Error

Derivation

1. Initial program 3.3

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

3. Applied p16-times-frac1.8

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

5. Applied associate-/l*1.6

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

6. Final simplification1.6

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 1)))
  (/.p16 (/.p16 (*.p16 (*.p16 i (+.p16 (+.p16 alpha beta) i)) (+.p16 (*.p16 beta alpha) (*.p16 i (+.p16 (+.p16 alpha beta) i)))) (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)))) (-.p16 (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (real->posit16 1.0))))