Average Error: 16.2 → 0.1
Time: 7.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999979903010765:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + {\left(\frac{\beta}{\alpha}\right)}^{3}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]
\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 -0.9999979903010765:\\
\;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + {\left(\frac{\beta}{\alpha}\right)}^{3}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{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)) -0.9999979903010765)
   (/
    (+
     (* 2.0 (+ (/ beta alpha) (pow (/ beta alpha) 3.0)))
     (-
      (+
       (/ 2.0 alpha)
       (+
        (/ 8.0 (pow alpha 3.0))
        (+
         (* 16.0 (/ beta (pow alpha 3.0)))
         (* 10.0 (* beta (/ beta (pow alpha 3.0)))))))
      (+
       (* 6.0 (/ beta (* alpha alpha)))
       (+ (/ 4.0 (* alpha alpha)) (* 2.0 (* (/ beta alpha) (/ beta alpha)))))))
    2.0)
   (/ (+ 1.0 (/ 1.0 (/ (+ (+ beta alpha) 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)) <= -0.9999979903010765) {
		tmp = ((2.0 * ((beta / alpha) + pow((beta / alpha), 3.0))) + (((2.0 / alpha) + ((8.0 / pow(alpha, 3.0)) + ((16.0 * (beta / pow(alpha, 3.0))) + (10.0 * (beta * (beta / pow(alpha, 3.0))))))) - ((6.0 * (beta / (alpha * alpha))) + ((4.0 / (alpha * alpha)) + (2.0 * ((beta / alpha) * (beta / alpha))))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((beta + alpha) + 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)) < -0.9999979903010765

    1. Initial program 59.6

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

    2. Taylor expanded around inf 5.6

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{{\beta}^{3}}{{\alpha}^{3}} + \left(2 \cdot \frac{\beta}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + \left(8 \cdot \frac{1}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \frac{{\beta}^{2}}{{\alpha}^{3}}\right)\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{{\alpha}^{2}} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + 2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)}}{2}\]

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{2 \cdot \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \frac{\beta}{\alpha}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\frac{\beta}{{\alpha}^{3}} \cdot \beta\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}}{2}\]
    4. Using strategy rm

    5. Applied pow1_binary640.0

      \[\leadsto \frac{2 \cdot \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \frac{\beta}{\alpha}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\frac{\beta}{{\alpha}^{3}} \cdot \color{blue}{{\beta}^{1}}\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\]

    6. Applied pow1_binary640.0

      \[\leadsto \frac{2 \cdot \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \frac{\beta}{\alpha}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\color{blue}{{\left(\frac{\beta}{{\alpha}^{3}}\right)}^{1}} \cdot {\beta}^{1}\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\]

    7. Applied pow-prod-down_binary640.0

      \[\leadsto \frac{2 \cdot \left({\left(\frac{\beta}{\alpha}\right)}^{3} + \frac{\beta}{\alpha}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \color{blue}{{\left(\frac{\beta}{{\alpha}^{3}} \cdot \beta\right)}^{1}}\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\]

    if -0.9999979903010765 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 0.1

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

    3. Applied clear-num_binary640.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999979903010765:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + {\left(\frac{\beta}{\alpha}\right)}^{3}\right) + \left(\left(\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} + \left(16 \cdot \frac{\beta}{{\alpha}^{3}} + 10 \cdot \left(\beta \cdot \frac{\beta}{{\alpha}^{3}}\right)\right)\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))