Average Error: 16.1 → 0.2
Time: 6.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999508754:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right) - \left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \frac{6 + \beta \cdot 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}}{2}\\ \end{array} \]
\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 -0.9999999999508754:\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right) - \left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \frac{6 + \beta \cdot 2}{\alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\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)) -0.9999999999508754)
   (/
    (-
     (+ (/ 2.0 alpha) (* 2.0 (/ beta alpha)))
     (+
      (/ 4.0 (* alpha alpha))
      (* (/ beta alpha) (/ (+ 6.0 (* beta 2.0)) alpha))))
    2.0)
   (/ (+ 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)) <= -0.9999999999508754) {
		tmp = (((2.0 / alpha) + (2.0 * (beta / alpha))) - ((4.0 / (alpha * alpha)) + ((beta / alpha) * ((6.0 + (beta * 2.0)) / alpha)))) / 2.0;
	} else {
		tmp = (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)) < -0.999999999950875407

    1. Initial program 60.1

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

    2. Taylor expanded in alpha around inf 2.9

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

    3. Simplified0.0

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

    4. Taylor expanded in alpha around 0 2.9

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

    5. Simplified0.0

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

    if -0.999999999950875407 < (/.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_binary640.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999508754:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right) - \left(\frac{4}{\alpha \cdot \alpha} + \frac{\beta}{\alpha} \cdot \frac{6 + \beta \cdot 2}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}}{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))