Average Error: 16.1 → 5.9
Time: 3.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 32596417.711401034:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left({1}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 32596417.711401034:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left({1}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\end{array}
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 32596417.711401034)
   (/
    (/
     (+
      (*
       beta
       (+
        (* 1.0 1.0)
        (*
         (/ alpha (+ alpha (+ beta 2.0)))
         (+ 1.0 (/ alpha (+ alpha (+ beta 2.0)))))))
      (*
       (+ alpha (+ beta 2.0))
       (- (pow 1.0 3.0) (pow (/ alpha (+ alpha (+ beta 2.0))) 3.0))))
     (*
      (+ alpha (+ beta 2.0))
      (+
       (* 1.0 1.0)
       (*
        (/ alpha (+ alpha (+ beta 2.0)))
        (+ 1.0 (/ alpha (+ alpha (+ beta 2.0))))))))
    2.0)
   (/
    (+
     (/ beta (+ 2.0 (+ alpha beta)))
     (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha))))
    2.0)))
double code(double alpha, double beta) {
	return (((double) ((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + 2.0))) + 1.0)) / 2.0);
}
↓
double code(double alpha, double beta) {
	double tmp;
	if ((alpha <= 32596417.711401034)) {
		tmp = ((((double) (((double) (beta * ((double) (((double) (1.0 * 1.0)) + ((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) * ((double) (1.0 + (alpha / ((double) (alpha + ((double) (beta + 2.0))))))))))))) + ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) (((double) pow(1.0, 3.0)) - ((double) pow((alpha / ((double) (alpha + ((double) (beta + 2.0))))), 3.0)))))))) / ((double) (((double) (alpha + ((double) (beta + 2.0)))) * ((double) (((double) (1.0 * 1.0)) + ((double) ((alpha / ((double) (alpha + ((double) (beta + 2.0))))) * ((double) (1.0 + (alpha / ((double) (alpha + ((double) (beta + 2.0)))))))))))))) / 2.0);
	} else {
		tmp = (((double) ((beta / ((double) (2.0 + ((double) (alpha + beta))))) + ((double) (((double) ((2.0 / alpha) + (8.0 / ((double) pow(alpha, 3.0))))) - (4.0 / ((double) (alpha * alpha))))))) / 2.0);
	}
	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 alpha < 32596417.7114010341

    1. Initial program Error: 0.1 bits

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

    3. Applied div-subError: 0.1 bits

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

    4. Applied associate-+l-Error: 0.1 bits

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

    5. SimplifiedError: 0.1 bits

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

    7. Applied flip3--Error: 0.1 bits

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

    8. Applied frac-subError: 0.1 bits

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

    9. SimplifiedError: 0.1 bits

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

    10. SimplifiedError: 0.1 bits

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

    if 32596417.7114010341 < alpha

    1. Initial program Error: 49.6 bits

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

    3. Applied div-subError: 49.6 bits

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

    4. Applied associate-+l-Error: 48.1 bits

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

    5. SimplifiedError: 48.1 bits

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

    6. Taylor expanded around inf Error: 18.0 bits

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    7. SimplifiedError: 18.0 bits

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 5.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 32596417.711401034:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right) + \left(\alpha + \left(\beta + 2\right)\right) \cdot \left({1}^{3} - {\left(\frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)}^{3}\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(1 \cdot 1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \frac{\alpha}{\alpha + \left(\beta + 2\right)}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} + \left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))