Average Error: 16.2 → 3.1
Time: 3.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998268807:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \frac{1}{\alpha} \cdot \left(\left(\frac{4}{\alpha} - 2\right) - \frac{8}{{\alpha}^{2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + 1 \cdot 1}}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998268807:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \frac{1}{\alpha} \cdot \left(\left(\frac{4}{\alpha} - 2\right) - \frac{8}{{\alpha}^{2}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + 1 \cdot 1}}{2}\\

\end{array}
double code(double alpha, double beta) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999999998268807)) {
		VAR = ((((cbrt(beta) * cbrt(beta)) / (((alpha + beta) + 2.0) / cbrt(beta))) - ((1.0 / alpha) * (((4.0 / alpha) - 2.0) - (8.0 / pow(alpha, 2.0))))) / 2.0);
	} else {
		VAR = (((pow(((beta - alpha) / ((alpha + beta) + 2.0)), 3.0) + pow(1.0, 3.0)) / ((((beta - alpha) / ((alpha + beta) + 2.0)) * (((beta - alpha) / ((alpha + beta) + 2.0)) - 1.0)) + (1.0 * 1.0))) / 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 (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999999998268807

    1. Initial program 60.0

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

    3. Applied div-sub60.0

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

    4. Applied associate-+l-58.1

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

    6. Applied add-cube-cbrt58.1

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

    7. Applied associate-/l*58.1

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

    8. Taylor expanded around inf 11.0

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

    9. Simplified11.0

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

    if -0.9999999998268807 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.2

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

    3. Applied flip3-+0.3

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

    4. Simplified0.3

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -0.9999999998268807:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\sqrt[3]{\beta}}} - \frac{1}{\alpha} \cdot \left(\left(\frac{4}{\alpha} - 2\right) - \frac{8}{{\alpha}^{2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} + {1}^{3}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + 1 \cdot 1}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))