Average Error: 9.8 → 0.1
Time: 20.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -109.157505507267487 \lor \neg \left(x \le 120.214345156184848\right):\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right) + \frac{2}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \end{array}\]
\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):\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right) + \frac{2}{{x}^{5}}\\

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

\end{array}
double f(double x) {
        double r56489 = 1.0;
        double r56490 = x;
        double r56491 = r56490 + r56489;
        double r56492 = r56489 / r56491;
        double r56493 = 2.0;
        double r56494 = r56493 / r56490;
        double r56495 = r56492 - r56494;
        double r56496 = r56490 - r56489;
        double r56497 = r56489 / r56496;
        double r56498 = r56495 + r56497;
        return r56498;
}
↓
double f(double x) {
        double r56499 = x;
        double r56500 = -109.15750550726749;
        bool r56501 = r56499 <= r56500;
        double r56502 = 120.21434515618485;
        bool r56503 = r56499 <= r56502;
        double r56504 = !r56503;
        bool r56505 = r56501 || r56504;
        double r56506 = 2.0;
        double r56507 = 7.0;
        double r56508 = pow(r56499, r56507);
        double r56509 = r56506 / r56508;
        double r56510 = r56506 / r56499;
        double r56511 = r56510 / r56499;
        double r56512 = r56511 / r56499;
        double r56513 = r56509 + r56512;
        double r56514 = 5.0;
        double r56515 = pow(r56499, r56514);
        double r56516 = r56506 / r56515;
        double r56517 = r56513 + r56516;
        double r56518 = 1.0;
        double r56519 = r56518 * r56499;
        double r56520 = r56499 + r56518;
        double r56521 = r56520 * r56506;
        double r56522 = r56519 - r56521;
        double r56523 = r56520 * r56499;
        double r56524 = r56522 / r56523;
        double r56525 = r56499 - r56518;
        double r56526 = r56518 / r56525;
        double r56527 = r56524 + r56526;
        double r56528 = r56505 ? r56517 : r56527;
        return r56528;
}

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. Using strategy rm

    3. Applied frac-sub52.6

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

    4. Taylor expanded around inf 0.5

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

    5. Simplified0.5

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

    7. Applied unpow30.5

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

    8. Applied associate-/r*0.1

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

    10. Applied associate-/r*0.1

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

    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. Using strategy rm

    3. Applied frac-sub0.0

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019199 
(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))))