Octave 3.8, jcobi/2

Average Error: 24.1 → 12.9

Time: 33.2s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.442037061945920645369018327902562528872 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 1.157236759942197475904943781835715492446 \cdot 10^{199}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\left(\sqrt[3]{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r96092 = alpha;
        double r96093 = beta;
        double r96094 = r96092 + r96093;
        double r96095 = r96093 - r96092;
        double r96096 = r96094 * r96095;
        double r96097 = 2.0;
        double r96098 = i;
        double r96099 = r96097 * r96098;
        double r96100 = r96094 + r96099;
        double r96101 = r96096 / r96100;
        double r96102 = r96100 + r96097;
        double r96103 = r96101 / r96102;
        double r96104 = 1.0;
        double r96105 = r96103 + r96104;
        double r96106 = r96105 / r96097;
        return r96106;
}

↓

double f(double alpha, double beta, double i) {
        double r96107 = alpha;
        double r96108 = 1.4420370619459206e+124;
        bool r96109 = r96107 <= r96108;
        double r96110 = 1.0;
        double r96111 = beta;
        double r96112 = r96107 + r96111;
        double r96113 = r96111 - r96107;
        double r96114 = 2.0;
        double r96115 = i;
        double r96116 = r96114 * r96115;
        double r96117 = r96112 + r96116;
        double r96118 = r96113 / r96117;
        double r96119 = cbrt(r96118);
        double r96120 = r96119 * r96119;
        double r96121 = cbrt(r96113);
        double r96122 = 1.0;
        double r96123 = r96122 / r96117;
        double r96124 = cbrt(r96123);
        double r96125 = r96121 * r96124;
        double r96126 = r96120 * r96125;
        double r96127 = r96117 + r96114;
        double r96128 = r96126 / r96127;
        double r96129 = r96112 * r96128;
        double r96130 = r96110 + r96129;
        double r96131 = 3.0;
        double r96132 = pow(r96130, r96131);
        double r96133 = cbrt(r96132);
        double r96134 = r96133 / r96114;
        double r96135 = 1.1572367599421975e+199;
        bool r96136 = r96107 <= r96135;
        double r96137 = 8.0;
        double r96138 = pow(r96107, r96131);
        double r96139 = r96137 / r96138;
        double r96140 = r96114 / r96107;
        double r96141 = r96139 + r96140;
        double r96142 = 4.0;
        double r96143 = r96107 * r96107;
        double r96144 = r96142 / r96143;
        double r96145 = r96141 - r96144;
        double r96146 = r96145 / r96114;
        double r96147 = sqrt(r96117);
        double r96148 = r96122 / r96147;
        double r96149 = cbrt(r96148);
        double r96150 = r96113 / r96147;
        double r96151 = cbrt(r96150);
        double r96152 = r96149 * r96151;
        double r96153 = r96152 * r96119;
        double r96154 = sqrt(r96127);
        double r96155 = r96153 / r96154;
        double r96156 = r96119 / r96154;
        double r96157 = r96155 * r96156;
        double r96158 = r96112 * r96157;
        double r96159 = r96158 + r96110;
        double r96160 = r96159 / r96114;
        double r96161 = r96136 ? r96146 : r96160;
        double r96162 = r96109 ? r96134 : r96161;
        return r96162;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 1.4420370619459206e+124

   1. Initial program 14.7

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

   3. Applied *-un-lft-identity 14.7

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

   4. Applied *-un-lft-identity 14.7

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

   5. Applied times-frac 4.3

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

   6. Applied times-frac 4.3

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

   7. Simplified 4.3

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

   9. Applied add-sqr-sqrt 4.3

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

   10. Applied add-cube-cbrt 4.3

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

   11. Applied times-frac 4.3

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

   13. Applied div-inv 4.3

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

   14. Applied cbrt-prod 4.4

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

   16. Applied add-cbrt-cube 4.3

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

   17. Simplified 4.3

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

   if 1.4420370619459206e+124 < alpha < 1.1572367599421975e+199

   1. Initial program 57.1

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

   2. Taylor expanded around inf 40.6

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

   3. Simplified 40.6

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

   if 1.1572367599421975e+199 < 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-identity 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 50.7

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

   6. Applied times-frac 50.8

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

   7. Simplified 50.8

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

   9. Applied add-sqr-sqrt 51.2

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

   10. Applied add-cube-cbrt 51.2

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

   11. Applied times-frac 51.4

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

   13. Applied add-sqr-sqrt 51.5

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

   14. Applied *-un-lft-identity 51.5

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

   15. Applied times-frac 51.5

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

   16. Applied cbrt-prod 51.4

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.442037061945920645369018327902562528872 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 1.157236759942197475904943781835715492446 \cdot 10^{199}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\left(\sqrt[3]{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))