Average Error: 45.8 → 0.0
Time: 23.9s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 223.45674042688236:\\ \;\;\;\;\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{{i}^{4}} + \left(\frac{1}{16} + \frac{\frac{0.015625}{i}}{i}\right)\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 223.45674042688236

    1. Initial program 45.0

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

    2. Simplified0.0

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

    if 223.45674042688236 < i

    1. Initial program 46.7

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

    2. Simplified30.9

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

    4. Applied flip--46.7

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

    5. Applied associate-/r/46.8

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

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{0.00390625}{{i}^{4}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 223.45674042688236:\\ \;\;\;\;\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{{i}^{4}} + \left(\frac{1}{16} + \frac{\frac{0.015625}{i}}{i}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +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)))