Average Error: 16.1 → 6.8
Time: 4.3s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 102400470.51088828:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot {\left(\left(\alpha + \beta\right) + 2\right)}^{0.3333333333333333}} \cdot \frac{\beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \leq 102400470.51088828:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot {\left(\left(\alpha + \beta\right) + 2\right)}^{0.3333333333333333}} \cdot \frac{\beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\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 (<= alpha 102400470.51088828)
   (/
    (-
     (*
      (/
       1.0
       (*
        (cbrt (+ (+ alpha beta) 2.0))
        (pow (+ (+ alpha beta) 2.0) 0.3333333333333333)))
      (/ beta (cbrt (+ (+ alpha beta) 2.0))))
     (- (/ alpha (+ (+ alpha beta) 2.0)) 1.0))
    2.0)
   (/
    (-
     (*
      (/
       (* (cbrt beta) (cbrt beta))
       (* (cbrt (+ (+ alpha beta) 2.0)) (cbrt (+ (+ alpha beta) 2.0))))
      (/ (cbrt beta) (cbrt (+ (+ alpha beta) 2.0))))
     (- (/ 4.0 (* alpha alpha)) (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0)))))
    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 (alpha <= 102400470.51088828) {
		tmp = (((1.0 / (cbrt((alpha + beta) + 2.0) * pow(((alpha + beta) + 2.0), 0.3333333333333333))) * (beta / cbrt((alpha + beta) + 2.0))) - ((alpha / ((alpha + beta) + 2.0)) - 1.0)) / 2.0;
	} else {
		tmp = ((((cbrt(beta) * cbrt(beta)) / (cbrt((alpha + beta) + 2.0) * cbrt((alpha + beta) + 2.0))) * (cbrt(beta) / cbrt((alpha + beta) + 2.0))) - ((4.0 / (alpha * alpha)) - ((2.0 / alpha) + (8.0 / pow(alpha, 3.0))))) / 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 alpha < 102400470.51088828

    1. Initial program 0.1

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

    3. Applied div-sub_binary640.1

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

    4. Applied associate-+l-_binary640.1

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

    5. Simplified0.1

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

    7. Applied add-cube-cbrt_binary640.3

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

    8. Applied *-un-lft-identity_binary640.3

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

    9. Applied times-frac_binary640.3

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

    11. Applied pow1/3_binary641.3

      \[\leadsto \frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \color{blue}{{\left(\left(\alpha + \beta\right) + 2\right)}^{0.3333333333333333}}} \cdot \frac{\beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2}\]

    if 102400470.51088828 < alpha

    1. Initial program 49.7

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

    3. Applied div-sub_binary6449.7

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

    4. Applied associate-+l-_binary6448.2

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

    5. Simplified48.2

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

    7. Applied add-cube-cbrt_binary6448.3

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

    8. Applied add-cube-cbrt_binary6448.2

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

    9. Applied times-frac_binary6448.2

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

    10. Taylor expanded around inf 18.1

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

    11. Simplified18.1

      \[\leadsto \frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 102400470.51088828:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot {\left(\left(\alpha + \beta\right) + 2\right)}^{0.3333333333333333}} \cdot \frac{\beta}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\frac{4}{\alpha \cdot \alpha} - \left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(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))