Average Error: 45.8 → 0.0
Time: 19.2s
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 183.69531661993628:\\ \;\;\;\;\frac{\frac{i}{2}}{(\left(i + i\right) \cdot \left(i + i\right) + \left(-1.0\right))_*} \cdot \frac{i}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 183.69531661993628

    1. Initial program 44.6

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

    2. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{\frac{i \cdot 1}{2}}{(\left(i + i\right) \cdot \left(i + i\right) + \left(-1.0\right))_*} \cdot \frac{i \cdot 1}{2}}\]

    if 183.69531661993628 < i

    1. Initial program 47.0

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

    2. Applied simplify30.9

      \[\leadsto \color{blue}{\frac{\frac{i \cdot 1}{2}}{(\left(i + i\right) \cdot \left(i + i\right) + \left(-1.0\right))_*} \cdot \frac{i \cdot 1}{2}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{16} + 0.015625 \cdot \frac{1}{{i}^{2}}\right)}\]

    4. Applied simplify0.0

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

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le 183.69531661993628:\\ \;\;\;\;\frac{\frac{i}{2}}{(\left(i + i\right) \cdot \left(i + i\right) + \left(-1.0\right))_*} \cdot \frac{i}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}}\]

Runtime

Time bar (total: 19.2s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))