Average Error: 45.4 → 0.0
Time: 27.8s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 509.0740230200442:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{\frac{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) - 1.0 \cdot 1.0}{1.0 + i \cdot \left(4 \cdot i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

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 < 509.0740230200442

    1. Initial program 44.2

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

    2. Initial simplification0.0

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

    4. Applied flip--0.0

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

    if 509.0740230200442 < i

    1. Initial program 46.6

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

    2. Initial simplification30.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 509.0740230200442:\\ \;\;\;\;\frac{\frac{i}{4} \cdot i}{\frac{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) - 1.0 \cdot 1.0}{1.0 + i \cdot \left(4 \cdot i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Runtime

Time bar (total: 27.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes23.90.00.023.9100%
herbie shell --seed 2018353 
(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)))