Average Error: 46.8 → 0.0
Time: 1.9s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 215.890116719305212:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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 215.890116719305212:\\
\;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\

\end{array}
double f(double i) {
        double r48969 = i;
        double r48970 = r48969 * r48969;
        double r48971 = r48970 * r48970;
        double r48972 = 2.0;
        double r48973 = r48972 * r48969;
        double r48974 = r48973 * r48973;
        double r48975 = r48971 / r48974;
        double r48976 = 1.0;
        double r48977 = r48974 - r48976;
        double r48978 = r48975 / r48977;
        return r48978;
}
↓
double f(double i) {
        double r48979 = i;
        double r48980 = 215.8901167193052;
        bool r48981 = r48979 <= r48980;
        double r48982 = r48979 * r48979;
        double r48983 = 2.0;
        double r48984 = 1.0;
        double r48985 = sqrt(r48984);
        double r48986 = fma(r48983, r48979, r48985);
        double r48987 = r48983 * r48979;
        double r48988 = r48987 - r48985;
        double r48989 = r48986 * r48988;
        double r48990 = r48983 * r48983;
        double r48991 = r48989 * r48990;
        double r48992 = r48982 / r48991;
        double r48993 = 0.00390625;
        double r48994 = 1.0;
        double r48995 = 4.0;
        double r48996 = pow(r48979, r48995);
        double r48997 = r48994 / r48996;
        double r48998 = 0.015625;
        double r48999 = 2.0;
        double r49000 = pow(r48979, r48999);
        double r49001 = r48994 / r49000;
        double r49002 = 0.0625;
        double r49003 = fma(r48998, r49001, r49002);
        double r49004 = fma(r48993, r48997, r49003);
        double r49005 = r48981 ? r48992 : r49004;
        return r49005;
}

Error

Bits error versus i

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

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

    6. Simplified0.0

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

    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. 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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))