Error

Derivation

1. Initial program 0.7

   \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
2. Using strategy rm

3. Applied p16-flip--1.3

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\frac{\beta}{\alpha}\right)}\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

4. Applied associate-/l/1.3

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\beta \cdot \beta\right) - \left(\alpha \cdot \alpha\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

5. Simplified1.2

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\alpha}{\beta}\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]
6. Using strategy rm

7. Applied *-commutative1.2

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

8. Applied p16-times-frac0.8

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\beta}{\alpha}\right)}\right)\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

9. Simplified0.8

   \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(real->posit(2.0)\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\alpha}{\beta}\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(real->posit(2.0)\right)}\]

10. Final simplification0.8

    \[\leadsto \frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\alpha + \beta} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))