Error

Derivation

1. Initial program 2.5

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

3. Applied difference-of-sqr-12.4

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

4. Applied p16-times-frac1.1

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

5. Applied p16-times-frac1.0

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

7. Applied p16-times-frac0.8

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

8. Simplified0.8

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

10. Applied p16-times-frac0.4

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

11. Simplified0.4

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

12. Final simplification0.4

    \[\leadsto \frac{\frac{i}{2} \cdot 1.0}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{2} \cdot 1.0}{2 \cdot i - 1.0}\]

Reproduce

herbie shell --seed 2019090 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (>.p16 i (real->posit16 0)))
  (/.p16 (/.p16 (*.p16 (*.p16 i i) (*.p16 i i)) (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i))) (-.p16 (*.p16 (*.p16 (real->posit16 2) i) (*.p16 (real->posit16 2) i)) (real->posit16 1.0))))