Average Error: 45.8 → 0.0
Time: 41.8s
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 223.45674042688236:\\ \;\;\;\;\frac{\frac{i}{2}}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*} \cdot \frac{i}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{0.0078125}{{i}^{5}}\right) \cdot i + \left((\left(\frac{1}{i}\right) \cdot \left(\frac{0.03125}{i}\right) + \frac{1}{8})_*\right))_*}{2}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 223.45674042688236

    1. Initial program 45.0

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

    2. Applied simplify0.0

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

    if 223.45674042688236 < i

    1. Initial program 46.7

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

    2. Applied simplify31.0

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

    3. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(0.0078125 \cdot \frac{1}{{i}^{5}} + \left(0.03125 \cdot \frac{1}{{i}^{3}} + \frac{1}{8} \cdot \frac{1}{i}\right)\right)} \cdot \frac{1 \cdot i}{2}\]

    4. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{(\left(\frac{0.0078125}{{i}^{5}}\right) \cdot i + \left((\left(\frac{1}{i}\right) \cdot \left(\frac{0.03125}{i}\right) + \frac{1}{8})_*\right))_*}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;i \le 223.45674042688236:\\ \;\;\;\;\frac{\frac{i}{2}}{(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + \left(-1.0\right))_*} \cdot \frac{i}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{0.0078125}{{i}^{5}}\right) \cdot i + \left((\left(\frac{1}{i}\right) \cdot \left(\frac{0.03125}{i}\right) + \frac{1}{8})_*\right))_*}{2}\\ \end{array}}\]

Runtime

Time bar (total: 41.8s)Debug logProfile

herbie shell --seed 2019053 +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)))