Average Error: 46.8 → 0.0
Time: 3.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1923.1709384006522:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]
\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 1923.1709384006522:\\
\;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\

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

\end{array}
double f(double i) {
        double r370 = i;
        double r371 = r370 * r370;
        double r372 = r371 * r371;
        double r373 = 2.0;
        double r374 = r373 * r370;
        double r375 = r374 * r374;
        double r376 = r372 / r375;
        double r377 = 1.0;
        double r378 = r375 - r377;
        double r379 = r376 / r378;
        return r379;
}
↓
double f(double i) {
        double r380 = i;
        double r381 = 1923.1709384006522;
        bool r382 = r380 <= r381;
        double r383 = r380 * r380;
        double r384 = 2.0;
        double r385 = r384 * r380;
        double r386 = 1.0;
        double r387 = sqrt(r386);
        double r388 = r385 + r387;
        double r389 = r385 - r387;
        double r390 = r388 * r389;
        double r391 = r384 * r384;
        double r392 = r390 * r391;
        double r393 = r383 / r392;
        double r394 = 0.0625;
        double r395 = 0.015625;
        double r396 = r395 * r386;
        double r397 = 2.0;
        double r398 = pow(r380, r397);
        double r399 = r396 / r398;
        double r400 = r394 + r399;
        double r401 = 0.00390625;
        double r402 = 4.0;
        double r403 = pow(r387, r402);
        double r404 = pow(r380, r402);
        double r405 = r403 / r404;
        double r406 = r401 * r405;
        double r407 = r400 + r406;
        double r408 = r382 ? r393 : r407;
        return r408;
}

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 < 1923.1709384006522

    1. Initial program 44.9

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if 1923.1709384006522 < i

    1. Initial program 48.9

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

    2. Simplified32.2

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt32.2

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

    5. Applied difference-of-squares32.2

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1923.1709384006522:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))