Average Error: 45.8 → 0.0
Time: 33.3s
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1861.475052632144:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot i\right) \cdot \left(\frac{1}{2} \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \left(\frac{0.00390625}{{i}^{4}} + \frac{1}{16}\right)\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1861.475052632144

    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

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

    if 1861.475052632144 < 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 simplify31.6

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

    3. Taylor expanded around inf 0.0

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

    4. Applied simplify0.0

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

Runtime

Time bar (total: 33.3s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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)))