Average Error: 46.2 → 0.0
Time: 34.7s
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 229.06302346572994:\\ \;\;\;\;\frac{{\left(\frac{i}{2}\right)}^{2}}{\left(2 \cdot 2\right) \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 229.06302346572994

    1. Initial program 45.2

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

    2. Initial simplification0.0

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

    4. Applied pow20.0

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

    5. Applied pow20.0

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

    6. Applied pow-prod-down0.0

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

    7. Simplified0.0

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

    if 229.06302346572994 < i

    1. Initial program 47.3

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

    2. Initial simplification32.1

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

    3. Taylor expanded around inf 0.0

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

    4. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 229.06302346572994:\\ \;\;\;\;\frac{{\left(\frac{i}{2}\right)}^{2}}{\left(2 \cdot 2\right) \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Runtime

Time bar (total: 34.7s)Debug logProfile

herbie shell --seed 2018221 
(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)))