Average Error: 16.3 → 0.4
Time: 4.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta - \alpha}}}{2}\\ \end{array} \]
\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 -1:\\
\;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{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)) -1.0)
   (/ (/ (+ 2.0 (* 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)) <= -1.0) {
		tmp = ((2.0 + (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)) < -1

    1. Initial program 60.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \beta + 2}{\alpha}}}{2} \]

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

    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 clear-num_binary640.5

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

    4. Simplified0.5

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2021198 
(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))