Average Error: 3.7 → 1.3
Time: 7.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)} \leq 0.08892208126538512:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \end{array}\]
\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}\;\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}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)} \leq 0.08892208126538512:\\
\;\;\;\;\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}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\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 (((((((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) (1.0 + ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0))))))) <= 0.08892208126538512)) {
		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) (1.0 + ((double) (((double) (alpha + beta)) + ((double) (1.0 * 2.0)))))));
	} else {
		VAR = (((double) (1.0 + ((double) ((2.0 / ((double) (alpha * alpha))) - (1.0 / alpha))))) / ((double) (((double) (alpha + ((double) (beta + ((double) (1.0 * 2.0)))))) * ((double) (alpha + ((double) (beta + ((double) (1.0 + ((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 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)) < 0.088922081265385117

    1. Initial program 0.1

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

    if 0.088922081265385117 < (/ (/ (/ (+ (+ (+ 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))

    1. Initial program 63.6

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

    3. Applied div-inv63.6

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

    4. Applied associate-/l*63.6

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

    5. Simplified63.6

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

    6. Taylor expanded around inf 21.5

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

    7. Simplified21.5

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)} \leq 0.08892208126538512:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 1 \cdot 2}}{\left(\alpha + \beta\right) + 1 \cdot 2}}{1 + \left(\left(\alpha + \beta\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{2}{\alpha \cdot \alpha} - \frac{1}{\alpha}\right)}{\left(\alpha + \left(\beta + 1 \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + \left(1 + 1 \cdot 2\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))