Average Error: 47.0 → 0.0
Time: 7.0s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1712.592311768091121848556213080883026123:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\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 1712.592311768091121848556213080883026123:\\
\;\;\;\;\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 r53391 = i;
        double r53392 = r53391 * r53391;
        double r53393 = r53392 * r53392;
        double r53394 = 2.0;
        double r53395 = r53394 * r53391;
        double r53396 = r53395 * r53395;
        double r53397 = r53393 / r53396;
        double r53398 = 1.0;
        double r53399 = r53396 - r53398;
        double r53400 = r53397 / r53399;
        return r53400;
}
↓
double f(double i) {
        double r53401 = i;
        double r53402 = 1712.5923117680911;
        bool r53403 = r53401 <= r53402;
        double r53404 = r53401 * r53401;
        double r53405 = 2.0;
        double r53406 = r53405 * r53401;
        double r53407 = r53406 * r53406;
        double r53408 = 1.0;
        double r53409 = r53407 - r53408;
        double r53410 = r53405 * r53405;
        double r53411 = r53409 * r53410;
        double r53412 = r53404 / r53411;
        double r53413 = 0.00390625;
        double r53414 = 1.0;
        double r53415 = 4.0;
        double r53416 = pow(r53401, r53415);
        double r53417 = r53414 / r53416;
        double r53418 = r53413 * r53417;
        double r53419 = 0.015625;
        double r53420 = 2.0;
        double r53421 = pow(r53401, r53420);
        double r53422 = r53414 / r53421;
        double r53423 = r53419 * r53422;
        double r53424 = 0.0625;
        double r53425 = r53423 + r53424;
        double r53426 = r53418 + r53425;
        double r53427 = r53403 ? r53412 : r53426;
        return r53427;
}

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 < 1712.5923117680911

    1. Initial program 45.3

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

    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 1712.5923117680911 < i

    1. Initial program 48.7

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

    2. Simplified33.2

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1712.592311768091121848556213080883026123:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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)))