Average Error: 23.9 → 11.1
Time: 1.3m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.277127249944820259658219751143281845182 \cdot 10^{219}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8}{\alpha \cdot \alpha}}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 3.277127249944820259658219751143281845182 \cdot 10^{219}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r4907649 = alpha;
        double r4907650 = beta;
        double r4907651 = r4907649 + r4907650;
        double r4907652 = r4907650 - r4907649;
        double r4907653 = r4907651 * r4907652;
        double r4907654 = 2.0;
        double r4907655 = i;
        double r4907656 = r4907654 * r4907655;
        double r4907657 = r4907651 + r4907656;
        double r4907658 = r4907653 / r4907657;
        double r4907659 = r4907657 + r4907654;
        double r4907660 = r4907658 / r4907659;
        double r4907661 = 1.0;
        double r4907662 = r4907660 + r4907661;
        double r4907663 = r4907662 / r4907654;
        return r4907663;
}
↓
double f(double alpha, double beta, double i) {
        double r4907664 = alpha;
        double r4907665 = 3.27712724994482e+219;
        bool r4907666 = r4907664 <= r4907665;
        double r4907667 = beta;
        double r4907668 = r4907667 + r4907664;
        double r4907669 = 2.0;
        double r4907670 = i;
        double r4907671 = r4907669 * r4907670;
        double r4907672 = r4907671 + r4907668;
        double r4907673 = r4907672 + r4907669;
        double r4907674 = sqrt(r4907673);
        double r4907675 = r4907668 / r4907674;
        double r4907676 = r4907667 - r4907664;
        double r4907677 = r4907676 / r4907672;
        double r4907678 = r4907677 / r4907674;
        double r4907679 = r4907675 * r4907678;
        double r4907680 = 1.0;
        double r4907681 = r4907679 + r4907680;
        double r4907682 = r4907681 * r4907681;
        double r4907683 = r4907681 * r4907682;
        double r4907684 = cbrt(r4907683);
        double r4907685 = r4907684 / r4907669;
        double r4907686 = 8.0;
        double r4907687 = r4907664 * r4907664;
        double r4907688 = r4907686 / r4907687;
        double r4907689 = r4907688 / r4907664;
        double r4907690 = 4.0;
        double r4907691 = r4907690 / r4907687;
        double r4907692 = r4907689 - r4907691;
        double r4907693 = r4907669 / r4907664;
        double r4907694 = r4907692 + r4907693;
        double r4907695 = r4907694 / r4907669;
        double r4907696 = r4907666 ? r4907685 : r4907695;
        return r4907696;
}

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 < 3.27712724994482e+219

    1. Initial program 19.4

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

    3. Applied add-sqr-sqrt19.4

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

    4. Applied *-un-lft-identity19.4

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

    5. Applied times-frac7.8

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

    6. Applied times-frac7.8

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

    8. Applied add-cbrt-cube7.7

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

    9. Simplified7.7

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

    if 3.27712724994482e+219 < 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 41.2

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    3. Simplified41.2

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.277127249944820259658219751143281845182 \cdot 10^{219}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{8}{\alpha \cdot \alpha}}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))