Average Error: 3.7 → 0.3
Time: 9.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 18400915820535212:\\ \;\;\;\;\frac{\frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1} \cdot \frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\beta + -1}{\left(\beta + \alpha\right) + 2}}}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]
\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}\;\beta \leq 18400915820535212:\\
\;\;\;\;\frac{\frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1} \cdot \frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\beta + -1}{\left(\beta + \alpha\right) + 2}}}}{\alpha + \left(\beta + 3\right)}\\

\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 18400915820535212.0)
   (/
    (/
     (*
      (sqrt (+ (+ (+ beta alpha) (* beta alpha)) 1.0))
      (/
       (sqrt (+ (+ (+ beta alpha) (* beta alpha)) 1.0))
       (+ (+ beta alpha) 2.0)))
     (+ (+ beta alpha) 2.0))
    (+ 1.0 (+ (+ beta alpha) 2.0)))
   (/
    (*
     (+ alpha 1.0)
     (/
      1.0
      (/ (+ (+ beta alpha) 2.0) (/ (+ beta -1.0) (+ (+ beta alpha) 2.0)))))
    (+ alpha (+ beta 3.0)))))
double code(double alpha, double beta) {
	return (((((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);
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 18400915820535212.0) {
		tmp = ((sqrt(((beta + alpha) + (beta * alpha)) + 1.0) * (sqrt(((beta + alpha) + (beta * alpha)) + 1.0) / ((beta + alpha) + 2.0))) / ((beta + alpha) + 2.0)) / (1.0 + ((beta + alpha) + 2.0));
	} else {
		tmp = ((alpha + 1.0) * (1.0 / (((beta + alpha) + 2.0) / ((beta + -1.0) / ((beta + alpha) + 2.0))))) / (alpha + (beta + 3.0));
	}
	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 < 18400915820535212

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity_binary640.2

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

    4. Applied add-sqr-sqrt_binary640.2

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

    5. Applied times-frac_binary640.2

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

    6. Simplified0.2

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

    7. Simplified0.2

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

    if 18400915820535212 < beta

    1. Initial program 11.4

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

    2. Simplified5.8

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

    4. Applied clear-num_binary645.8

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

    5. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 18400915820535212:\\ \;\;\;\;\frac{\frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1} \cdot \frac{\sqrt{\left(\left(\beta + \alpha\right) + \beta \cdot \alpha\right) + 1}}{\left(\beta + \alpha\right) + 2}}{\left(\beta + \alpha\right) + 2}}{1 + \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\beta + -1}{\left(\beta + \alpha\right) + 2}}}}{\alpha + \left(\beta + 3\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(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)))