Average Error: 16.2 → 6.3
Time: 4.5s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1192146593148.475:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) + \left(\beta - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2\right)}^{3}} - 1\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 1192146593148.475:\\
\;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) + \left(\beta - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2\right)}^{3}} - 1\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\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 1192146593148.475)
   (/
    (/
     (+
      (*
       beta
       (*
        (/ alpha (+ (+ alpha beta) 2.0))
        (+ (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))
      (-
       beta
       (*
        (+ (+ alpha beta) 2.0)
        (-
         (* (* alpha alpha) (/ alpha (pow (+ (+ alpha beta) 2.0) 3.0)))
         1.0))))
     (*
      (+ (+ alpha beta) 2.0)
      (+
       1.0
       (*
        (/ alpha (+ (+ 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 <= 1192146593148.475) {
		tmp = (((beta * ((alpha / ((alpha + beta) + 2.0)) * ((alpha / ((alpha + beta) + 2.0)) + 1.0))) + (beta - (((alpha + beta) + 2.0) * (((alpha * alpha) * (alpha / pow(((alpha + beta) + 2.0), 3.0))) - 1.0)))) / (((alpha + beta) + 2.0) * (1.0 + ((alpha / ((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 < 1192146593148.4751

    1. Initial program 0.2

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

    3. Applied div-sub_binary640.2

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

    4. Applied associate-+l-_binary640.2

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

    5. Simplified0.2

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

    7. Applied flip3--_binary640.2

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

    8. Applied frac-sub_binary640.2

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

    9. Simplified0.2

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

    10. Simplified0.2

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

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

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

    13. Applied add-cube-cbrt_binary640.3

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

    14. Applied times-frac_binary640.3

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

    15. Applied unpow-prod-down_binary640.3

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

    16. Simplified0.3

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

    17. Simplified0.2

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

    if 1192146593148.4751 < alpha

    1. Initial program 49.8

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

    3. Applied div-sub_binary6449.8

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

    4. Applied associate-+l-_binary6448.1

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

    5. Simplified48.1

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

    6. Taylor expanded around inf 18.9

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

    7. Simplified18.9

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1192146593148.475:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) + \left(\beta - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2\right)}^{3}} - 1\right)\right)}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))