Average Error: 24.4 → 11.2
Time: 13.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.1912244088489427 \cdot 10^{177}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \sqrt[3]{{\left(\left(-\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \left(\alpha + \beta\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \left(\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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 6.1912244088489427 \cdot 10^{177}:\\
\;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \sqrt[3]{{\left(\left(-\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \left(\alpha + \beta\right) + 1\right)}^{3}}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r115810 = alpha;
        double r115811 = beta;
        double r115812 = r115810 + r115811;
        double r115813 = r115811 - r115810;
        double r115814 = r115812 * r115813;
        double r115815 = 2.0;
        double r115816 = i;
        double r115817 = r115815 * r115816;
        double r115818 = r115812 + r115817;
        double r115819 = r115814 / r115818;
        double r115820 = r115818 + r115815;
        double r115821 = r115819 / r115820;
        double r115822 = 1.0;
        double r115823 = r115821 + r115822;
        double r115824 = r115823 / r115815;
        return r115824;
}
↓
double f(double alpha, double beta, double i) {
        double r115825 = alpha;
        double r115826 = 6.191224408848943e+177;
        bool r115827 = r115825 <= r115826;
        double r115828 = beta;
        double r115829 = r115825 + r115828;
        double r115830 = 2.0;
        double r115831 = i;
        double r115832 = r115830 * r115831;
        double r115833 = r115829 + r115832;
        double r115834 = r115828 / r115833;
        double r115835 = r115833 + r115830;
        double r115836 = r115834 / r115835;
        double r115837 = r115829 * r115836;
        double r115838 = r115825 / r115833;
        double r115839 = r115838 / r115835;
        double r115840 = -r115839;
        double r115841 = r115840 * r115829;
        double r115842 = 1.0;
        double r115843 = r115841 + r115842;
        double r115844 = 3.0;
        double r115845 = pow(r115843, r115844);
        double r115846 = cbrt(r115845);
        double r115847 = r115837 + r115846;
        double r115848 = r115847 / r115830;
        double r115849 = 1.0;
        double r115850 = r115849 / r115825;
        double r115851 = r115830 * r115850;
        double r115852 = 8.0;
        double r115853 = pow(r115825, r115844);
        double r115854 = r115849 / r115853;
        double r115855 = r115852 * r115854;
        double r115856 = r115851 + r115855;
        double r115857 = 4.0;
        double r115858 = 2.0;
        double r115859 = pow(r115825, r115858);
        double r115860 = r115849 / r115859;
        double r115861 = r115857 * r115860;
        double r115862 = r115856 - r115861;
        double r115863 = r115837 + r115862;
        double r115864 = r115863 / r115830;
        double r115865 = r115827 ? r115848 : r115864;
        return r115865;
}

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 < 6.191224408848943e+177

    1. Initial program 17.9

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

    3. Applied *-un-lft-identity17.9

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

    4. Applied *-un-lft-identity17.9

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

    5. Applied times-frac6.7

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

    6. Applied times-frac6.6

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

    7. Simplified6.6

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

    9. Applied div-sub6.6

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

    10. Applied div-sub6.6

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

    12. Applied sub-neg6.6

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

    13. Applied distribute-lft-in6.6

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

    14. Applied associate-+l+6.5

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

    15. Simplified6.5

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

    17. Applied add-cbrt-cube6.5

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

    18. Simplified6.5

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

    if 6.191224408848943e+177 < 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. Using strategy rm

    3. Applied *-un-lft-identity64.0

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

    4. Applied *-un-lft-identity64.0

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

    5. Applied times-frac49.0

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

    6. Applied times-frac49.0

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

    7. Simplified49.0

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

    9. Applied div-sub49.0

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

    10. Applied div-sub49.0

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

    12. Applied sub-neg49.0

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

    13. Applied distribute-lft-in49.0

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

    14. Applied associate-+l+48.6

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

    15. Simplified48.6

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

    16. Taylor expanded around inf 39.3

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.1912244088489427 \cdot 10^{177}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \sqrt[3]{{\left(\left(-\frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right) \cdot \left(\alpha + \beta\right) + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + \left(\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))