Average Error: 15.5 → 5.8
Time: 3.6s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 114971.60375650562:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\beta + \left(\alpha + 2\right)}}}\right) + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 114971.60375650562:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\beta + \left(\alpha + 2\right)}}}\right) + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\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(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\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 <= 114971.60375650562)) {
		VAR = ((double) (((double) (((double) (((double) (((double) cbrt(((double) (beta / ((double) (alpha + ((double) (beta + 2.0)))))))) * ((double) cbrt(((double) (beta / ((double) (alpha + ((double) (beta + 2.0)))))))))) * ((double) log(((double) exp(((double) cbrt(((double) (beta / ((double) (beta + ((double) (alpha + 2.0)))))))))))))) + ((double) (1.0 - ((double) (alpha / ((double) (beta + ((double) (alpha + 2.0)))))))))) / 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) (((double) (2.0 / alpha)) + ((double) (8.0 / ((double) pow(alpha, 3.0)))))) - ((double) (4.0 / ((double) (alpha * alpha)))))))) / 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 < 114971.60375650562

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Applied associate-+l-0.0

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

    5. Simplified0.0

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

    7. Applied add-cube-cbrt0.0

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    11. Applied add-log-exp0.0

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

    12. Simplified0.0

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

    if 114971.60375650562 < alpha

    1. Initial program 48.8

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

    3. Applied div-sub48.8

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

    4. Applied associate-+l-47.3

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

    5. Simplified47.3

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

    7. Applied add-cube-cbrt47.3

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

    8. Simplified47.3

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

    9. Simplified47.3

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

    10. Taylor expanded around inf 18.3

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

    11. Simplified18.3

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 114971.60375650562:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) \cdot \log \left(e^{\sqrt[3]{\frac{\beta}{\beta + \left(\alpha + 2\right)}}}\right) + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))