Octave 3.8, jcobi/2

Average Error: 24.0 → 13.2

Time: 28.8s

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 9.76982660809475993097637758871956020598 \cdot 10^{62}:\\ \;\;\;\;\frac{1 + \sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.266980258015373608283608969920590572301 \cdot 10^{166}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)}^{3} + {1}^{3}}{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) + \left(1 \cdot 1 - \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot 1\right)}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4385099 = alpha;
        double r4385100 = beta;
        double r4385101 = r4385099 + r4385100;
        double r4385102 = r4385100 - r4385099;
        double r4385103 = r4385101 * r4385102;
        double r4385104 = 2.0;
        double r4385105 = i;
        double r4385106 = r4385104 * r4385105;
        double r4385107 = r4385101 + r4385106;
        double r4385108 = r4385103 / r4385107;
        double r4385109 = r4385107 + r4385104;
        double r4385110 = r4385108 / r4385109;
        double r4385111 = 1.0;
        double r4385112 = r4385110 + r4385111;
        double r4385113 = r4385112 / r4385104;
        return r4385113;
}

↓

double f(double alpha, double beta, double i) {
        double r4385114 = alpha;
        double r4385115 = 9.76982660809476e+62;
        bool r4385116 = r4385114 <= r4385115;
        double r4385117 = 1.0;
        double r4385118 = beta;
        double r4385119 = r4385118 + r4385114;
        double r4385120 = r4385118 - r4385114;
        double r4385121 = i;
        double r4385122 = 2.0;
        double r4385123 = r4385121 * r4385122;
        double r4385124 = r4385123 + r4385119;
        double r4385125 = r4385120 / r4385124;
        double r4385126 = r4385125 * r4385125;
        double r4385127 = r4385125 * r4385126;
        double r4385128 = cbrt(r4385127);
        double r4385129 = r4385122 + r4385124;
        double r4385130 = r4385128 / r4385129;
        double r4385131 = r4385119 * r4385130;
        double r4385132 = r4385131 * r4385131;
        double r4385133 = r4385131 * r4385132;
        double r4385134 = cbrt(r4385133);
        double r4385135 = r4385117 + r4385134;
        double r4385136 = r4385135 / r4385122;
        double r4385137 = 2.2669802580153736e+166;
        bool r4385138 = r4385114 <= r4385137;
        double r4385139 = r4385122 / r4385114;
        double r4385140 = 4.0;
        double r4385141 = r4385114 * r4385114;
        double r4385142 = r4385140 / r4385141;
        double r4385143 = r4385139 - r4385142;
        double r4385144 = 8.0;
        double r4385145 = r4385114 * r4385141;
        double r4385146 = r4385144 / r4385145;
        double r4385147 = r4385143 + r4385146;
        double r4385148 = r4385147 / r4385122;
        double r4385149 = 3.0;
        double r4385150 = pow(r4385131, r4385149);
        double r4385151 = pow(r4385117, r4385149);
        double r4385152 = r4385150 + r4385151;
        double r4385153 = r4385117 * r4385117;
        double r4385154 = r4385131 * r4385117;
        double r4385155 = r4385153 - r4385154;
        double r4385156 = r4385132 + r4385155;
        double r4385157 = r4385152 / r4385156;
        double r4385158 = r4385157 / r4385122;
        double r4385159 = r4385138 ? r4385148 : r4385158;
        double r4385160 = r4385116 ? r4385136 : r4385159;
        return r4385160;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 9.76982660809476e+62

   1. Initial program 12.1

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

   3. Applied *-un-lft-identity 12.1

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

   4. Applied *-un-lft-identity 12.1

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

   5. Applied times-frac 1.6

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

   6. Applied times-frac 1.6

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

   7. Simplified 1.6

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

   9. Applied add-cbrt-cube 1.6

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

   11. Applied add-cbrt-cube 1.6

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

   if 9.76982660809476e+62 < alpha < 2.2669802580153736e+166

   1. Initial program 45.7

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

   3. Applied *-un-lft-identity 45.7

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

   4. Applied *-un-lft-identity 45.7

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

   5. Applied times-frac 32.3

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

   6. Applied times-frac 32.2

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

   7. Simplified 32.2

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

   9. Applied add-log-exp 32.2

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

   10. Taylor expanded around inf 39.9

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

   11. Simplified 39.9

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

   if 2.2669802580153736e+166 < 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 47.4

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

   6. Applied times-frac 47.4

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

   7. Simplified 47.4

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

   9. Applied add-cbrt-cube 47.4

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

   11. Applied flip3-+ 47.5

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.76982660809475993097637758871956020598 \cdot 10^{62}:\\ \;\;\;\;\frac{1 + \sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.266980258015373608283608969920590572301 \cdot 10^{166}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right)}^{3} + {1}^{3}}{\left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) + \left(1 \cdot 1 - \left(\left(\beta + \alpha\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}\right) \cdot 1\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))