Average Error: 45.4 → 0.2
Time: 41.8s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{2}}{8 \cdot i - \frac{2.0}{i}}\]

Error

Bits error versus i

Derivation

  1. Initial program 45.4

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

  2. Simplified15.9

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

  3. Taylor expanded around inf 0.2

    \[\leadsto \frac{\frac{i}{2}}{\color{blue}{8 \cdot i - 2.0 \cdot \frac{1}{i}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\frac{i}{2}}{\color{blue}{8 \cdot i - \frac{2.0}{i}}}\]

  5. Final simplification0.2

    \[\leadsto \frac{\frac{i}{2}}{8 \cdot i - \frac{2.0}{i}}\]

Reproduce

herbie shell --seed 2019051 
(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)))