Average Error: 46.1 → 0.0
Time: 8.9s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 5247.199702919035:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot i\right) \cdot \left(\frac{1}{2} \cdot i\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right)\\ \end{array}\]

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 < 5247.199702919035

    1. Initial program 45.5

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

    2. Initial simplification0.0

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

    if 5247.199702919035 < i

    1. Initial program 46.8

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

    2. Initial simplification31.2

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

    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. Simplified0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 5247.199702919035:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot i\right) \cdot \left(\frac{1}{2} \cdot i\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right)\\ \end{array}\]

Runtime

Time bar (total: 8.9s)Debug logProfile

herbie shell --seed 2018215 
(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)))