Average Error: 15.9 → 6.6
Time: 3.6s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.74995439910408 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - 1}{1 + \frac{\alpha \cdot \left(-1 + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + -1\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 1.74995439910408 \cdot 10^{+24}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - 1}{1 + \frac{\alpha \cdot \left(-1 + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + -1\right)}}}{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 1.74995439910408e+24)
   (/
    (-
     (/ beta (+ (+ alpha beta) 2.0))
     (/
      (- (pow (/ alpha (+ (+ alpha beta) 2.0)) 3.0) 1.0)
      (+
       1.0
       (/
        (* alpha (+ -1.0 (pow (/ alpha (+ (+ alpha beta) 2.0)) 2.0)))
        (*
         (+ (+ alpha beta) 2.0)
         (+ (/ 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 <= 1.74995439910408e+24) {
		tmp = ((beta / ((alpha + beta) + 2.0)) - ((pow((alpha / ((alpha + beta) + 2.0)), 3.0) - 1.0) / (1.0 + ((alpha * (-1.0 + pow((alpha / ((alpha + beta) + 2.0)), 2.0))) / (((alpha + beta) + 2.0) * ((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 < 1.7499543991040801e24

    1. Initial program 1.0

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

    3. Applied div-sub_binary641.0

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

    4. Applied associate-+l-_binary641.0

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

    5. Simplified1.0

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

    7. Applied flip3--_binary641.1

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

    8. Simplified1.1

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

    9. Simplified1.1

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

    11. Applied flip-+_binary641.1

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

    12. Applied frac-times_binary641.1

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

    13. Simplified1.1

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

    14. Simplified1.1

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

    if 1.7499543991040801e24 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub_binary6450.2

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

    4. Applied associate-+l-_binary6448.5

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

    5. Simplified48.5

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

    6. Taylor expanded around inf 19.2

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

    7. Simplified19.2

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.74995439910408 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - 1}{1 + \frac{\alpha \cdot \left(-1 + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + -1\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))