Average Error: 23.9 → 12.6
Time: 15.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.800621932398946 \cdot 10^{+75}:\\ \;\;\;\;\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{\sqrt[3]{1}}{\sqrt{\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{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1228122934029902 \cdot 10^{+171}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + \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)}}} \cdot \left(\sqrt[3]{1 + \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)}}} \cdot \sqrt[3]{1 + \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)}}}\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 \leq 1.800621932398946 \cdot 10^{+75}:\\
\;\;\;\;\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{\sqrt[3]{1}}{\sqrt{\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{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right)}^{3}}}{2}\\

\mathbf{elif}\;\alpha \leq 3.1228122934029902 \cdot 10^{+171}:\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{1 + \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)}}} \cdot \left(\sqrt[3]{1 + \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)}}} \cdot \sqrt[3]{1 + \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)}}}\right)}{2}\\

\end{array}
(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 1.800621932398946e+75)
   (/
    (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)))))))
         (/ (cbrt 1.0) (sqrt (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))))
        (/
         (/ (cbrt (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
         (sqrt (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))
      3.0))
    2.0)
   (if (<= alpha 3.1228122934029902e+171)
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (*
       (cbrt
        (+
         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)))))))
          (/
           (/ (cbrt (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
           (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))
       (*
        (cbrt
         (+
          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)))))))
           (/
            (/ (cbrt (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
            (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))
        (cbrt
         (+
          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)))))))
           (/
            (/ (cbrt (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
            (cbrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))))))
      2.0))))
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 <= 1.800621932398946e+75) {
		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)))))) * (cbrt(1.0) / sqrt(cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * ((cbrt(beta - alpha) / ((alpha + beta) + (2.0 * i))) / sqrt(cbrt(2.0 + ((alpha + beta) + (2.0 * i))))))), 3.0)) / 2.0;
	} else if (alpha <= 3.1228122934029902e+171) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = (cbrt(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)))))) * ((cbrt(beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * (cbrt(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)))))) * ((cbrt(beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))) * cbrt(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)))))) * ((cbrt(beta - alpha) / ((alpha + beta) + (2.0 * i))) / cbrt(2.0 + ((alpha + beta) + (2.0 * i)))))))) / 2.0;
	}
	return tmp;
}

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 3 regimes

  2. if alpha < 1.8006219323989459e75

    1. Initial program 13.2

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

    3. Applied *-un-lft-identity_binary6413.2

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

    4. Applied *-un-lft-identity_binary6413.2

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

    5. Applied times-frac_binary642.2

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

    6. Applied times-frac_binary642.2

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

    7. Simplified2.2

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

    8. Simplified2.2

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

    10. Applied add-cube-cbrt_binary642.4

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

    11. Applied *-un-lft-identity_binary642.4

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

    12. Applied add-cube-cbrt_binary642.2

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

    13. Applied times-frac_binary642.2

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

    14. Applied times-frac_binary642.2

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

    15. Applied associate-*r*_binary642.2

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

    16. Simplified2.2

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

    18. Applied add-cbrt-cube_binary642.3

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

    19. Simplified2.3

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

    21. Applied add-sqr-sqrt_binary642.3

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

    22. Applied *-un-lft-identity_binary642.3

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

    23. Applied *-un-lft-identity_binary642.3

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

    24. Applied cbrt-prod_binary642.3

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

    25. Applied times-frac_binary642.3

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

    26. Applied times-frac_binary642.3

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

    27. Applied associate-*r*_binary642.3

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

    28. Simplified2.3

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

    if 1.8006219323989459e75 < alpha < 3.12281229340299024e171

    1. Initial program 47.1

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

    2. Taylor expanded around inf 39.5

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

    3. Simplified39.5

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

    if 3.12281229340299024e171 < 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_binary6464.0

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

    4. Applied *-un-lft-identity_binary6464.0

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

    5. Applied times-frac_binary6447.8

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

    6. Applied times-frac_binary6447.9

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

    7. Simplified47.9

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

    8. Simplified47.9

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

    10. Applied add-cube-cbrt_binary6448.6

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

    11. Applied *-un-lft-identity_binary6448.6

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

    12. Applied add-cube-cbrt_binary6448.2

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

    13. Applied times-frac_binary6448.2

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

    14. Applied times-frac_binary6448.1

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

    15. Applied associate-*r*_binary6448.1

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

    16. Simplified48.1

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

    18. Applied add-cube-cbrt_binary6448.2

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.800621932398946 \cdot 10^{+75}:\\ \;\;\;\;\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{\sqrt[3]{1}}{\sqrt{\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{\sqrt[3]{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1228122934029902 \cdot 10^{+171}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + \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)}}} \cdot \left(\sqrt[3]{1 + \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)}}} \cdot \sqrt[3]{1 + \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)}}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020274 
(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))