Average Error: 16.2 → 6.1
Time: 3.6s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 31220247461162088:\\ \;\;\;\;\frac{\frac{\frac{\beta}{\sqrt{\alpha + \left(\beta + 2\right)}}}{\sqrt{2 + \left(\alpha + \beta\right)}} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}}\right) - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 31220247461162088:\\
\;\;\;\;\frac{\frac{\frac{\beta}{\sqrt{\alpha + \left(\beta + 2\right)}}}{\sqrt{2 + \left(\alpha + \beta\right)}} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}}\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 <= 3.122024746116209e+16)) {
		VAR = ((double) (((double) (((double) (((double) (beta / ((double) sqrt(((double) (alpha + ((double) (beta + 2.0)))))))) / ((double) sqrt(((double) (2.0 + ((double) (alpha + beta)))))))) - ((double) (((double) (alpha / ((double) (beta + ((double) (alpha + 2.0)))))) - 1.0)))) / 2.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) cbrt(beta)) / ((double) cbrt(((double) (alpha + ((double) (beta + 2.0)))))))) * ((double) (((double) (((double) cbrt(beta)) / ((double) cbrt(((double) (alpha + ((double) (beta + 2.0)))))))) * ((double) (((double) cbrt(beta)) / ((double) cbrt(((double) (alpha + ((double) (beta + 2.0)))))))))))) - ((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 < 31220247461162088

    1. Initial program 0.5

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

    3. Applied div-sub0.5

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

    4. Applied associate-+l-0.5

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

    5. Simplified0.5

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

    7. Applied add-sqr-sqrt0.5

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

    8. Applied associate-/r*0.5

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

    9. Simplified0.5

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

    if 31220247461162088 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.1

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

    4. Applied associate-+l-48.5

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

    5. Simplified48.5

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

    7. Applied add-cube-cbrt48.7

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

    8. Applied add-cube-cbrt48.5

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

    9. Applied times-frac48.5

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

    10. Simplified48.5

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

    11. Simplified48.5

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

    12. Taylor expanded around inf 18.2

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

    13. Simplified18.2

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 31220247461162088:\\ \;\;\;\;\frac{\frac{\frac{\beta}{\sqrt{\alpha + \left(\beta + 2\right)}}}{\sqrt{2 + \left(\alpha + \beta\right)}} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \left(\frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\alpha + \left(\beta + 2\right)}}\right) - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))