Average Error: 9.1 → 0.2
Time: 35.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -113.57820944408793:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 111.69425311390617:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{x \cdot \left(x \cdot x\right)}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -113.57820944408793:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 111.69425311390617:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r3235352 = 1.0;
        double r3235353 = x;
        double r3235354 = r3235353 + r3235352;
        double r3235355 = r3235352 / r3235354;
        double r3235356 = 2.0;
        double r3235357 = r3235356 / r3235353;
        double r3235358 = r3235355 - r3235357;
        double r3235359 = r3235353 - r3235352;
        double r3235360 = r3235352 / r3235359;
        double r3235361 = r3235358 + r3235360;
        return r3235361;
}
↓
double f(double x) {
        double r3235362 = x;
        double r3235363 = -113.57820944408793;
        bool r3235364 = r3235362 <= r3235363;
        double r3235365 = 2.0;
        double r3235366 = 7.0;
        double r3235367 = pow(r3235362, r3235366);
        double r3235368 = r3235365 / r3235367;
        double r3235369 = r3235362 * r3235362;
        double r3235370 = r3235365 / r3235369;
        double r3235371 = r3235370 / r3235362;
        double r3235372 = 5.0;
        double r3235373 = pow(r3235362, r3235372);
        double r3235374 = r3235365 / r3235373;
        double r3235375 = r3235371 + r3235374;
        double r3235376 = r3235368 + r3235375;
        double r3235377 = 111.69425311390617;
        bool r3235378 = r3235362 <= r3235377;
        double r3235379 = 1.0;
        double r3235380 = r3235379 + r3235362;
        double r3235381 = r3235379 / r3235380;
        double r3235382 = r3235365 / r3235362;
        double r3235383 = r3235381 - r3235382;
        double r3235384 = r3235362 - r3235379;
        double r3235385 = r3235379 / r3235384;
        double r3235386 = r3235383 + r3235385;
        double r3235387 = r3235362 * r3235369;
        double r3235388 = r3235365 / r3235387;
        double r3235389 = r3235374 + r3235388;
        double r3235390 = r3235389 + r3235368;
        double r3235391 = r3235378 ? r3235386 : r3235390;
        double r3235392 = r3235364 ? r3235376 : r3235391;
        return r3235392;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -113.57820944408793

    1. Initial program 18.4

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

    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

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

    4. Taylor expanded around 0 0.1

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

    if -113.57820944408793 < x < 111.69425311390617

    1. Initial program 0.1

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

    if 111.69425311390617 < x

    1. Initial program 18.7

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.1

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

    4. Taylor expanded around 0 0.1

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

    6. Applied div-inv0.1

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

    7. Applied associate-/l*0.4

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

    8. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -113.57820944408793:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x \cdot x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 111.69425311390617:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{x \cdot \left(x \cdot x\right)}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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