Average Error: 24.4 → 11.2
Time: 23.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.72289282377475551167372926245308152867 \cdot 10^{218}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.72289282377475551167372926245308152867 \cdot 10^{218}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) + 1\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r85543 = alpha;
        double r85544 = beta;
        double r85545 = r85543 + r85544;
        double r85546 = r85544 - r85543;
        double r85547 = r85545 * r85546;
        double r85548 = 2.0;
        double r85549 = i;
        double r85550 = r85548 * r85549;
        double r85551 = r85545 + r85550;
        double r85552 = r85547 / r85551;
        double r85553 = r85551 + r85548;
        double r85554 = r85552 / r85553;
        double r85555 = 1.0;
        double r85556 = r85554 + r85555;
        double r85557 = r85556 / r85548;
        return r85557;
}
↓
double f(double alpha, double beta, double i) {
        double r85558 = alpha;
        double r85559 = 3.7228928237747555e+218;
        bool r85560 = r85558 <= r85559;
        double r85561 = beta;
        double r85562 = r85558 + r85561;
        double r85563 = r85561 - r85558;
        double r85564 = cbrt(r85563);
        double r85565 = r85564 * r85564;
        double r85566 = 2.0;
        double r85567 = i;
        double r85568 = r85566 * r85567;
        double r85569 = r85562 + r85568;
        double r85570 = r85569 + r85566;
        double r85571 = cbrt(r85570);
        double r85572 = r85565 / r85571;
        double r85573 = r85572 / r85571;
        double r85574 = r85564 / r85569;
        double r85575 = r85571 * r85571;
        double r85576 = cbrt(r85575);
        double r85577 = cbrt(r85571);
        double r85578 = r85576 * r85577;
        double r85579 = r85574 / r85578;
        double r85580 = r85573 * r85579;
        double r85581 = r85562 * r85580;
        double r85582 = 1.0;
        double r85583 = r85581 + r85582;
        double r85584 = 3.0;
        double r85585 = pow(r85583, r85584);
        double r85586 = cbrt(r85585);
        double r85587 = r85586 / r85566;
        double r85588 = 1.0;
        double r85589 = r85588 / r85558;
        double r85590 = r85566 * r85589;
        double r85591 = 8.0;
        double r85592 = pow(r85558, r85584);
        double r85593 = r85588 / r85592;
        double r85594 = r85591 * r85593;
        double r85595 = r85590 + r85594;
        double r85596 = 4.0;
        double r85597 = 2.0;
        double r85598 = pow(r85558, r85597);
        double r85599 = r85588 / r85598;
        double r85600 = r85596 * r85599;
        double r85601 = r85595 - r85600;
        double r85602 = r85601 / r85566;
        double r85603 = r85560 ? r85587 : r85602;
        return r85603;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.7228928237747555e+218

    1. Initial program 20.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity20.2

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

    4. Applied *-un-lft-identity20.2

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

    5. Applied times-frac7.9

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

    6. Applied times-frac7.9

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

    7. Simplified7.9

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube7.9

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

    10. Simplified7.9

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{3}}}}{2}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt8.0

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

    13. Applied *-un-lft-identity8.0

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

    14. Applied add-cube-cbrt8.0

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

    15. Applied times-frac7.9

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

    16. Applied times-frac8.0

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

    17. Simplified8.0

      \[\leadsto \frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\color{blue}{\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1\right)}^{3}}}{2}\]
    18. Using strategy rm

    19. Applied add-cube-cbrt7.9

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

    20. Applied cbrt-prod7.9

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

    if 3.7228928237747555e+218 < alpha

    1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Taylor expanded around inf 42.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.72289282377475551167372926245308152867 \cdot 10^{218}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))