Average Error: 16.3 → 3.3
Time: 4.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \le -1:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}} + 1}{2}\\ \end{array}\]
\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 -1:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \mathsf{fma}\left(4, \frac{1}{{\alpha}^{2}}, -\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}} + 1}{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 ((((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))) <= -1.0)) {
		VAR = ((double) (((double) (((double) (beta / ((double) (((double) (alpha + beta)) + 2.0)))) - ((double) fma(4.0, ((double) (1.0 / ((double) pow(alpha, 2.0)))), ((double) -(((double) fma(2.0, ((double) (1.0 / alpha)), ((double) (8.0 * ((double) (1.0 / ((double) pow(alpha, 3.0)))))))))))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) cbrt(((double) pow(((double) (((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0)))), 3.0)))) + 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)) < -1.0

    1. Initial program 60.6

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

    3. Applied div-sub60.6

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

    4. Applied associate-+l-58.7

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

    5. Taylor expanded around inf 11.0

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

    6. Simplified11.0

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

    if -1.0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.6

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

    3. Applied add-cbrt-cube15.3

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

    4. Applied add-cbrt-cube18.6

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

    5. Applied cbrt-undiv18.6

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

    6. Simplified0.6

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))