Average Error: 45.8 → 0.2
Time: 39.7s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[i \cdot \frac{\frac{1}{4}}{\frac{i}{\frac{1}{4}} - \frac{1.0}{i}}\]

Error

Bits error versus i

Derivation

  1. Initial program 45.8

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

  2. Simplified15.5

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

  4. Applied *-un-lft-identity15.5

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

  5. Applied div-inv15.5

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

  6. Applied times-frac15.5

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

  7. Simplified15.5

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

  8. Simplified0.2

    \[\leadsto i \cdot \color{blue}{\frac{\frac{1}{4}}{\frac{i}{\frac{1}{4}} - \frac{1.0}{i}}}\]

  9. Final simplification0.2

    \[\leadsto i \cdot \frac{\frac{1}{4}}{\frac{i}{\frac{1}{4}} - \frac{1.0}{i}}\]

Reproduce

herbie shell --seed 2019053 
(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)))