Average Error: 46.8 → 0.0
Time: 2.5s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 260.729187563373387:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;i \le 260.729187563373387:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\

\end{array}
double f(double i) {
        double r52240 = i;
        double r52241 = r52240 * r52240;
        double r52242 = r52241 * r52241;
        double r52243 = 2.0;
        double r52244 = r52243 * r52240;
        double r52245 = r52244 * r52244;
        double r52246 = r52242 / r52245;
        double r52247 = 1.0;
        double r52248 = r52245 - r52247;
        double r52249 = r52246 / r52248;
        return r52249;
}
↓
double f(double i) {
        double r52250 = i;
        double r52251 = 260.7291875633734;
        bool r52252 = r52250 <= r52251;
        double r52253 = r52250 * r52250;
        double r52254 = 2.0;
        double r52255 = r52254 * r52250;
        double r52256 = r52255 * r52255;
        double r52257 = 1.0;
        double r52258 = r52256 - r52257;
        double r52259 = r52254 * r52254;
        double r52260 = r52258 * r52259;
        double r52261 = r52253 / r52260;
        double r52262 = 0.00390625;
        double r52263 = 1.0;
        double r52264 = 4.0;
        double r52265 = pow(r52250, r52264);
        double r52266 = r52263 / r52265;
        double r52267 = r52262 * r52266;
        double r52268 = 0.015625;
        double r52269 = 2.0;
        double r52270 = pow(r52250, r52269);
        double r52271 = r52263 / r52270;
        double r52272 = r52268 * r52271;
        double r52273 = 0.0625;
        double r52274 = r52272 + r52273;
        double r52275 = r52267 + r52274;
        double r52276 = r52252 ? r52261 : r52275;
        return r52276;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 260.7291875633734

    1. Initial program 45.1

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified0.0

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

    if 260.7291875633734 < i

    1. Initial program 48.5

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified32.6

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 260.729187563373387:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))