Average Error: 46.0 → 0.0
Time: 3.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 245.497035223853601:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right) + 0.0625\\ \end{array}\]
\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 245.497035223853601:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right) + 0.0625\\

\end{array}
double f(double i) {
        double r73265 = i;
        double r73266 = r73265 * r73265;
        double r73267 = r73266 * r73266;
        double r73268 = 2.0;
        double r73269 = r73268 * r73265;
        double r73270 = r73269 * r73269;
        double r73271 = r73267 / r73270;
        double r73272 = 1.0;
        double r73273 = r73270 - r73272;
        double r73274 = r73271 / r73273;
        return r73274;
}
↓
double f(double i) {
        double r73275 = i;
        double r73276 = 245.4970352238536;
        bool r73277 = r73275 <= r73276;
        double r73278 = r73275 * r73275;
        double r73279 = 2.0;
        double r73280 = r73279 * r73275;
        double r73281 = r73280 * r73280;
        double r73282 = 1.0;
        double r73283 = r73281 - r73282;
        double r73284 = r73279 * r73279;
        double r73285 = r73283 * r73284;
        double r73286 = r73278 / r73285;
        double r73287 = 0.015625;
        double r73288 = 2.0;
        double r73289 = pow(r73275, r73288);
        double r73290 = r73287 / r73289;
        double r73291 = 0.00390625;
        double r73292 = 4.0;
        double r73293 = pow(r73275, r73292);
        double r73294 = r73291 / r73293;
        double r73295 = r73290 + r73294;
        double r73296 = 0.0625;
        double r73297 = r73295 + r73296;
        double r73298 = r73277 ? r73286 : r73297;
        return r73298;
}

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 < 245.4970352238536

    1. Initial program 43.9

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

    2. 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 245.4970352238536 < i

    1. Initial program 48.0

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

    2. Simplified31.5

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

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]

    4. Simplified0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 245.497035223853601:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.015625}{{i}^{2}} + \frac{0.00390625}{{i}^{4}}\right) + 0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))