Average Error: 46.9 → 0.0
Time: 1.7s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 3729.0156333395453:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]
\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 3729.0156333395453:\\
\;\;\;\;\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}:\\
\;\;\;\;\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 r64903 = i;
        double r64904 = r64903 * r64903;
        double r64905 = r64904 * r64904;
        double r64906 = 2.0;
        double r64907 = r64906 * r64903;
        double r64908 = r64907 * r64907;
        double r64909 = r64905 / r64908;
        double r64910 = 1.0;
        double r64911 = r64908 - r64910;
        double r64912 = r64909 / r64911;
        return r64912;
}
↓
double f(double i) {
        double r64913 = i;
        double r64914 = 3729.0156333395453;
        bool r64915 = r64913 <= r64914;
        double r64916 = r64913 * r64913;
        double r64917 = 2.0;
        double r64918 = r64917 * r64913;
        double r64919 = r64918 * r64918;
        double r64920 = 1.0;
        double r64921 = r64919 - r64920;
        double r64922 = r64917 * r64917;
        double r64923 = r64921 * r64922;
        double r64924 = r64916 / r64923;
        double r64925 = 0.00390625;
        double r64926 = 1.0;
        double r64927 = 4.0;
        double r64928 = pow(r64913, r64927);
        double r64929 = r64926 / r64928;
        double r64930 = 0.015625;
        double r64931 = 2.0;
        double r64932 = pow(r64913, r64931);
        double r64933 = r64926 / r64932;
        double r64934 = 0.0625;
        double r64935 = fma(r64930, r64933, r64934);
        double r64936 = fma(r64925, r64929, r64935);
        double r64937 = r64915 ? r64924 : r64936;
        return r64937;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 3729.0156333395453

    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

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

    if 3729.0156333395453 < 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. Simplified33.1

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 3729.0156333395453:\\ \;\;\;\;\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.00390625, \frac{1}{{i}^{4}}, \mathsf{fma}\left(0.015625, \frac{1}{{i}^{2}}, 0.0625\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))