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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 241.8575048288039:\\ \;\;\;\;\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i + i\right) \cdot \left(i + i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}\]

Derivation

Split input into 2 regimes
if i < 241.8575048288039
1. Initial program 45.4
  \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
2. Applied simplify0.0
  \[\leadsto \color{blue}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}\]
if 241.8575048288039 < i
1. Initial program 46.6
  \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
2. Applied simplify30.9
  \[\leadsto \color{blue}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}\]
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
  \[\leadsto \color{blue}{\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}}\]
Recombined 2 regimes into one program.
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))

Error

Derivation

`if i < 241.8575048288039`

`if 241.8575048288039 < i`

Runtime