Error

Derivation

1. Split input into 2 regimes

2. if i < 226.15481187430868

   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 simplify0.1

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

   4. Applied log1p-expm1-u0.1

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

   5. Applied simplify0.0

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

   if 226.15481187430868 < i

   1. Initial program 46.1

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

   2. Applied simplify30.1

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

   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.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018199 +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)))