Average Error: 23.0 → 11.2
Time: 4.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.1721802170861066 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + {\alpha}^{-2} \cdot \left(\frac{8}{\alpha} - 4\right)}{2}\\ \end{array}\]
\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 2.1721802170861066 \cdot 10^{+114}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)} + 1\right)}^{3}}}{2}\\

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

\end{array}
double code(double alpha, double beta, double i) {
	return (((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 <= 2.1721802170861066e+114)) {
		VAR = (((double) cbrt(((double) pow(((double) (((double) (((double) (alpha + beta)) * ((((double) (beta - alpha)) / ((double) (alpha + ((double) (beta + ((double) (2.0 * i))))))) / ((double) (alpha + ((double) (beta + ((double) (2.0 + ((double) (2.0 * i))))))))))) + 1.0)), 3.0)))) / 2.0);
	} else {
		VAR = (((double) ((2.0 / alpha) + ((double) (((double) pow(alpha, -2.0)) * ((double) ((8.0 / alpha) - 4.0)))))) / 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 < 2.17218021708610656e114

    1. Initial program 13.4

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

    2. Simplified10.7

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

    4. Applied add-cube-cbrt10.7

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

    5. Applied times-frac3.5

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

    6. Simplified3.5

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

    7. Simplified3.5

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

    9. Applied add-cbrt-cube3.5

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

    10. Simplified3.4

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

    if 2.17218021708610656e114 < alpha

    1. Initial program 60.2

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

    2. Simplified50.8

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

    4. Applied add-cube-cbrt50.8

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

    5. Applied times-frac45.4

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

    6. Simplified45.4

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

    7. Simplified45.4

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

    8. Taylor expanded around inf 41.4

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

    9. Simplified41.4

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.1721802170861066 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2 \cdot i\right)}}{\alpha + \left(\beta + \left(2 + 2 \cdot i\right)\right)} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + {\alpha}^{-2} \cdot \left(\frac{8}{\alpha} - 4\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))