Average Error: 3.5 → 2.1
Time: 5.4s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.064453308972152 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 1 \cdot 2}}{\left(\beta + \alpha\right) + 1 \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 \leq 6.064453308972152 \cdot 10^{+195}:\\
\;\;\;\;\frac{\frac{\left(\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 1 \cdot 2}}{\left(\beta + \alpha\right) + 1 \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + 1 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ 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)))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 6.064453308972152e+195)
   (/
    (/
     (*
      (+ (+ (+ beta alpha) (* beta alpha)) 1.0)
      (/ 1.0 (+ (+ beta alpha) (* 1.0 2.0))))
     (+ (+ beta alpha) (* 1.0 2.0)))
    (+ 1.0 (+ (+ beta alpha) (* 1.0 2.0))))
   0.0))
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 tmp;
	if ((beta <= 6.064453308972152e+195)) {
		tmp = ((((double) (((double) (((double) (((double) (beta + alpha)) + ((double) (beta * alpha)))) + 1.0)) * (1.0 / ((double) (((double) (beta + alpha)) + ((double) (1.0 * 2.0))))))) / ((double) (((double) (beta + alpha)) + ((double) (1.0 * 2.0))))) / ((double) (1.0 + ((double) (((double) (beta + alpha)) + ((double) (1.0 * 2.0)))))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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 < 6.06445330897215189e195

    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 div-inv_binary641.5

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

    4. Simplified1.5

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

    if 6.06445330897215189e195 < beta

    1. Initial program 17.2

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

    2. Taylor expanded around inf 6.3

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.064453308972152 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 1 \cdot 2}}{\left(\beta + \alpha\right) + 1 \cdot 2}}{1 + \left(\left(\beta + \alpha\right) + 1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))