Average Error: 3.8 → 2.9
Time: 59.9s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.9107204585361316 \cdot 10^{197}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \sqrt{0.5} + \left(\left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \end{array}\]
\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 1.9107204585361316 \cdot 10^{197}:\\
\;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\alpha + \left(\beta + 3\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\beta \cdot \sqrt{0.5} + \left(\left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\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 <= 1.9107204585361316e+197)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) (alpha + beta)) + ((double) (alpha * beta)))) + 1.0)) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0)))))) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0)))))) / ((double) (alpha + ((double) (beta + 3.0))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) (beta * ((double) sqrt(0.5)))) + ((double) (((double) (((double) (0.75 * ((double) (alpha * ((double) sqrt(0.5)))))) + ((double) (1.0 * ((double) sqrt(0.5)))))) - ((double) (0.125 * ((double) (beta / ((double) sqrt(0.5)))))))))) / ((double) sqrt(((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0)))))))) / ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0)))))) / ((double) (1.0 + ((double) (((double) (alpha + beta)) + ((double) (1.0 * 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 < 1.9107204585361316e197

    1. Initial program 1.8

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

    2. Taylor expanded around 0 1.8

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

    3. Simplified1.8

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

    if 1.9107204585361316e197 < alpha

    1. Initial program 18.5

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

    3. Applied add-sqr-sqrt18.5

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

    4. Applied associate-/r*18.5

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

    5. Simplified18.5

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

    6. Taylor expanded around 0 10.8

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right)\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    7. Simplified10.8

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{0.5} \cdot \beta + \left(\left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.9107204585361316 \cdot 10^{197}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta \cdot \sqrt{0.5} + \left(\left(0.75 \cdot \left(\alpha \cdot \sqrt{0.5}\right) + 1 \cdot \sqrt{0.5}\right) - 0.125 \cdot \frac{\beta}{\sqrt{0.5}}\right)}{\sqrt{\left(\alpha + \beta\right) + 1 \cdot 2}}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \end{array}\]

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)))