Average Error: 46.8 → 0.0
Time: 2.4s
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 r55479 = i;
        double r55480 = r55479 * r55479;
        double r55481 = r55480 * r55480;
        double r55482 = 2.0;
        double r55483 = r55482 * r55479;
        double r55484 = r55483 * r55483;
        double r55485 = r55481 / r55484;
        double r55486 = 1.0;
        double r55487 = r55484 - r55486;
        double r55488 = r55485 / r55487;
        return r55488;
}
↓
double f(double i) {
        double r55489 = i;
        double r55490 = 1923.1709384006522;
        bool r55491 = r55489 <= r55490;
        double r55492 = r55489 * r55489;
        double r55493 = 2.0;
        double r55494 = r55493 * r55489;
        double r55495 = 1.0;
        double r55496 = sqrt(r55495);
        double r55497 = r55494 + r55496;
        double r55498 = r55494 - r55496;
        double r55499 = r55497 * r55498;
        double r55500 = r55493 * r55493;
        double r55501 = r55499 * r55500;
        double r55502 = r55492 / r55501;
        double r55503 = 0.0625;
        double r55504 = 0.015625;
        double r55505 = r55504 * r55495;
        double r55506 = 2.0;
        double r55507 = pow(r55489, r55506);
        double r55508 = r55505 / r55507;
        double r55509 = r55503 + r55508;
        double r55510 = 0.00390625;
        double r55511 = 4.0;
        double r55512 = pow(r55496, r55511);
        double r55513 = pow(r55489, r55511);
        double r55514 = r55512 / r55513;
        double r55515 = r55510 * r55514;
        double r55516 = r55509 + r55515;
        double r55517 = r55491 ? r55502 : r55516;
        return r55517;
}

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)))