Average Error: 16.4 → 16.1
Time: 7.0s
Precision: binary64
\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \frac{\frac{\alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2} + -1}{\frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt{\left(\beta + \alpha\right) + 2}} - \frac{\frac{\alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\alpha}{\left(\beta + \alpha\right) + 2} + -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
 (/
  (-
   (*
    (/ (* (cbrt beta) (cbrt beta)) (sqrt (+ (+ beta alpha) 2.0)))
    (/ (cbrt beta) (sqrt (+ (+ beta alpha) 2.0))))
   (/
    (+
     (* (/ alpha (+ (+ beta alpha) 2.0)) (/ alpha (+ (+ beta alpha) 2.0)))
     -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 ((((cbrt(beta) * cbrt(beta)) / sqrt((beta + alpha) + 2.0)) * (cbrt(beta) / sqrt((beta + alpha) + 2.0))) - ((((alpha / ((beta + alpha) + 2.0)) * (alpha / ((beta + alpha) + 2.0))) + -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 16.4

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

  3. Applied div-sub_binary6416.3

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

  4. Applied associate-+l-_binary6415.9

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

  5. Simplified15.9

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

  7. Applied add-sqr-sqrt_binary6415.9

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

  8. Applied add-cube-cbrt_binary6416.1

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

  9. Applied times-frac_binary6416.1

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

  11. Applied flip--_binary6416.1

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

  12. Simplified16.1

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

  13. Simplified16.1

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

  14. Final simplification16.1

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

Reproduce

herbie shell --seed 2020268 
(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))