Average Error: 24.4 → 11.5
Time: 35.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.681747304865176575191504176776135729897 \cdot 10^{206}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 2.681747304865176575191504176776135729897 \cdot 10^{206}:\\
\;\;\;\;\frac{\frac{\beta + \alpha}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} + 1}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r5451763 = alpha;
        double r5451764 = beta;
        double r5451765 = r5451763 + r5451764;
        double r5451766 = r5451764 - r5451763;
        double r5451767 = r5451765 * r5451766;
        double r5451768 = 2.0;
        double r5451769 = i;
        double r5451770 = r5451768 * r5451769;
        double r5451771 = r5451765 + r5451770;
        double r5451772 = r5451767 / r5451771;
        double r5451773 = r5451771 + r5451768;
        double r5451774 = r5451772 / r5451773;
        double r5451775 = 1.0;
        double r5451776 = r5451774 + r5451775;
        double r5451777 = r5451776 / r5451768;
        return r5451777;
}
↓
double f(double alpha, double beta, double i) {
        double r5451778 = alpha;
        double r5451779 = 2.6817473048651766e+206;
        bool r5451780 = r5451778 <= r5451779;
        double r5451781 = beta;
        double r5451782 = r5451781 + r5451778;
        double r5451783 = 2.0;
        double r5451784 = i;
        double r5451785 = r5451783 * r5451784;
        double r5451786 = r5451782 + r5451785;
        double r5451787 = r5451783 + r5451786;
        double r5451788 = sqrt(r5451787);
        double r5451789 = r5451782 / r5451788;
        double r5451790 = r5451781 - r5451778;
        double r5451791 = r5451790 / r5451786;
        double r5451792 = r5451791 / r5451788;
        double r5451793 = r5451789 * r5451792;
        double r5451794 = 1.0;
        double r5451795 = r5451793 + r5451794;
        double r5451796 = r5451795 / r5451783;
        double r5451797 = 8.0;
        double r5451798 = r5451778 * r5451778;
        double r5451799 = r5451778 * r5451798;
        double r5451800 = r5451797 / r5451799;
        double r5451801 = 4.0;
        double r5451802 = r5451801 / r5451798;
        double r5451803 = r5451800 - r5451802;
        double r5451804 = r5451783 / r5451778;
        double r5451805 = r5451803 + r5451804;
        double r5451806 = r5451805 / r5451783;
        double r5451807 = r5451780 ? r5451796 : r5451806;
        return r5451807;
}

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 < 2.6817473048651766e+206

    1. Initial program 19.6

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

    3. Applied add-sqr-sqrt19.5

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

    4. Applied *-un-lft-identity19.5

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

    5. Applied times-frac7.9

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

    6. Applied times-frac7.9

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

    if 2.6817473048651766e+206 < 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 40.9

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

    3. Simplified40.9

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.681747304865176575191504176776135729897 \cdot 10^{206}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))