Average Error: 16.5 → 7.5
Time: 3.0s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.71478932058979567 \cdot 10^{69}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.71478932058979567 \cdot 10^{69}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\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) {
	return ((double) (((double) (((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))) + 1.0)) / 2.0));
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 2.7147893205897957e+69)) {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (alpha + beta)) + 2.0)))) - ((double) log(((double) exp(((double) (((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))) - 1.0)))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) cbrt(((double) (beta / ((double) (alpha + ((double) (beta + 2.0)))))))) * ((double) (((double) cbrt(((double) (beta / ((double) (alpha + ((double) (beta + 2.0)))))))) * ((double) cbrt(((double) (beta / ((double) (alpha + ((double) (beta + 2.0)))))))))))) - ((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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.71478932058979567e69

    1. Initial program 3.6

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

    3. Applied div-sub3.6

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

    4. Applied associate-+l-3.6

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

    5. Simplified3.6

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

    7. Applied add-log-exp3.6

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

    8. Applied add-log-exp3.6

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

    9. Applied diff-log3.6

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

    10. Simplified3.6

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

    if 2.71478932058979567e69 < alpha

    1. Initial program 52.0

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

    3. Applied div-sub52.0

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

    4. Applied associate-+l-50.1

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

    5. Simplified50.1

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

    7. Applied add-cube-cbrt50.1

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

    8. Simplified50.1

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

    9. Simplified50.1

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

    10. Taylor expanded around inf 18.2

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

    11. Simplified18.2

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.71478932058979567 \cdot 10^{69}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \left(e^{\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))