Average Error: 3.8 → 1.1
Time: 1.7m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 4.03573551752264356 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}\;\beta \le 4.03573551752264356 \cdot 10^{156}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

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

\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 ((beta <= 4.0357355175226436e+156)) {
		VAR = ((double) (((double) (((double) (1.0 / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (beta * alpha)))) + 1.0)))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0))));
	} else {
		VAR = ((double) (((double) (((double) (1.0 / ((double) (((double) (((double) (1.0 / alpha)) + ((double) (1.0 / beta)))) - ((double) (1.0 / ((double) pow(alpha, 2.0)))))))) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))))) / ((double) (((double) (((double) (alpha + beta)) + ((double) (2.0 * 1.0)))) + 1.0))));
	}
	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 beta < 4.0357355175226436e+156

    1. Initial program 1.3

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

    3. Applied clear-num1.3

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

    if 4.0357355175226436e+156 < beta

    1. Initial program 16.2

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

    3. Applied clear-num16.2

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

    4. Taylor expanded around inf 0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 4.03573551752264356 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020131 
(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)))