Average Error: 23.9 → 10.9
Time: 6.2s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.6630243519946244 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(1 + \frac{2 + 2 \cdot i}{\alpha + \beta}\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(1 + \frac{2 + 2 \cdot i}{\alpha + \beta}\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\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 \le 2.6630243519946244 \cdot 10^{156}:\\
\;\;\;\;\frac{\frac{1}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(1 + \frac{2 + 2 \cdot i}{\alpha + \beta}\right)} + 1}{2}\\

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

\end{array}
double code(double alpha, double beta, double i) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) * ((double) (beta - alpha)))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))) + 1.0)) / 2.0));
}
↓
double code(double alpha, double beta, double i) {
	double VAR;
	if ((alpha <= 2.6630243519946244e+156)) {
		VAR = ((double) (((double) (((double) (1.0 / ((double) (((double) (((double) (beta + ((double) (alpha + ((double) (2.0 * i)))))) / ((double) (beta - alpha)))) * ((double) (1.0 + ((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))))))))) + 1.0)) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (1.0 + ((double) (((double) (2.0 + ((double) (2.0 * i)))) / ((double) (alpha + beta)))))) * ((double) (alpha + ((double) (beta + ((double) (2.0 * i)))))))))) - ((double) (((double) (4.0 / ((double) (alpha * alpha)))) - ((double) (((double) (2.0 / alpha)) + ((double) (8.0 / ((double) pow(alpha, 3.0)))))))))) / 2.0));
	}
	return VAR;
}

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 < 2.6630243519946244e156

    1. Initial program 15.9

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

    2. Simplified5.3

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

    4. Applied clear-num5.3

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

    5. Simplified5.3

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

    if 2.6630243519946244e156 < 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. Simplified47.2

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

    4. Applied div-sub47.2

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

    5. Applied associate-+l-45.7

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

    6. Simplified45.7

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

    7. Taylor expanded around inf 38.9

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

    8. Simplified38.9

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.6630243519946244 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{1}{\frac{\beta + \left(\alpha + 2 \cdot i\right)}{\beta - \alpha} \cdot \left(1 + \frac{2 + 2 \cdot i}{\alpha + \beta}\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(1 + \frac{2 + 2 \cdot i}{\alpha + \beta}\right) \cdot \left(\alpha + \left(\beta + 2 \cdot i\right)\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))