Average Error: 45.6 → 0.0
Time: 24.0s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 759.4631749369508:\\ \;\;\;\;\frac{i}{\sqrt[3]{\left(4 \cdot i\right) \cdot i - 1.0} \cdot \sqrt[3]{\left(4 \cdot i\right) \cdot i - 1.0}} \cdot \frac{\frac{i}{4}}{\frac{\sqrt[3]{16 \cdot {i}^{4} - 1.0 \cdot 1.0}}{\sqrt{\sqrt[3]{\left(4 \cdot i\right) \cdot i + 1.0}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot i\right) \cdot i + 1.0}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

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 < 759.4631749369508

    1. Initial program 44.1

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\color{blue}{\left(\sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0} \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}\right) \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}}}\]

    5. Applied times-frac0.0

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

    7. Applied flip--0.0

      \[\leadsto \frac{i}{\sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0} \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}} \cdot \frac{\frac{i}{4}}{\sqrt[3]{\color{blue}{\frac{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) - 1.0 \cdot 1.0}{i \cdot \left(4 \cdot i\right) + 1.0}}}}\]

    8. Applied cbrt-div0.0

      \[\leadsto \frac{i}{\sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0} \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}} \cdot \frac{\frac{i}{4}}{\color{blue}{\frac{\sqrt[3]{\left(i \cdot \left(4 \cdot i\right)\right) \cdot \left(i \cdot \left(4 \cdot i\right)\right) - 1.0 \cdot 1.0}}{\sqrt[3]{i \cdot \left(4 \cdot i\right) + 1.0}}}}\]

    9. Simplified0.0

      \[\leadsto \frac{i}{\sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0} \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}} \cdot \frac{\frac{i}{4}}{\frac{\color{blue}{\sqrt[3]{{i}^{4} \cdot 16 - 1.0 \cdot 1.0}}}{\sqrt[3]{i \cdot \left(4 \cdot i\right) + 1.0}}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{i}{\sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0} \cdot \sqrt[3]{i \cdot \left(4 \cdot i\right) - 1.0}} \cdot \frac{\frac{i}{4}}{\frac{\sqrt[3]{{i}^{4} \cdot 16 - 1.0 \cdot 1.0}}{\color{blue}{\sqrt{\sqrt[3]{i \cdot \left(4 \cdot i\right) + 1.0}} \cdot \sqrt{\sqrt[3]{i \cdot \left(4 \cdot i\right) + 1.0}}}}}\]

    if 759.4631749369508 < i

    1. Initial program 47.1

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

    2. Simplified31.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 759.4631749369508:\\ \;\;\;\;\frac{i}{\sqrt[3]{\left(4 \cdot i\right) \cdot i - 1.0} \cdot \sqrt[3]{\left(4 \cdot i\right) \cdot i - 1.0}} \cdot \frac{\frac{i}{4}}{\frac{\sqrt[3]{16 \cdot {i}^{4} - 1.0 \cdot 1.0}}{\sqrt{\sqrt[3]{\left(4 \cdot i\right) \cdot i + 1.0}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot i\right) \cdot i + 1.0}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{16} + \frac{0.00390625}{{i}^{4}}\right) + \frac{\frac{0.015625}{i}}{i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(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)))