Average Error: 3.4 → 1.1
Time: 7.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.370230282252863 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]
\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 \leq 6.370230282252863 \cdot 10^{+161}:\\
\;\;\;\;\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{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) (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 <= 6.370230282252863e+161)) {
		VAR = (((((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 = (1.0 / ((double) (((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))) * ((double) (2.0 + ((double) ((beta / alpha) + (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 < 6.37023028225286271e161

    1. Initial program Error: 1.2 bits

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

    2. Taylor expanded around 0 Error: 1.2 bits

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

    3. SimplifiedError: 1.2 bits

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

    if 6.37023028225286271e161 < alpha

    1. Initial program Error: 15.4 bits

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

    2. Taylor expanded around 0 Error: 15.4 bits

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

    3. SimplifiedError: 15.4 bits

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

    5. Applied clear-numError: 15.5 bits

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

    6. SimplifiedError: 15.5 bits

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

    7. Taylor expanded around inf Error: 0.7 bits

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

  4. Final simplificationError: 1.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6.370230282252863 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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)))