Average Error: 3.5 → 1.2
Time: 6.1s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.7632698702504957 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)} \cdot \frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}}{\alpha + \left(\beta + 1 \cdot 2\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 4.7632698702504957 \cdot 10^{156}:\\
\;\;\;\;\frac{\frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)} \cdot \frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}}{\alpha + \left(\beta + 1 \cdot 2\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\

\end{array}
double code(double alpha, double beta) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0))));
}
↓
double code(double alpha, double beta) {
	double VAR;
	if ((alpha <= 4.763269870250496e+156)) {
		VAR = ((double) (((double) (((double) (((double) sqrt(((double) (alpha + ((double) (beta + ((double) (((double) (alpha * beta)) + 1.0)))))))) * ((double) (((double) sqrt(((double) (alpha + ((double) (beta + ((double) (((double) (alpha * beta)) + 1.0)))))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta)))))) / ((double) (1.0 + ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))))));
	} else {
		VAR = ((double) (1.0 / ((double) (((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))) * ((double) (2.0 + ((double) (((double) (beta / alpha)) + ((double) (alpha / beta))))))))));
	}
	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 < 4.7632698702504957e156

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity1.2

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

    4. Applied add-sqr-sqrt1.3

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

    5. Applied times-frac1.3

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

    6. Simplified1.3

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

    7. Simplified1.3

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

    if 4.7632698702504957e156 < alpha

    1. Initial program 14.5

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

    3. Applied clear-num15.0

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

    4. Simplified15.0

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

    5. Taylor expanded around inf 0.8

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.7632698702504957 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)} \cdot \frac{\sqrt{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}}{\alpha + \left(\beta + 1 \cdot 2\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))