Average Error: 46.0 → 0.1
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{i}{(i \cdot 2 + \left(\sqrt{1.0}\right))_*} \cdot \frac{i}{(-4 \cdot \left(\sqrt{1.0}\right) + \left(8 \cdot i\right))_*}\]

Error

Bits error versus i

Derivation

  1. Initial program 46.0

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

  2. Simplified16.2

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

  4. Applied add-sqr-sqrt16.2

    \[\leadsto \frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}\right) \cdot 4}\]

  5. Applied *-commutative16.2

    \[\leadsto \frac{i \cdot i}{\left(4 \cdot \color{blue}{\left(i \cdot i\right)} - \sqrt{1.0} \cdot \sqrt{1.0}\right) \cdot 4}\]

  6. Applied add-sqr-sqrt16.2

    \[\leadsto \frac{i \cdot i}{\left(\color{blue}{\left(\sqrt{4} \cdot \sqrt{4}\right)} \cdot \left(i \cdot i\right) - \sqrt{1.0} \cdot \sqrt{1.0}\right) \cdot 4}\]

  7. Applied unswap-sqr16.2

    \[\leadsto \frac{i \cdot i}{\left(\color{blue}{\left(\sqrt{4} \cdot i\right) \cdot \left(\sqrt{4} \cdot i\right)} - \sqrt{1.0} \cdot \sqrt{1.0}\right) \cdot 4}\]

  8. Applied difference-of-squares16.2

    \[\leadsto \frac{i \cdot i}{\color{blue}{\left(\left(\sqrt{4} \cdot i + \sqrt{1.0}\right) \cdot \left(\sqrt{4} \cdot i - \sqrt{1.0}\right)\right)} \cdot 4}\]

  9. Applied associate-*l*16.2

    \[\leadsto \frac{i \cdot i}{\color{blue}{\left(\sqrt{4} \cdot i + \sqrt{1.0}\right) \cdot \left(\left(\sqrt{4} \cdot i - \sqrt{1.0}\right) \cdot 4\right)}}\]

  10. Applied *-un-lft-identity16.2

    \[\leadsto \frac{i \cdot \color{blue}{\left(1 \cdot i\right)}}{\left(\sqrt{4} \cdot i + \sqrt{1.0}\right) \cdot \left(\left(\sqrt{4} \cdot i - \sqrt{1.0}\right) \cdot 4\right)}\]

  11. Applied associate-*r*16.2

    \[\leadsto \frac{\color{blue}{\left(i \cdot 1\right) \cdot i}}{\left(\sqrt{4} \cdot i + \sqrt{1.0}\right) \cdot \left(\left(\sqrt{4} \cdot i - \sqrt{1.0}\right) \cdot 4\right)}\]

  12. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{i \cdot 1}{\sqrt{4} \cdot i + \sqrt{1.0}} \cdot \frac{i}{\left(\sqrt{4} \cdot i - \sqrt{1.0}\right) \cdot 4}}\]

  13. Simplified0.1

    \[\leadsto \color{blue}{\frac{i}{(i \cdot 2 + \left(\sqrt{1.0}\right))_*}} \cdot \frac{i}{\left(\sqrt{4} \cdot i - \sqrt{1.0}\right) \cdot 4}\]

  14. Simplified0.1

    \[\leadsto \frac{i}{(i \cdot 2 + \left(\sqrt{1.0}\right))_*} \cdot \color{blue}{\frac{i}{(-4 \cdot \left(\sqrt{1.0}\right) + \left(8 \cdot i\right))_*}}\]

  15. Final simplification0.1

    \[\leadsto \frac{i}{(i \cdot 2 + \left(\sqrt{1.0}\right))_*} \cdot \frac{i}{(-4 \cdot \left(\sqrt{1.0}\right) + \left(8 \cdot i\right))_*}\]

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))