Average Error: 45.4 → 0.0
Time: 2.1m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1901.6558299178805:\\ \;\;\;\;\frac{i}{2} \cdot \frac{\frac{i}{2}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1901.6558299178805

    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

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot i\right) \cdot \left(\frac{1}{2} \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Applied simplify0.0

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

    7. Applied simplify0.0

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

    if 1901.6558299178805 < i

    1. Initial program 46.9

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

    2. Applied simplify31.7

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

    3. Taylor expanded around inf 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

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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)))