Average Error: 16.3 → 5.7
Time: 3.8s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 389840199.85676336:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) - \frac{\alpha \cdot \frac{\frac{\alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)} - 1 \cdot 1}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 389840199.85676336:\\
\;\;\;\;\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) - \frac{\alpha \cdot \frac{\frac{\alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)} - 1 \cdot 1}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\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 <= 389840199.85676336)) {
		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) (alpha * ((double) (((double) (alpha / ((double) (beta + ((double) (alpha + 2.0)))))) / ((double) (beta + ((double) (alpha + 2.0)))))))) - ((double) (1.0 * 1.0)))) / ((double) (((double) (alpha / ((double) (beta + ((double) (alpha + 2.0)))))) + 1.0)))))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (beta / ((double) (2.0 + ((double) (alpha + beta)))))) - ((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 < 389840199.85676336

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    5. Simplified0.1

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

    7. Applied add-cube-cbrt0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    11. Applied flip--0.1

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

    12. Simplified0.1

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

    if 389840199.85676336 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.8

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

    5. Simplified48.8

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

    6. Taylor expanded around inf 17.3

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

    7. Simplified17.3

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 389840199.85676336:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \left(\sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}} \cdot \sqrt[3]{\frac{\beta}{\alpha + \left(\beta + 2\right)}}\right) - \frac{\alpha \cdot \frac{\frac{\alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)} - 1 \cdot 1}{\frac{\alpha}{\beta + \left(\alpha + 2\right)} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))