Average Error: 45.5 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 265.6473276634458:\\ \;\;\;\;\frac{\frac{i}{2}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \cdot \frac{\frac{\frac{i}{2}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

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 < 265.6473276634458

    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

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    7. Simplified0.0

      \[\leadsto \frac{\frac{\frac{i}{2}}{\sqrt[3]{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}}{\sqrt[3]{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}} \cdot \color{blue}{\frac{\frac{i}{2}}{\sqrt[3]{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) - 1.0}}}\]

    if 265.6473276634458 < 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 simplification31.6

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 265.6473276634458:\\ \;\;\;\;\frac{\frac{i}{2}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}} \cdot \frac{\frac{\frac{i}{2}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}}}{\sqrt[3]{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018220 
(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)))