Average Error: 16.3 → 3.3
Time: 1.7min
Precision: binary64
Cost: 2305
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right) - \frac{\beta}{-2 - \left(\beta + \alpha\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(-1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\
\;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}\right) - \frac{\beta}{-2 - \left(\beta + \alpha\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(-1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\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 (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
   (/
    (-
     (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
     (/ beta (- -2.0 (+ beta alpha))))
    2.0)
   (/
    (-
     (/ beta (+ (+ beta alpha) 2.0))
     (+ -1.0 (/ 1.0 (/ (+ (+ beta alpha) 2.0) alpha))))
    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 (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = ((((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) - (beta / (-2.0 - (beta + alpha)))) / 2.0;
	} else {
		tmp = ((beta / ((beta + alpha) + 2.0)) - (-1.0 + (1.0 / (((beta + alpha) + 2.0) / 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
Alternative 1
Accuracy15.8
Cost1344
\[\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha}} - 1\right)}{2}\]
Alternative 2
Accuracy15.8
Cost2368
\[\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\frac{\alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2} + -1}{\frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1}}{2}\]
Alternative 3
Accuracy16.3
Cost1856
\[\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{1}{\sqrt{\left(\beta + \alpha\right) + 2} \cdot \frac{\sqrt{\left(\beta + \alpha\right) + 2}}{\alpha}} - 1\right)}{2}\]
Alternative 4
Accuracy15.9
Cost2177
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1018942208242 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\beta \cdot 0.5 + \left(\beta \cdot -0.25\right) \cdot \left(\beta + \alpha\right)\right) - \left(\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\alpha}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]
Alternative 5
Accuracy22.9
Cost2305
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1018942208242 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{1}{\frac{\sqrt{\left(\beta + \alpha\right) + 2}}{\sqrt{\alpha}} \cdot \frac{\sqrt{\left(\beta + \alpha\right) + 2}}{\sqrt{\alpha}}} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]
Alternative 6
Accuracy23.0
Cost2177
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1018942208242 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\sqrt{\alpha} \cdot \frac{\sqrt{\alpha}}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\\ \end{array}\]
Alternative 7
Accuracy30.6
Cost1984
\[\frac{\frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -1

    1. Initial program 60.6

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

    3. Applied div-sub_binary64_76560.6

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

    4. Applied associate-+l-_binary64_69558.6

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

    5. Simplified58.6

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

    7. Applied clear-num_binary64_75958.6

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

    8. Simplified58.6

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

    10. Applied frac-2neg_binary64_77158.6

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

    11. Simplified58.6

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

    12. Taylor expanded around inf 11.3

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

    13. Simplified11.3

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

    if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.5

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

    3. Applied div-sub_binary64_7650.5

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

    4. Applied associate-+l-_binary64_6950.5

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

    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 clear-num_binary64_7590.5

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

    8. Simplified0.5

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

    10. Applied pow1_binary64_8210.5

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2020322 
(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))