Average Error: 15.8 → 0.2
Time: 6.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999975506229:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}\right)}^{3}}{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} \cdot \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} + \left(1 - \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}\right)}}{2}\\ \end{array}\]
\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 -0.9999999975506229:\\
\;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 + {\left(\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}\right)}^{3}}{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} \cdot \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} + \left(1 - \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \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)) -0.9999999975506229)
   (/
    (+
     (-
      (+ (/ 2.0 alpha) (* (/ beta alpha) (- 2.0 (/ 6.0 alpha))))
      (/ 4.0 (* alpha alpha)))
     (* (* (/ beta alpha) (/ beta alpha)) -2.0))
    2.0)
   (/
    (/
     (+ 1.0 (pow (/ 1.0 (/ (+ alpha (+ beta 2.0)) (- beta alpha))) 3.0))
     (+
      (*
       (/ 1.0 (/ (+ alpha (+ beta 2.0)) (- beta alpha)))
       (/ 1.0 (/ (+ alpha (+ beta 2.0)) (- beta alpha))))
      (- 1.0 (/ 1.0 (/ (+ alpha (+ beta 2.0)) (- beta 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)) <= -0.9999999975506229) {
		tmp = ((((2.0 / alpha) + ((beta / alpha) * (2.0 - (6.0 / alpha)))) - (4.0 / (alpha * alpha))) + (((beta / alpha) * (beta / alpha)) * -2.0)) / 2.0;
	} else {
		tmp = ((1.0 + pow((1.0 / ((alpha + (beta + 2.0)) / (beta - alpha))), 3.0)) / (((1.0 / ((alpha + (beta + 2.0)) / (beta - alpha))) * (1.0 / ((alpha + (beta + 2.0)) / (beta - alpha)))) + (1.0 - (1.0 / ((alpha + (beta + 2.0)) / (beta - 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 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99999999755062285

    1. Initial program 60.0

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

    2. Taylor expanded around inf 3.0

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}\right) - \left(6 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + 2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)}}{2}\]

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}}{2}\]

    if -0.99999999755062285 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.2

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

    3. Applied clear-num_binary64_17820.2

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

    4. Simplified0.2

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

    6. Applied flip3-+_binary64_17860.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999975506229:\\ \;\;\;\;\frac{\left(\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha \cdot \alpha}\right) + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}\right)}^{3}}{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} \cdot \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}} + \left(1 - \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))