Average Error: 16.3 → 6.1
Time: 3.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.101691238264905 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}^{3}}{\frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(\frac{\beta}{\alpha + \left(\beta + 2\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 9.101691238264905 \cdot 10^{+21}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}^{3}}{\frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(\frac{\beta}{\alpha + \left(\beta + 2\right)} - 1\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
double code(double alpha, double beta) {
	return (((double) ((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0))) + 1.0)) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 9.101691238264905e+21)) {
		VAR = ((((double) (((double) pow((beta / ((double) (alpha + ((double) (beta + 2.0))))), 3.0)) - ((double) pow(((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) - 1.0)), 3.0)))) / ((double) (((double) ((beta / ((double) (alpha + ((double) (beta + 2.0))))) * (beta / ((double) (alpha + ((double) (beta + 2.0))))))) + ((double) (((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) - 1.0)) * ((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) + ((double) ((beta / ((double) (alpha + ((double) (beta + 2.0))))) - 1.0))))))))) / 2.0);
	} else {
		VAR = (((double) ((beta / ((double) (2.0 + ((double) (alpha + beta))))) + ((double) (((double) ((2.0 / alpha) + (8.0 / ((double) pow(alpha, 3.0))))) - (4.0 / ((double) (alpha * alpha))))))) / 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 < 9.1016912382649046e21

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Applied associate-+l-0.7

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

    5. Simplified0.7

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

    7. Applied flip3--0.7

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

    8. Simplified0.7

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

    9. Simplified0.7

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

    if 9.1016912382649046e21 < alpha

    1. Initial program 50.9

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

    3. Applied div-sub50.9

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

    4. Applied associate-+l-49.2

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

    5. Simplified49.2

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

    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 simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.101691238264905 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta}{\alpha + \left(\beta + 2\right)}\right)}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right)}^{3}}{\frac{\beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} - 1\right) \cdot \left(\frac{\alpha}{\alpha + \left(\beta + 2\right)} + \left(\frac{\beta}{\alpha + \left(\beta + 2\right)} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))