Average Error: 9.8 → 0.0
Time: 12.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\
\;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\

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

\end{array}
double f(double x) {
        double r144356 = 1.0;
        double r144357 = x;
        double r144358 = r144357 + r144356;
        double r144359 = r144356 / r144358;
        double r144360 = 2.0;
        double r144361 = r144360 / r144357;
        double r144362 = r144359 - r144361;
        double r144363 = r144357 - r144356;
        double r144364 = r144356 / r144363;
        double r144365 = r144362 + r144364;
        return r144365;
}
↓
double f(double x) {
        double r144366 = x;
        double r144367 = -118.86664556304571;
        bool r144368 = r144366 <= r144367;
        double r144369 = 111.79023992954416;
        bool r144370 = r144366 <= r144369;
        double r144371 = !r144370;
        bool r144372 = r144368 || r144371;
        double r144373 = 2.0;
        double r144374 = 1.0;
        double r144375 = 5.0;
        double r144376 = pow(r144366, r144375);
        double r144377 = r144374 / r144376;
        double r144378 = -3.0;
        double r144379 = pow(r144366, r144378);
        double r144380 = r144377 + r144379;
        double r144381 = 7.0;
        double r144382 = pow(r144366, r144381);
        double r144383 = r144374 / r144382;
        double r144384 = r144380 + r144383;
        double r144385 = r144373 * r144384;
        double r144386 = 1.0;
        double r144387 = r144366 + r144386;
        double r144388 = r144386 / r144387;
        double r144389 = r144373 / r144366;
        double r144390 = r144388 - r144389;
        double r144391 = r144366 * r144366;
        double r144392 = r144386 * r144386;
        double r144393 = r144391 - r144392;
        double r144394 = r144386 / r144393;
        double r144395 = r144394 * r144387;
        double r144396 = r144390 + r144395;
        double r144397 = r144372 ? r144385 : r144396;
        return r144397;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -118.86664556304571 or 111.79023992954416 < x

    1. Initial program 19.7

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

    3. Applied flip--49.8

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

    4. Applied associate-/r/52.8

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

    5. Taylor expanded around inf 0.5

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

    6. Simplified0.5

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

    8. Applied pow-flip0.0

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

    9. Simplified0.0

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

    if -118.86664556304571 < x < 111.79023992954416

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))