Average Error: 9.8 → 0.5
Time: 16.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -87.41395378367815283127129077911376953125 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -87.41395378367815283127129077911376953125 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r99551 = 1.0;
        double r99552 = x;
        double r99553 = r99552 + r99551;
        double r99554 = r99551 / r99553;
        double r99555 = 2.0;
        double r99556 = r99555 / r99552;
        double r99557 = r99554 - r99556;
        double r99558 = r99552 - r99551;
        double r99559 = r99551 / r99558;
        double r99560 = r99557 + r99559;
        return r99560;
}
↓
double f(double x) {
        double r99561 = 1.0;
        double r99562 = x;
        double r99563 = r99562 + r99561;
        double r99564 = r99561 / r99563;
        double r99565 = 2.0;
        double r99566 = r99565 / r99562;
        double r99567 = r99564 - r99566;
        double r99568 = r99562 - r99561;
        double r99569 = r99561 / r99568;
        double r99570 = r99567 + r99569;
        double r99571 = -87.41395378367815;
        bool r99572 = r99570 <= r99571;
        double r99573 = 0.0;
        bool r99574 = r99570 <= r99573;
        double r99575 = !r99574;
        bool r99576 = r99572 || r99575;
        double r99577 = 7.0;
        double r99578 = pow(r99562, r99577);
        double r99579 = r99565 / r99578;
        double r99580 = r99562 * r99562;
        double r99581 = r99565 / r99580;
        double r99582 = r99581 / r99562;
        double r99583 = 5.0;
        double r99584 = pow(r99562, r99583);
        double r99585 = r99565 / r99584;
        double r99586 = r99582 + r99585;
        double r99587 = r99579 + r99586;
        double r99588 = r99576 ? r99570 : r99587;
        return r99588;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.5
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -87.41395378367815 or 0.0 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

    1. Initial program 0.6

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

    if -87.41395378367815 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 0.0

    1. Initial program 19.5

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

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

    5. Applied unpow30.8

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

    6. Applied associate-/r*0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -87.41395378367815283127129077911376953125 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 0.0\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(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))))