Average Error: 16.3 → 6.0
Time: 4.9s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1550527369.59974074:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) - \left(\alpha + \left(\beta + 2\right)\right) \cdot \log \left(e^{\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2} + 1 \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1550527369.59974074:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) - \left(\alpha + \left(\beta + 2\right)\right) \cdot \log \left(e^{\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2} + 1 \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{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 <= 1550527369.5997407)) {
		VAR = ((double) (((double) (((double) (((double) (beta * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))) * ((double) (1.0 + ((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))))))))))) - ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) log(((double) exp(((double) (((double) (((double) pow(((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))), 2.0)) + ((double) (1.0 * ((double) (1.0 + ((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))))))))) * ((double) (((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))) - 1.0)))))))))))) / ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))) * ((double) (1.0 + ((double) (alpha / ((double) (alpha + ((double) (beta + 2.0)))))))))))))))) / 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 < 1550527369.59974074

    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 flip3--0.1

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

    8. Applied frac-sub0.1

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

    9. Simplified0.1

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

    10. Simplified0.1

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

    12. Applied add-log-exp0.1

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

    13. Applied add-log-exp0.1

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

    14. Applied diff-log0.1

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

    15. Simplified0.1

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

    17. Applied difference-cubes0.1

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

    18. Simplified0.1

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

    if 1550527369.59974074 < alpha

    1. Initial program 49.7

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

    3. Applied div-sub49.7

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

    4. Applied associate-+l-48.2

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

    5. Simplified48.2

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

    6. Taylor expanded around inf 18.0

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

    7. Simplified18.0

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1550527369.59974074:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) - \left(\alpha + \left(\beta + 2\right)\right) \cdot \log \left(e^{\left({\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{2} + 1 \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 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))