Average Error: 9.7 → 0.1
Time: 8.0m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -111.2811882400747:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 111.29608611545548:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left((\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) + \left(\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{x}\right))_* + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -111.2811882400747:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 111.29608611545548:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left((\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) + \left(\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{x}\right))_* + \frac{1}{x - 1}\right)\\

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

\end{array}
double f(double x) {
        double r57791463 = 1.0;
        double r57791464 = x;
        double r57791465 = r57791464 + r57791463;
        double r57791466 = r57791463 / r57791465;
        double r57791467 = 2.0;
        double r57791468 = r57791467 / r57791464;
        double r57791469 = r57791466 - r57791468;
        double r57791470 = r57791464 - r57791463;
        double r57791471 = r57791463 / r57791470;
        double r57791472 = r57791469 + r57791471;
        return r57791472;
}
↓
double f(double x) {
        double r57791473 = x;
        double r57791474 = -111.2811882400747;
        bool r57791475 = r57791473 <= r57791474;
        double r57791476 = 2.0;
        double r57791477 = 7.0;
        double r57791478 = pow(r57791473, r57791477);
        double r57791479 = r57791476 / r57791478;
        double r57791480 = 5.0;
        double r57791481 = pow(r57791473, r57791480);
        double r57791482 = r57791476 / r57791481;
        double r57791483 = r57791476 / r57791473;
        double r57791484 = r57791473 * r57791473;
        double r57791485 = r57791483 / r57791484;
        double r57791486 = r57791482 + r57791485;
        double r57791487 = r57791479 + r57791486;
        double r57791488 = 111.29608611545548;
        bool r57791489 = r57791473 <= r57791488;
        double r57791490 = 1.0;
        double r57791491 = r57791490 + r57791473;
        double r57791492 = r57791490 / r57791491;
        double r57791493 = r57791492 - r57791483;
        double r57791494 = cbrt(r57791476);
        double r57791495 = r57791494 / r57791473;
        double r57791496 = -r57791495;
        double r57791497 = r57791494 * r57791494;
        double r57791498 = r57791497 * r57791495;
        double r57791499 = fma(r57791496, r57791497, r57791498);
        double r57791500 = r57791473 - r57791490;
        double r57791501 = r57791490 / r57791500;
        double r57791502 = r57791499 + r57791501;
        double r57791503 = r57791493 + r57791502;
        double r57791504 = r57791483 / r57791473;
        double r57791505 = r57791504 / r57791473;
        double r57791506 = r57791505 + r57791482;
        double r57791507 = r57791479 + r57791506;
        double r57791508 = r57791489 ? r57791503 : r57791507;
        double r57791509 = r57791475 ? r57791487 : r57791508;
        return r57791509;
}

Error

Bits error versus x

Target

Original9.7
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -111.2811882400747

    1. Initial program 19.0

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

    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

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

    if -111.2811882400747 < x < 111.29608611545548

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-cube-cbrt1.9

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

    5. Applied times-frac2.2

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

    6. Applied *-un-lft-identity2.2

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

    7. Applied *-un-lft-identity2.2

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

    8. Applied times-frac2.2

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

    9. Applied prod-diff2.2

      \[\leadsto \color{blue}{\left((\left(\frac{1}{1}\right) \cdot \left(\frac{1}{x + 1}\right) + \left(-\frac{\sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right))_* + (\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right) + \left(\frac{\sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right))_*\right)} + \frac{1}{x - 1}\]

    10. Applied associate-+l+2.2

      \[\leadsto \color{blue}{(\left(\frac{1}{1}\right) \cdot \left(\frac{1}{x + 1}\right) + \left(-\frac{\sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right))_* + \left((\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right) + \left(\frac{\sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right))_* + \frac{1}{x - 1}\right)}\]

    11. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{1}{1 + x} - \frac{2}{x}\right)} + \left((\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right) + \left(\frac{\sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right))_* + \frac{1}{x - 1}\right)\]

    if 111.29608611545548 < x

    1. Initial program 19.9

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

    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

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

    5. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -111.2811882400747:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 111.29608611545548:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left((\left(-\frac{\sqrt[3]{2}}{x}\right) \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) + \left(\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{x}\right))_* + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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))))