Average Error: 24.2 → 12.4
Time: 27.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.093291601906169352105417227277985833818 \cdot 10^{100}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right)}^{3}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 1.093291601906169352105417227277985833818 \cdot 10^{100}:\\
\;\;\;\;\frac{\sqrt[3]{{\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}}\right)}^{3}} + 1}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r106134 = alpha;
        double r106135 = beta;
        double r106136 = r106134 + r106135;
        double r106137 = r106135 - r106134;
        double r106138 = r106136 * r106137;
        double r106139 = 2.0;
        double r106140 = i;
        double r106141 = r106139 * r106140;
        double r106142 = r106136 + r106141;
        double r106143 = r106138 / r106142;
        double r106144 = r106142 + r106139;
        double r106145 = r106143 / r106144;
        double r106146 = 1.0;
        double r106147 = r106145 + r106146;
        double r106148 = r106147 / r106139;
        return r106148;
}
↓
double f(double alpha, double beta, double i) {
        double r106149 = alpha;
        double r106150 = 1.0932916019061694e+100;
        bool r106151 = r106149 <= r106150;
        double r106152 = beta;
        double r106153 = r106149 + r106152;
        double r106154 = 2.0;
        double r106155 = i;
        double r106156 = r106154 * r106155;
        double r106157 = r106153 + r106156;
        double r106158 = r106157 + r106154;
        double r106159 = sqrt(r106158);
        double r106160 = r106153 / r106159;
        double r106161 = r106152 - r106149;
        double r106162 = r106161 / r106157;
        double r106163 = r106162 / r106159;
        double r106164 = r106160 * r106163;
        double r106165 = 3.0;
        double r106166 = pow(r106164, r106165);
        double r106167 = cbrt(r106166);
        double r106168 = 1.0;
        double r106169 = r106167 + r106168;
        double r106170 = r106169 / r106154;
        double r106171 = r106154 / r106149;
        double r106172 = 8.0;
        double r106173 = pow(r106149, r106165);
        double r106174 = r106172 / r106173;
        double r106175 = r106171 + r106174;
        double r106176 = 4.0;
        double r106177 = r106149 * r106149;
        double r106178 = r106176 / r106177;
        double r106179 = r106175 - r106178;
        double r106180 = r106179 / r106154;
        double r106181 = r106151 ? r106170 : r106180;
        return r106181;
}

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 < 1.0932916019061694e+100

    1. Initial program 13.7

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

    3. Applied add-sqr-sqrt13.7

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

    4. Applied *-un-lft-identity13.7

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

    5. Applied times-frac3.2

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

    6. Applied times-frac3.2

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

    7. Simplified3.2

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

    9. Applied add-cbrt-cube8.2

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

    10. Applied add-cbrt-cube18.1

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

    11. Applied add-cbrt-cube22.8

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

    12. Applied cbrt-undiv22.8

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

    13. Applied cbrt-undiv22.8

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

    14. Applied add-cbrt-cube22.8

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

    15. Applied add-cbrt-cube22.8

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

    16. Applied cbrt-undiv22.8

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

    17. Applied cbrt-unprod22.8

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

    18. Simplified3.2

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

    if 1.0932916019061694e+100 < alpha

    1. Initial program 58.7

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

    3. Applied add-sqr-sqrt58.8

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

    4. Applied *-un-lft-identity58.8

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

    5. Applied times-frac43.6

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

    6. Applied times-frac43.5

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

    7. Simplified43.5

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

    8. Taylor expanded around inf 42.6

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

    9. Simplified42.6

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.093291601906169352105417227277985833818 \cdot 10^{100}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right)}^{3}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))