Error

Derivation

1. Initial program 0.4

   \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
2. Using strategy rm

3. Applied associate-+l+0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
4. Using strategy rm

5. Applied associate-/l/0.4

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\beta \cdot \alpha\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]

6. Simplified0.4

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \left(\frac{\alpha}{\left(real->posit(1.0)\right)}\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]
7. Using strategy rm

8. Applied associate-+l+0.4

   \[\leadsto \frac{\left(\frac{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \left(\frac{\alpha}{\left(real->posit(1.0)\right)}\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}{\left(\frac{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}}{\left(real->posit(1.0)\right)}\right)}\]
9. Using strategy rm

10. Applied distribute-lft-in0.4

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \alpha\right)}{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \left(real->posit(1.0)\right)\right)}\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]

11. Simplified0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\frac{\beta}{\left(real->posit(1.0)\right)}\right) \cdot \alpha\right)}{\color{blue}{\left(\frac{\left(real->posit(1.0)\right)}{\beta}\right)}}\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(\left(real->posit(2)\right) \cdot \left(real->posit(1)\right)\right)}\right)}\right)}{\left(real->posit(1.0)\right)}\right)}\]

12. Final simplification0.4

    \[\leadsto \frac{\frac{\left(\beta + 1.0\right) \cdot \alpha + \left(1.0 + \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \left(\beta + 2 \cdot 1\right)\right) + 1.0}\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))