Average Error: 45.3 → 0.0
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 238.59754101754152:\\ \;\;\;\;\frac{\frac{i}{2} \cdot \frac{i}{2}}{{\left(\left(i + i\right) \cdot \left(i + i\right)\right)}^{3} - {1.0}^{3}} \cdot \left(\left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot \left(\left(i + i\right) \cdot \left(i + i\right)\right) + \left(1.0 \cdot 1.0 + \left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot 1.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 238.59754101754152

    1. Initial program 44.1

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

    2. Applied simplify0.0

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

    4. Applied flip3--0.0

      \[\leadsto \frac{\frac{i}{2} \cdot \frac{i}{2}}{\color{blue}{\frac{{\left(\left(i + i\right) \cdot \left(i + i\right)\right)}^{3} - {1.0}^{3}}{\left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot \left(\left(i + i\right) \cdot \left(i + i\right)\right) + \left(1.0 \cdot 1.0 + \left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot 1.0\right)}}}\]

    5. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{{\left(\left(i + i\right) \cdot \left(i + i\right)\right)}^{3} - {1.0}^{3}} \cdot \left(\left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot \left(\left(i + i\right) \cdot \left(i + i\right)\right) + \left(1.0 \cdot 1.0 + \left(\left(i + i\right) \cdot \left(i + i\right)\right) \cdot 1.0\right)\right)}\]

    if 238.59754101754152 < i

    1. Initial program 46.4

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

    2. Applied simplify31.0

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

    3. Taylor expanded around inf 0.0

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

    4. Applied simplify0.0

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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)))