Average Error: 16.2 → 15.8
Time: 22.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{e^{\sqrt[3]{{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)}^{3}}}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{e^{\sqrt[3]{{\log \left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)\right)}^{3}}}}{2}
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (/
  (exp
   (cbrt
    (pow
     (log
      (-
       (/ beta (+ (+ beta alpha) 2.0))
       (- (/ alpha (+ (+ beta alpha) 2.0)) 1.0)))
     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) {
	return exp(cbrt(pow(log((beta / ((beta + alpha) + 2.0)) - ((alpha / ((beta + alpha) + 2.0)) - 1.0)), 3.0))) / 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 16.2

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

  3. Applied div-sub_binary6416.2

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

  4. Applied associate-+l-_binary6415.7

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

  5. Simplified15.7

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

  7. Applied add-exp-log_binary6415.8

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

  8. Simplified15.8

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

  10. Applied add-cbrt-cube_binary6415.8

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

  11. Simplified15.8

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

  12. Final simplification15.8

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

Reproduce

herbie shell --seed 2020288 
(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))