Average Error: 16.1 → 13.0
Time: 5.6s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\left(\frac{\alpha}{\alpha + 2} + 1\right) \cdot \frac{\beta}{\alpha + 2} + \left(1 - \frac{\alpha}{\alpha + 2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}\right)}{2}\\ \end{array}\]
\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 -1:\\
\;\;\;\;\frac{\left(\frac{\alpha}{\alpha + 2} + 1\right) \cdot \frac{\beta}{\alpha + 2} + \left(1 - \frac{\alpha}{\alpha + 2}\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)) -1.0)
   (/
    (+
     (* (+ (/ alpha (+ alpha 2.0)) 1.0) (/ beta (+ alpha 2.0)))
     (- 1.0 (/ alpha (+ 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)) <= -1.0) {
		tmp = ((((alpha / (alpha + 2.0)) + 1.0) * (beta / (alpha + 2.0))) + (1.0 - (alpha / (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)) < -1

    1. Initial program 60.6

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

    2. Taylor expanded around 0 60.9

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

    3. Simplified48.7

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

    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 add-log-exp_binary64_18220.5

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

    4. Applied add-log-exp_binary64_18220.5

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

    5. Applied sum-log_binary64_18740.6

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

    6. Simplified0.6

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

  4. Final simplification13.0

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

Reproduce

herbie shell --seed 2020344 
(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))