Average Error: 3.5 → 1.3
Time: 12.9s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.30054433857241922 \cdot 10^{164}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\alpha + \left(\beta + 1 \cdot 2\right)}{\alpha + \left(\beta + \left(1 + \alpha \cdot \beta\right)\right)}}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\beta} + \left(\frac{1}{\alpha} - \frac{1}{\alpha \cdot \alpha}\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]
\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 7.30054433857241922 \cdot 10^{164}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{\alpha + \left(\beta + 1 \cdot 2\right)}{\alpha + \left(\beta + \left(1 + \alpha \cdot \beta\right)\right)}}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\beta} + \left(\frac{1}{\alpha} - \frac{1}{\alpha \cdot \alpha}\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\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 <= 7.300544338572419e+164)) {
		VAR = (((1.0 / (((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))) / ((double) (alpha + ((double) (beta + ((double) (1.0 + ((double) (alpha * beta)))))))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))) / ((double) (1.0 + ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta)))))));
	} else {
		VAR = (((1.0 / ((double) ((1.0 / beta) + ((double) ((1.0 / alpha) - (1.0 / ((double) (alpha * alpha)))))))) / ((double) (((double) (1.0 * 2.0)) + ((double) (alpha + beta))))) / ((double) (1.0 + ((double) (((double) (1.0 * 2.0)) + ((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 < 7.30054433857241922e164

    1. Initial program 1.5

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

    3. Applied clear-num1.5

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

    4. Simplified1.5

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

    if 7.30054433857241922e164 < alpha

    1. Initial program 15.0

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

    3. Applied clear-num15.0

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

    4. Simplified15.0

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.30054433857241922 \cdot 10^{164}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{\alpha + \left(\beta + 1 \cdot 2\right)}{\alpha + \left(\beta + \left(1 + \alpha \cdot \beta\right)\right)}}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\beta} + \left(\frac{1}{\alpha} - \frac{1}{\alpha \cdot \alpha}\right)}}{1 \cdot 2 + \left(\alpha + \beta\right)}}{1 + \left(1 \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))