Average Error: 46.8 → 0.0
Time: 2.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]
\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 215.890116719305212:\\
\;\;\;\;\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 r138103 = i;
        double r138104 = r138103 * r138103;
        double r138105 = r138104 * r138104;
        double r138106 = 2.0;
        double r138107 = r138106 * r138103;
        double r138108 = r138107 * r138107;
        double r138109 = r138105 / r138108;
        double r138110 = 1.0;
        double r138111 = r138108 - r138110;
        double r138112 = r138109 / r138111;
        return r138112;
}
↓
double f(double i) {
        double r138113 = i;
        double r138114 = 215.8901167193052;
        bool r138115 = r138113 <= r138114;
        double r138116 = r138113 * r138113;
        double r138117 = 2.0;
        double r138118 = r138117 * r138113;
        double r138119 = 1.0;
        double r138120 = sqrt(r138119);
        double r138121 = r138118 + r138120;
        double r138122 = r138118 - r138120;
        double r138123 = r138121 * r138122;
        double r138124 = r138117 * r138117;
        double r138125 = r138123 * r138124;
        double r138126 = r138116 / r138125;
        double r138127 = 0.0625;
        double r138128 = 0.015625;
        double r138129 = r138128 * r138119;
        double r138130 = 2.0;
        double r138131 = pow(r138113, r138130);
        double r138132 = r138129 / r138131;
        double r138133 = r138127 + r138132;
        double r138134 = 0.00390625;
        double r138135 = 4.0;
        double r138136 = pow(r138120, r138135);
        double r138137 = pow(r138113, r138135);
        double r138138 = r138136 / r138137;
        double r138139 = r138134 * r138138;
        double r138140 = r138133 + r138139;
        double r138141 = r138115 ? r138126 : r138140;
        return r138141;
}

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 < 215.8901167193052

    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

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    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 215.8901167193052 < i

    1. Initial program 48.5

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

    2. Simplified32.0

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

    4. Applied add-sqr-sqrt32.0

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

    5. Applied difference-of-squares32.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i + \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right) + 0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))