Average Error: 45.8 → 0.0
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 217.79103415086342:\\ \;\;\;\;i \cdot \left(\frac{\frac{1}{4}}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 217.79103415086342

    1. Initial program 44.4

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

    2. Initial simplification0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]
    3. Using strategy rm

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

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{1 \cdot \left(i \cdot \left(4 \cdot i\right) - 1.0\right)}}\]

    5. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{i}{1} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}}\]

    6. Simplified0.0

      \[\leadsto \color{blue}{i} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]
    7. Using strategy rm

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

      \[\leadsto i \cdot \frac{\frac{i}{4}}{\color{blue}{1 \cdot \left(i \cdot \left(4 \cdot i\right) - 1.0\right)}}\]

    9. Applied div-inv0.0

      \[\leadsto i \cdot \frac{\color{blue}{i \cdot \frac{1}{4}}}{1 \cdot \left(i \cdot \left(4 \cdot i\right) - 1.0\right)}\]

    10. Applied times-frac0.0

      \[\leadsto i \cdot \color{blue}{\left(\frac{i}{1} \cdot \frac{\frac{1}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\right)}\]

    11. Simplified0.0

      \[\leadsto i \cdot \left(\color{blue}{i} \cdot \frac{\frac{1}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\right)\]

    12. Simplified0.0

      \[\leadsto i \cdot \left(i \cdot \color{blue}{\frac{\frac{1}{4}}{4 \cdot \left(i \cdot i\right) - 1.0}}\right)\]

    if 217.79103415086342 < i

    1. Initial program 47.2

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

    2. Initial simplification31.7

      \[\leadsto \frac{i \cdot \frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity31.7

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{1 \cdot \left(i \cdot \left(4 \cdot i\right) - 1.0\right)}}\]

    5. Applied times-frac31.9

      \[\leadsto \color{blue}{\frac{i}{1} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}}\]

    6. Simplified31.9

      \[\leadsto \color{blue}{i} \cdot \frac{\frac{i}{4}}{i \cdot \left(4 \cdot i\right) - 1.0}\]

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 217.79103415086342:\\ \;\;\;\;i \cdot \left(\frac{\frac{1}{4}}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Runtime

Time bar (total: 18.1s)Debug logProfile

herbie shell --seed 2018346 
(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)))