Average Error: 46.1 → 0.0
Time: 14.5s
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 193.52074249686544:\\ \;\;\;\;\frac{\frac{i}{4}}{\sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}} \cdot \frac{i}{\sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*} \cdot \sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{{i}^{4}} + (\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_*\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 193.52074249686544

    1. Initial program 45.1

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

    2. Initial simplification0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied times-frac0.0

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

    if 193.52074249686544 < 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. Initial simplification32.8

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 193.52074249686544:\\ \;\;\;\;\frac{\frac{i}{4}}{\sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}} \cdot \frac{i}{\sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*} \cdot \sqrt[3]{(\left(4 \cdot i\right) \cdot i + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.00390625}{{i}^{4}} + (\left(\frac{0.015625}{i}\right) \cdot \left(\frac{1}{i}\right) + \frac{1}{16})_*\\ \end{array}\]

Runtime

Time bar (total: 14.5s)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(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)))