Average Error: 3.7 → 2.3
Time: 25.7s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.3992159678067332 \cdot 10^{192}:\\ \;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 4.3992159678067332 \cdot 10^{192}:\\
\;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 <= 4.399215967806733e+192)) {
		VAR = ((double) (((double) (((double) sqrt(((double) (((double) (alpha + ((double) (beta + ((double) (((double) (alpha * beta)) + 1.0)))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))))) / ((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))))) * ((double) (((double) sqrt(((double) (((double) (alpha + ((double) (beta + ((double) (((double) (alpha * beta)) + 1.0)))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))))) / ((double) (((double) sqrt(((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))))) * ((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (1.0 * 2.0))))))))))))));
	} else {
		VAR = 0.0;
	}
	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 < 4.3992159678067332e192

    1. Initial program 1.7

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

    3. Applied *-un-lft-identity1.7

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

    4. Applied add-sqr-sqrt2.2

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

    5. Applied add-sqr-sqrt1.8

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

    6. Applied times-frac1.8

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

    7. Applied times-frac1.8

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

    8. Simplified1.8

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

    9. Simplified1.8

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

    if 4.3992159678067332e192 < alpha

    1. Initial program 17.6

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

    2. Taylor expanded around inf 5.6

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.3992159678067332 \cdot 10^{192}:\\ \;\;\;\;\frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)}} \cdot \frac{\sqrt{\frac{\alpha + \left(\beta + \left(\alpha \cdot \beta + 1\right)\right)}{\alpha + \left(\beta + 1 \cdot 2\right)}}}{\sqrt{\alpha + \left(\beta + 1 \cdot 2\right)} \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))