Average Error: 45.2 → 0.1
Time: 48.8s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\frac{\frac{i}{2}}{8 \cdot i - \frac{1.0}{\frac{i}{2}}}\]

Error

Bits error versus i

Derivation

  1. Initial program 45.2

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

  2. Simplified15.2

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

  4. Applied div-sub15.2

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

  5. Simplified0.1

    \[\leadsto \frac{\frac{i}{2}}{\color{blue}{8 \cdot i} - \frac{1.0}{\frac{i}{2}}}\]

  6. Final simplification0.1

    \[\leadsto \frac{\frac{i}{2}}{8 \cdot i - \frac{1.0}{\frac{i}{2}}}\]

Reproduce

herbie shell --seed 2019093 
(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)))