Average Error: 46.5 → 0.0
Time: 2.0s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 631.9542587844484842207748442888259887695:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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 631.9542587844484842207748442888259887695:\\
\;\;\;\;\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 r124183 = i;
        double r124184 = r124183 * r124183;
        double r124185 = r124184 * r124184;
        double r124186 = 2.0;
        double r124187 = r124186 * r124183;
        double r124188 = r124187 * r124187;
        double r124189 = r124185 / r124188;
        double r124190 = 1.0;
        double r124191 = r124188 - r124190;
        double r124192 = r124189 / r124191;
        return r124192;
}
↓
double f(double i) {
        double r124193 = i;
        double r124194 = 631.9542587844485;
        bool r124195 = r124193 <= r124194;
        double r124196 = r124193 * r124193;
        double r124197 = 2.0;
        double r124198 = r124197 * r124193;
        double r124199 = r124198 * r124198;
        double r124200 = 1.0;
        double r124201 = r124199 - r124200;
        double r124202 = r124197 * r124197;
        double r124203 = r124201 * r124202;
        double r124204 = r124196 / r124203;
        double r124205 = 0.00390625;
        double r124206 = 1.0;
        double r124207 = 4.0;
        double r124208 = pow(r124193, r124207);
        double r124209 = r124206 / r124208;
        double r124210 = r124205 * r124209;
        double r124211 = 0.015625;
        double r124212 = 2.0;
        double r124213 = pow(r124193, r124212);
        double r124214 = r124206 / r124213;
        double r124215 = r124211 * r124214;
        double r124216 = 0.0625;
        double r124217 = r124215 + r124216;
        double r124218 = r124210 + r124217;
        double r124219 = r124195 ? r124204 : r124218;
        return r124219;
}

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 < 631.9542587844485

    1. Initial program 45.2

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

    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 631.9542587844485 < i

    1. Initial program 47.9

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

    2. Simplified32.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 631.9542587844484842207748442888259887695:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))