Average Error: 24.2 → 11.5
Time: 8.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.13159331103118907 \cdot 10^{144}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]
\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 \le 6.13159331103118907 \cdot 10^{144}:\\
\;\;\;\;\frac{\left(\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\

\end{array}
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 temp;
	if ((alpha <= 6.131593311031189e+144)) {
		temp = (((((((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) * sqrt((1.0 / (((alpha + beta) + (2.0 * i)) + 2.0)))) * sqrt(sqrt((1.0 / (((alpha + beta) + (2.0 * i)) + 2.0))))) * sqrt(sqrt((1.0 / (((alpha + beta) + (2.0 * i)) + 2.0))))) + 1.0) / 2.0);
	} else {
		temp = ((((8.0 * (1.0 / pow(alpha, 3.0))) - (4.0 * (1.0 / pow(alpha, 2.0)))) + (2.0 / alpha)) / 2.0);
	}
	return temp;
}

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 < 6.131593311031189e+144

    1. Initial program 15.7

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

    3. Applied *-un-lft-identity15.7

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

    4. Applied times-frac4.8

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

    5. Simplified4.8

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

    7. Applied div-inv4.8

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

    9. Applied add-sqr-sqrt4.8

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

    10. Applied associate-*r*4.8

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

    12. Applied add-sqr-sqrt4.8

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

    13. Applied sqrt-prod4.8

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

    14. Applied associate-*r*4.8

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

    if 6.131593311031189e+144 < alpha

    1. Initial program 63.2

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

    3. Applied *-un-lft-identity63.2

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

    4. Applied times-frac45.6

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

    5. Simplified45.6

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

    6. Taylor expanded around inf 42.7

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

    7. Simplified42.7

      \[\leadsto \frac{\color{blue}{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.13159331103118907 \cdot 10^{144}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))