Average Error: 15.5 → 3.1
Time: 5.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999107464:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]
\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 -0.9999999999107464:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\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)) -0.9999999999107464)
   (/
    (-
     (/
      (/
       1.0
       (*
        (cbrt (/ (+ (+ beta alpha) 2.0) beta))
        (cbrt (/ (+ (+ beta alpha) 2.0) beta))))
      (cbrt (/ (+ (+ beta alpha) 2.0) beta)))
     (- (/ 4.0 (* alpha alpha)) (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0)))))
    2.0)
   (/ (+ 1.0 (/ 1.0 (/ (+ (+ beta alpha) 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)) <= -0.9999999999107464) {
		tmp = (((1.0 / (cbrt(((beta + alpha) + 2.0) / beta) * cbrt(((beta + alpha) + 2.0) / beta))) / cbrt(((beta + alpha) + 2.0) / beta)) - ((4.0 / (alpha * alpha)) - ((2.0 / alpha) + (8.0 / pow(alpha, 3.0))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 / (((beta + alpha) + 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)) < -0.999999999910746395

    1. Initial program 60.0

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

    3. Applied div-sub_binary64_212960.0

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

    4. Applied associate-+l-_binary64_205958.1

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

    5. Simplified58.1

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2}\]
    6. Using strategy rm

    7. Applied clear-num_binary64_212358.1

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary64_215958.1

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}\right) \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\]

    10. Applied associate-/r*_binary64_206858.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\]

    11. Taylor expanded around inf 11.5

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    12. Simplified11.5

      \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2}{\beta}}} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]

    if -0.999999999910746395 < (/.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_binary64_21230.2

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

    4. Simplified0.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999999107464:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}} \cdot \sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}}}{\sqrt[3]{\frac{\left(\beta + \alpha\right) + 2}{\beta}}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta - \alpha}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020352 
(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))