Average Error: 16.2 → 0.1
Time: 7.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999066498653:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot {\left(\frac{\beta}{\alpha}\right)}^{2}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}\right)}{2}\\ \end{array}\]
\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 -0.9999999066498653:\\
\;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot {\left(\frac{\beta}{\alpha}\right)}^{2}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}\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)) -0.9999999066498653)
   (/
    (-
     (+ (* 2.0 (/ beta alpha)) (/ 2.0 alpha))
     (+
      (* 6.0 (/ beta (* alpha alpha)))
      (+ (/ 4.0 (* alpha alpha)) (* 2.0 (pow (/ beta alpha) 2.0)))))
    2.0)
   (/ (log (exp (+ (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 1.0))) 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 (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999066498653) {
		tmp = (((2.0 * (beta / alpha)) + (2.0 / alpha)) - ((6.0 * (beta / (alpha * alpha))) + ((4.0 / (alpha * alpha)) + (2.0 * pow((beta / alpha), 2.0))))) / 2.0;
	} else {
		tmp = log(exp(((beta - alpha) / ((beta + alpha) + 2.0)) + 1.0)) / 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 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.9999999066498653

    1. Initial program 59.8

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified0.0

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

    5. Applied pow2_binary64_18640.0

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

    if -0.9999999066498653 < (/.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 add-log-exp_binary64_18220.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999066498653:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \frac{2}{\alpha}\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot {\left(\frac{\beta}{\alpha}\right)}^{2}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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))