Average Error: 16.8 → 5.8
Time: 12.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 999297775.850383162:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\alpha}^{3} \cdot {\left(\frac{1}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 999297775.850383162:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\alpha}^{3} \cdot {\left(\frac{1}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\

\end{array}
double code(double alpha, double beta) {
	return ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0);
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 999297775.8503832)) {
		VAR = ((((beta * (((alpha / ((alpha + beta) + 2.0)) * (alpha / ((alpha + beta) + 2.0))) + ((1.0 * 1.0) + ((alpha / ((alpha + beta) + 2.0)) * 1.0)))) - (((alpha + beta) + 2.0) * ((pow(alpha, 3.0) * pow((1.0 / ((alpha + beta) + 2.0)), 3.0)) - pow(1.0, 3.0)))) / (((1.0 * (1.0 + (alpha / ((alpha + beta) + 2.0)))) + ((alpha / ((alpha + beta) + 2.0)) * (alpha / ((alpha + beta) + 2.0)))) * ((alpha + beta) + 2.0))) / 2.0);
	} else {
		VAR = (((beta / ((alpha + beta) + 2.0)) - ((4.0 * (1.0 / pow(alpha, 2.0))) - ((2.0 * (1.0 / alpha)) + (8.0 * (1.0 / 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 < 999297775.8503832

    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. Using strategy rm

    6. Applied flip3--0.2

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

    7. Applied frac-sub0.2

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

    8. Simplified0.2

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

    10. Applied div-inv0.2

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

    11. Applied unpow-prod-down0.2

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

    if 999297775.8503832 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.0

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

    4. Applied associate-+l-49.4

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

    5. Taylor expanded around inf 17.2

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 999297775.850383162:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} + \left(1 \cdot 1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot 1\right)\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\alpha}^{3} \cdot {\left(\frac{1}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{\left(1 \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \left(\left(\alpha + \beta\right) + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))