Average Error: 23.8 → 10.9
Time: 6.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.078461969447068 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]
\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 \leq 6.078461969447068 \cdot 10^{+166}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{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 6.078461969447068e+166)
   (/
    (cbrt
     (pow
      (+
       (*
        (+ alpha beta)
        (/
         (/
          (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
          (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
         (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
       1.0)
      3.0))
    2.0)
   (/
    (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
    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 <= 6.078461969447068e+166) {
		tmp = cbrt(pow((((alpha + beta) * ((((beta - alpha) / ((alpha + beta) + (2.0 * i))) / sqrt(2.0 + ((alpha + beta) + (2.0 * i)))) / sqrt(2.0 + ((alpha + beta) + (2.0 * i))))) + 1.0), 3.0)) / 2.0;
	} else {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 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 2 regimes

  2. if alpha < 6.0784619694470683e166

    1. Initial program 16.6

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

    3. Applied *-un-lft-identity_binary6416.6

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

    4. Applied *-un-lft-identity_binary6416.6

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

    5. Applied times-frac_binary645.6

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

    6. Applied times-frac_binary645.6

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

    7. Simplified5.6

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

    8. Simplified5.6

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

    10. Applied add-cbrt-cube_binary645.6

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

    11. Simplified5.6

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

    13. Applied add-sqr-sqrt_binary645.6

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

    14. Applied *-un-lft-identity_binary645.6

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

    15. Applied *-un-lft-identity_binary645.6

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

    16. Applied times-frac_binary645.6

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

    17. Applied times-frac_binary645.6

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

    18. Applied associate-*r*_binary645.6

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

    19. Simplified5.6

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

    21. Applied div-inv_binary645.6

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

    22. Applied associate-*l*_binary645.6

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

    23. Simplified5.6

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

    if 6.0784619694470683e166 < 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. Taylor expanded around inf 41.3

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

    3. Simplified41.3

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.078461969447068 \cdot 10^{+166}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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))