Average Error: 44.9 → 0.2
Time: 44.2s
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 44.9

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

  2. Simplified14.9

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

  4. Applied times-frac14.9

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

  6. Applied pow114.9

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

  7. Applied pow114.9

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

  8. Applied pow-prod-down14.9

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

  9. Simplified0.2

    \[\leadsto {\color{blue}{\left(\frac{i}{(\left(\frac{1.0}{i}\right) \cdot -4 + \left(16 \cdot i\right))_*}\right)}}^{1}\]

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019091 +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)))