Average Error: 9.5 → 0.1
Time: 57.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -119.2186693862327:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \le 109.3629970410856:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -119.2186693862327:\\
\;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\

\mathbf{elif}\;x \le 109.3629970410856:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r3239474 = 1.0;
        double r3239475 = x;
        double r3239476 = r3239475 + r3239474;
        double r3239477 = r3239474 / r3239476;
        double r3239478 = 2.0;
        double r3239479 = r3239478 / r3239475;
        double r3239480 = r3239477 - r3239479;
        double r3239481 = r3239475 - r3239474;
        double r3239482 = r3239474 / r3239481;
        double r3239483 = r3239480 + r3239482;
        return r3239483;
}
↓
double f(double x) {
        double r3239484 = x;
        double r3239485 = -119.2186693862327;
        bool r3239486 = r3239484 <= r3239485;
        double r3239487 = 2.0;
        double r3239488 = 5.0;
        double r3239489 = pow(r3239484, r3239488);
        double r3239490 = r3239487 / r3239489;
        double r3239491 = 7.0;
        double r3239492 = pow(r3239484, r3239491);
        double r3239493 = r3239487 / r3239492;
        double r3239494 = r3239484 * r3239484;
        double r3239495 = r3239487 / r3239494;
        double r3239496 = r3239495 / r3239484;
        double r3239497 = r3239493 + r3239496;
        double r3239498 = r3239490 + r3239497;
        double r3239499 = 109.3629970410856;
        bool r3239500 = r3239484 <= r3239499;
        double r3239501 = 1.0;
        double r3239502 = r3239501 + r3239484;
        double r3239503 = r3239501 / r3239502;
        double r3239504 = r3239487 / r3239484;
        double r3239505 = r3239503 - r3239504;
        double r3239506 = r3239484 - r3239501;
        double r3239507 = r3239501 / r3239506;
        double r3239508 = r3239505 + r3239507;
        double r3239509 = r3239501 / r3239494;
        double r3239510 = r3239509 * r3239504;
        double r3239511 = r3239510 + r3239493;
        double r3239512 = r3239511 + r3239490;
        double r3239513 = r3239500 ? r3239508 : r3239512;
        double r3239514 = r3239486 ? r3239498 : r3239513;
        return r3239514;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -119.2186693862327

    1. Initial program 19.5

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

    3. Applied flip-+19.5

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

    4. Taylor expanded around -inf 0.4

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

    5. Simplified0.5

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

    7. Applied associate-/r*0.1

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

    if -119.2186693862327 < x < 109.3629970410856

    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-+29.0

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

    5. Applied flip--30.0

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

    6. Applied associate-/r/30.0

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

    7. Simplified0.1

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

    if 109.3629970410856 < x

    1. Initial program 19.1

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

    3. Applied flip-+19.1

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

    4. Taylor expanded around -inf 0.6

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

    5. Simplified0.7

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

    7. Applied *-un-lft-identity0.7

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

    8. Applied times-frac0.1

      \[\leadsto \left(\frac{2}{{x}^{7}} + \color{blue}{\frac{1}{x \cdot x} \cdot \frac{2}{x}}\right) + \frac{2}{{x}^{5}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -119.2186693862327:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \le 109.3629970410856:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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))))