Average Error: 46.2 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
\[\frac{i}{(\left(\frac{1.0}{i}\right) \cdot -4 + \left(16 \cdot i\right))_*}\]

Error

Bits error versus i

Derivation

  1. Initial program 46.2

    \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
  2. Simplified15.9

    \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
  3. Using strategy rm
  4. Applied *-un-lft-identity15.9

    \[\leadsto \frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - \color{blue}{1 \cdot 1.0}\right) \cdot 4}\]
  5. Applied *-un-lft-identity15.9

    \[\leadsto \frac{i \cdot i}{\left(\color{blue}{\left(1 \cdot 4\right)} \cdot \left(i \cdot i\right) - 1 \cdot 1.0\right) \cdot 4}\]
  6. Applied associate-*l*15.9

    \[\leadsto \frac{i \cdot i}{\left(\color{blue}{1 \cdot \left(4 \cdot \left(i \cdot i\right)\right)} - 1 \cdot 1.0\right) \cdot 4}\]
  7. Applied distribute-lft-out--15.9

    \[\leadsto \frac{i \cdot i}{\color{blue}{\left(1 \cdot \left(4 \cdot \left(i \cdot i\right) - 1.0\right)\right)} \cdot 4}\]
  8. Applied associate-*l*15.9

    \[\leadsto \frac{i \cdot i}{\color{blue}{1 \cdot \left(\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4\right)}}\]
  9. Applied *-un-lft-identity15.9

    \[\leadsto \frac{\color{blue}{1 \cdot \left(i \cdot i\right)}}{1 \cdot \left(\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4\right)}\]
  10. Applied times-frac15.9

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
  11. Simplified15.9

    \[\leadsto \color{blue}{1} \cdot \frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}\]
  12. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\frac{i}{(\left(\frac{1.0}{i}\right) \cdot -4 + \left(i \cdot 16\right))_*}}\]
  13. Final simplification0.2

    \[\leadsto \frac{i}{(\left(\frac{1.0}{i}\right) \cdot -4 + \left(16 \cdot i\right))_*}\]

Reproduce

herbie shell --seed 2019088 +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)))