Average Error: 15.8 → 15.3
Time: 1.1min
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\sqrt[3]{{\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}^{6}} + -1}{\frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\sqrt[3]{{\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}^{6}} + -1}{\frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1}}{2}
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (/
  (-
   (/ beta (+ (+ beta alpha) 2.0))
   (/
    (+ (cbrt (pow (/ alpha (+ (+ beta alpha) 2.0)) 6.0)) -1.0)
    (+ (/ 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) {
	return ((beta / ((beta + alpha) + 2.0)) - ((cbrt(pow((alpha / ((beta + alpha) + 2.0)), 6.0)) + -1.0) / ((alpha / ((beta + alpha) + 2.0)) + 1.0))) / 2.0;
}

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. Initial program 15.8

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

  3. Applied div-sub_binary6415.8

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

  4. Applied associate-+l-_binary6415.3

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

  5. Simplified15.3

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

  7. Applied add-cube-cbrt_binary6415.4

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

  8. Simplified15.4

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

  9. Simplified15.4

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

  11. Applied flip--_binary6415.4

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

  12. Simplified15.4

    \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\color{blue}{{\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)}^{6} + -1}}{\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}} + 1}}{2}\]

  13. Simplified15.4

    \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{{\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)}^{6} + -1}{\color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}}{2}\]
  14. Using strategy rm

  15. Applied add-cbrt-cube_binary6415.4

    \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\color{blue}{\sqrt[3]{\left({\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)}^{6} \cdot {\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right) \cdot {\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}} + -1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\]

  16. Simplified15.3

    \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\sqrt[3]{\color{blue}{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{6}}} + -1}{\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1}}{2}\]

  17. Final simplification15.3

    \[\leadsto \frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\sqrt[3]{{\left(\frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)}^{6}} + -1}{\frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1}}{2}\]

Reproduce

herbie shell --seed 2020285 
(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))