Average Error: 16.3 → 6.0
Time: 3.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.533781242299976 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(\alpha \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + -1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 7.533781242299976 \cdot 10^{+17}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(\alpha \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + -1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\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 7.533781242299976e+17)
   (/
    (-
     (/ beta (+ (+ alpha beta) 2.0))
     (/
      (+
       (*
        (* alpha (/ alpha (+ (+ alpha beta) 2.0)))
        (/ 1.0 (+ (+ alpha beta) 2.0)))
       -1.0)
      (+ (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))
    2.0)
   (/
    (-
     (/ beta (+ (+ alpha beta) 2.0))
     (- (/ 4.0 (* alpha alpha)) (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0)))))
    2.0)))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.533781242299976e+17) {
		tmp = ((beta / ((alpha + beta) + 2.0)) - ((((alpha * (alpha / ((alpha + beta) + 2.0))) * (1.0 / ((alpha + beta) + 2.0))) + -1.0) / ((alpha / ((alpha + beta) + 2.0)) + 1.0))) / 2.0;
	} else {
		tmp = ((beta / ((alpha + beta) + 2.0)) - ((4.0 / (alpha * alpha)) - ((2.0 / alpha) + (8.0 / pow(alpha, 3.0))))) / 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 < 753378124229997570

    1. Initial program 0.5

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

    3. Applied div-sub_binary640.5

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

    4. Applied associate-+l-_binary640.5

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

    5. Simplified0.5

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

    7. Applied flip--_binary640.5

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

    8. Simplified0.5

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

    9. Simplified0.5

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

    11. Applied div-inv_binary640.5

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

    12. Applied associate-*r*_binary640.5

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

    13. Simplified0.5

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

    if 753378124229997570 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub_binary6451.1

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

    4. Applied associate-+l-_binary6449.5

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

    5. Simplified49.5

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

    6. Taylor expanded around inf 18.1

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

    7. Simplified18.1

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.533781242299976 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\left(\alpha \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2} + -1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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))