Average Error: 16.2 → 5.9
Time: 15.4s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 18187930964.3279305:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\beta + 2\right) + \alpha}{\beta - \alpha}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 18187930964.3279305:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\beta + 2\right) + \alpha}{\beta - \alpha}} + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\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 <= 18187930964.32793)) {
		VAR = (((double) ((1.0 / (((double) (((double) (beta + 2.0)) + alpha)) / ((double) (beta - alpha)))) + 1.0)) / 2.0);
	} else {
		VAR = (((double) ((beta / ((double) (((double) (alpha + beta)) + 2.0))) - ((double) (((double) ((4.0 / ((double) (alpha * alpha))) - (8.0 / ((double) pow(alpha, 3.0))))) - (2.0 / 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 < 18187930964.3279305

    1. Initial program 0.1

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

    3. Applied clear-num0.2

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

    4. Simplified0.2

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

    if 18187930964.3279305 < alpha

    1. Initial program 50.4

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

    3. Applied div-sub50.4

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

    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. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 18187930964.3279305:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\beta + 2\right) + \alpha}{\beta - \alpha}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))