Average Error: 15.9 → 6.2
Time: 4.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.9114798649862423 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\log \left(e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}}\right)}{1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 5.9114798649862423 \cdot 10^{26}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\log \left(e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}}\right)}{1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\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 <= 5.911479864986242e+26)) {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (alpha + beta)) + 2.0)))) - ((double) (((double) log(((double) exp(((double) (((double) pow(((double) (alpha / ((double) (((double) (alpha + beta)) + 2.0)))), 3.0)) - ((double) pow(1.0, 3.0)))))))) / ((double) (((double) (1.0 * ((double) (1.0 + ((double) (alpha / ((double) (((double) (alpha + beta)) + 2.0)))))))) + ((double) (((double) (alpha / ((double) (((double) (alpha + beta)) + 2.0)))) * ((double) (alpha / ((double) (((double) (alpha + beta)) + 2.0)))))))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (alpha + beta)) + 2.0)))) - ((double) (((double) (4.0 * ((double) (1.0 / ((double) pow(alpha, 2.0)))))) - ((double) (((double) (2.0 * ((double) (1.0 / alpha)))) + ((double) (8.0 * ((double) (1.0 / ((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 < 5.911479864986242e+26

    1. Initial program 1.1

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

    3. Applied div-sub1.1

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

    4. Applied associate-+l-1.1

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

    6. Applied flip3--1.1

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

    7. Simplified1.1

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

    9. Applied add-log-exp1.1

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

    10. Applied add-log-exp1.1

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

    11. Applied diff-log1.1

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

    12. Simplified1.1

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

    if 5.911479864986242e+26 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.4

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

    4. Applied associate-+l-48.9

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

    5. Taylor expanded around inf 18.0

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.9114798649862423 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\log \left(e^{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}}\right)}{1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))