 1} + \frac{1}{1 + x}\right) - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{7}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))