Octave 3.8, jcobi/2

Average Error: 23.8 → 11.5

Time: 16.0s

Precision: binary64

\[\alpha > -1 \land \beta > -1 \land i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.449160688824685 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.765248146023335 \cdot 10^{+135} \lor \neg \left(\alpha \leq 1.0163560453529555 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right)}^{3}}}{2}\\ \end{array}\]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 4.449160688824685e+97)
   (/
    (+
     (*
      (+ alpha beta)
      (/
       (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
     1.0)
    2.0)
   (if (or (<= alpha 3.765248146023335e+135)
           (not (<= alpha 1.0163560453529555e+192)))
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (cbrt
       (pow
        (+
         1.0
         (*
          (*
           (*
            (+ alpha beta)
            (/
             (* (cbrt (- beta alpha)) (cbrt (- beta alpha)))
             (*
              (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
              (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))))
           (/
            (/ 1.0 (sqrt (+ (+ alpha beta) (* 2.0 i))))
            (sqrt (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))))
          (/
           (/ (cbrt (- beta alpha)) (sqrt (+ (+ alpha beta) (* 2.0 i))))
           (sqrt (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))
        3.0))
      2.0))))

C

double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 4.449160688824685e+97) {
		tmp = (((alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i))))) + 1.0) / 2.0;
	} else if ((alpha <= 3.765248146023335e+135) || !(alpha <= 1.0163560453529555e+192)) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = cbrt(pow((1.0 + ((((alpha + beta) * ((cbrt(beta - alpha) * cbrt(beta - alpha)) / (cbrt(2.0 + ((alpha + beta) + (2.0 * i))) * cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * ((1.0 / sqrt((alpha + beta) + (2.0 * i))) / sqrt(cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * ((cbrt(beta - alpha) / sqrt((alpha + beta) + (2.0 * i))) / sqrt(cbrt(2.0 + ((alpha + beta) + (2.0 * i))))))), 3.0)) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 4.4491606888246854e97

   1. Initial program 13.6

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

   3. Applied *-un-lft-identity_binary64_1799 13.6

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

   4. Applied *-un-lft-identity_binary64_1799 13.6

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

   5. Applied times-frac_binary64_1805 3.1

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

   6. Applied times-frac_binary64_1805 3.1

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

   7. Simplified 3.1

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

   8. Simplified 3.1

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

   if 4.4491606888246854e97 < alpha < 3.76524814602333502e135 or 1.01635604535295545e192 < alpha

   1. Initial program 58.9

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

   2. Taylor expanded around inf 41.2

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

   3. Simplified 41.2

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

   if 3.76524814602333502e135 < alpha < 1.01635604535295545e192

   1. Initial program 58.2

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

   3. Applied *-un-lft-identity_binary64_1799 58.2

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

   4. Applied *-un-lft-identity_binary64_1799 58.2

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

   5. Applied times-frac_binary64_1805 38.4

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

   6. Applied times-frac_binary64_1805 38.3

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

   7. Simplified 38.3

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

   8. Simplified 38.3

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

   10. Applied add-cbrt-cube_binary64_1835 38.3

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

   11. Simplified 38.3

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

   13. Applied add-cube-cbrt_binary64_1834 38.3

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

   14. Applied *-un-lft-identity_binary64_1799 38.3

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

   15. Applied add-cube-cbrt_binary64_1834 38.5

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

   16. Applied times-frac_binary64_1805 38.4

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

   17. Applied times-frac_binary64_1805 38.4

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

   18. Applied associate-*r*_binary64_1739 38.4

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

   19. Simplified 38.4

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

   21. Applied add-sqr-sqrt_binary64_1821 38.5

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

   22. Applied add-sqr-sqrt_binary64_1821 38.5

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

   23. Applied *-un-lft-identity_binary64_1799 38.5

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

   24. Applied times-frac_binary64_1805 38.5

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

   25. Applied times-frac_binary64_1805 38.5

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

   26. Applied associate-*r*_binary64_1739 38.5

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

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.449160688824685 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 3.765248146023335 \cdot 10^{+135} \lor \neg \left(\alpha \leq 1.0163560453529555 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :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))