Average Error: 23.9 → 11.3
Time: 36.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.6401313163804604 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.6401313163804604 \cdot 10^{+197}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r3309555 = alpha;
        double r3309556 = beta;
        double r3309557 = r3309555 + r3309556;
        double r3309558 = r3309556 - r3309555;
        double r3309559 = r3309557 * r3309558;
        double r3309560 = 2.0;
        double r3309561 = i;
        double r3309562 = r3309560 * r3309561;
        double r3309563 = r3309557 + r3309562;
        double r3309564 = r3309559 / r3309563;
        double r3309565 = 2.0;
        double r3309566 = r3309563 + r3309565;
        double r3309567 = r3309564 / r3309566;
        double r3309568 = 1.0;
        double r3309569 = r3309567 + r3309568;
        double r3309570 = r3309569 / r3309565;
        return r3309570;
}
↓
double f(double alpha, double beta, double i) {
        double r3309571 = alpha;
        double r3309572 = 4.6401313163804604e+197;
        bool r3309573 = r3309571 <= r3309572;
        double r3309574 = beta;
        double r3309575 = r3309574 + r3309571;
        double r3309576 = r3309574 - r3309571;
        double r3309577 = cbrt(r3309576);
        double r3309578 = r3309577 * r3309577;
        double r3309579 = i;
        double r3309580 = 2.0;
        double r3309581 = r3309579 * r3309580;
        double r3309582 = r3309581 + r3309575;
        double r3309583 = r3309577 / r3309582;
        double r3309584 = 2.0;
        double r3309585 = r3309584 + r3309582;
        double r3309586 = cbrt(r3309585);
        double r3309587 = r3309583 / r3309586;
        double r3309588 = r3309578 * r3309587;
        double r3309589 = r3309586 * r3309586;
        double r3309590 = r3309588 / r3309589;
        double r3309591 = r3309575 * r3309590;
        double r3309592 = 1.0;
        double r3309593 = r3309591 + r3309592;
        double r3309594 = r3309593 * r3309593;
        double r3309595 = r3309594 * r3309593;
        double r3309596 = cbrt(r3309595);
        double r3309597 = r3309596 / r3309584;
        double r3309598 = 8.0;
        double r3309599 = r3309571 * r3309571;
        double r3309600 = r3309571 * r3309599;
        double r3309601 = r3309598 / r3309600;
        double r3309602 = 4.0;
        double r3309603 = r3309602 / r3309599;
        double r3309604 = r3309601 - r3309603;
        double r3309605 = r3309584 / r3309571;
        double r3309606 = r3309604 + r3309605;
        double r3309607 = r3309606 / r3309584;
        double r3309608 = r3309573 ? r3309597 : r3309607;
        return r3309608;
}

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 < 4.6401313163804604e+197

    1. Initial program 18.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity18.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity18.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac7.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac7.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Simplified7.3

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt7.4

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    10. Applied *-un-lft-identity7.4

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    11. Applied add-cube-cbrt7.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    12. Applied times-frac7.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    13. Applied times-frac7.3

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]

    14. Applied associate-*r*7.3

      \[\leadsto \frac{\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
    15. Using strategy rm

    16. Applied add-cbrt-cube7.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]

    17. Simplified7.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]

    if 4.6401313163804604e+197 < alpha

    1. Initial program 63.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 41.8

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified41.8

      \[\leadsto \frac{\color{blue}{\left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.6401313163804604 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}{\sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \sqrt[3]{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} + 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))