Average Error: 9.9 → 0.2
Time: 46.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -110.47721459746352:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 108.81613413709378:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -110.47721459746352:\\
\;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 108.81613413709378:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r2953552 = 1.0;
        double r2953553 = x;
        double r2953554 = r2953553 + r2953552;
        double r2953555 = r2953552 / r2953554;
        double r2953556 = 2.0;
        double r2953557 = r2953556 / r2953553;
        double r2953558 = r2953555 - r2953557;
        double r2953559 = r2953553 - r2953552;
        double r2953560 = r2953552 / r2953559;
        double r2953561 = r2953558 + r2953560;
        return r2953561;
}
↓
double f(double x) {
        double r2953562 = x;
        double r2953563 = -110.47721459746352;
        bool r2953564 = r2953562 <= r2953563;
        double r2953565 = 2.0;
        double r2953566 = r2953565 / r2953562;
        double r2953567 = r2953562 * r2953562;
        double r2953568 = r2953566 / r2953567;
        double r2953569 = 7.0;
        double r2953570 = pow(r2953562, r2953569);
        double r2953571 = r2953565 / r2953570;
        double r2953572 = 5.0;
        double r2953573 = pow(r2953562, r2953572);
        double r2953574 = r2953565 / r2953573;
        double r2953575 = r2953571 + r2953574;
        double r2953576 = r2953568 + r2953575;
        double r2953577 = 108.81613413709378;
        bool r2953578 = r2953562 <= r2953577;
        double r2953579 = 1.0;
        double r2953580 = r2953579 + r2953562;
        double r2953581 = r2953579 / r2953580;
        double r2953582 = r2953581 - r2953566;
        double r2953583 = r2953562 - r2953579;
        double r2953584 = r2953579 / r2953583;
        double r2953585 = r2953582 + r2953584;
        double r2953586 = r2953567 * r2953562;
        double r2953587 = r2953565 / r2953586;
        double r2953588 = r2953587 + r2953575;
        double r2953589 = r2953578 ? r2953585 : r2953588;
        double r2953590 = r2953564 ? r2953576 : r2953589;
        return r2953590;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -110.47721459746352

    1. Initial program 19.6

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    4. Taylor expanded around 0 0.6

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

    5. Simplified0.1

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

    if -110.47721459746352 < x < 108.81613413709378

    1. Initial program 0.1

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

    3. Applied flip-+29.5

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

    5. Applied flip--30.6

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

    6. Applied associate-/r/30.6

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

    7. Simplified0.1

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

    if 108.81613413709378 < x

    1. Initial program 19.3

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

    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. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -110.47721459746352:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 108.81613413709378:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot x} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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