Average Error: 24.0 → 13.5
Time: 8.0s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.70176466292540245 \cdot 10^{137}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\alpha + \beta}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log 2 + \left(1 \cdot \frac{1}{\beta} + \log \left(\frac{1}{\alpha}\right)\right)\right) - \left(2 \cdot \frac{1}{\alpha} + \log \left(\frac{1}{\beta}\right)\right)}}{2}\\ \end{array}\]
\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 2.70176466292540245 \cdot 10^{137}:\\
\;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\alpha + \beta}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\left(\log 2 + \left(1 \cdot \frac{1}{\beta} + \log \left(\frac{1}{\alpha}\right)\right)\right) - \left(2 \cdot \frac{1}{\alpha} + \log \left(\frac{1}{\beta}\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 <= 2.7017646629254024e+137)) {
		VAR = ((double) (((double) exp(((double) cbrt(((double) pow(((double) log(((double) (((double) (((double) (((double) (alpha + beta)) / ((double) (((double) (alpha + ((double) (((double) (2.0 * i)) + beta)))) / ((double) (beta - alpha)))))) * ((double) (1.0 / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * i)))) + 2.0)))))) + 1.0)))), 3.0)))))) / 2.0));
	} else {
		VAR = ((double) (((double) exp(((double) (((double) (((double) log(2.0)) + ((double) (((double) (1.0 * ((double) (1.0 / beta)))) + ((double) log(((double) (1.0 / alpha)))))))) - ((double) (((double) (2.0 * ((double) (1.0 / alpha)))) + ((double) log(((double) (1.0 / beta)))))))))) / 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 < 2.70176466292540245e137

    1. Initial program 15.4

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

    3. Applied associate-/l*4.5

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

    4. Simplified4.5

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

    6. Applied div-inv4.5

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

    8. Applied add-exp-log4.5

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

    10. Applied add-cbrt-cube4.5

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

    11. Simplified4.5

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

    if 2.70176466292540245e137 < alpha

    1. Initial program 62.9

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

    3. Applied associate-/l*46.1

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

    4. Simplified46.1

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

    6. Applied div-inv46.1

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

    8. Applied add-exp-log46.1

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

    9. Taylor expanded around inf 54.2

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.70176466292540245 \cdot 10^{137}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\alpha + \beta}{\frac{\alpha + \left(2 \cdot i + \beta\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log 2 + \left(1 \cdot \frac{1}{\beta} + \log \left(\frac{1}{\alpha}\right)\right)\right) - \left(2 \cdot \frac{1}{\alpha} + \log \left(\frac{1}{\beta}\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(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))