Average Error: 16.6 → 0.1
Time: 8.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999996845006388:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) + \left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2 + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2}\\ \end{array}\]
\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 -0.9999996845006388:\\
\;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) + \left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2 + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{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)) -0.9999996845006388)
   (/
    (+
     (+ (/ 2.0 alpha) (* (/ beta alpha) (- 2.0 (/ 6.0 alpha))))
     (+ (* (* (/ beta alpha) (/ beta alpha)) -2.0) (/ -4.0 (* alpha alpha))))
    2.0)
   (/
    (+
     (cbrt
      (*
       (/ (- beta alpha) (+ (+ beta alpha) 2.0))
       (*
        (/ (- beta alpha) (+ (+ beta alpha) 2.0))
        (/ (- 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)) <= -0.9999996845006388) {
		tmp = (((2.0 / alpha) + ((beta / alpha) * (2.0 - (6.0 / alpha)))) + ((((beta / alpha) * (beta / alpha)) * -2.0) + (-4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = (cbrt(((beta - alpha) / ((beta + alpha) + 2.0)) * (((beta - alpha) / ((beta + alpha) + 2.0)) * ((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)) < -0.999999684500638808

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.9

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

    3. Simplified0.1

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

    if -0.999999684500638808 < (/.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-cbrt-cube_binary64_18190.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999996845006388:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{\beta}{\alpha} \cdot \left(2 - \frac{6}{\alpha}\right)\right) + \left(\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2 + \frac{-4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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))