Average Error: 2.4 → 0.4
Time: 36.8s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{\frac{2}{\frac{i}{i}}}}{2 \cdot i + 1.0} \cdot \frac{\frac{i}{\frac{2}{\frac{i}{i}}}}{2 \cdot i - 1.0}\]

Error

Bits error versus i

Derivation

  1. Initial program 2.4

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

  3. Applied associate-/l*1.0

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

  5. Applied difference-of-sqr-10.9

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

  6. Applied p16-times-frac0.8

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

  7. Applied p16-times-frac0.9

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

  8. Applied p16-times-frac0.5

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

  10. Applied associate-/l*0.4

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

  12. Applied associate-/l*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019051 
(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))))