Average Error: 45.4 → 0.0
Time: 14.1s
Precision: 64
\[i \gt 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1645.4224050168862:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)\\ \end{array}\]
\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) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 1645.4224050168862:\\
\;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)\\

\end{array}
double f(double i) {
        double r1187778 = i;
        double r1187779 = r1187778 * r1187778;
        double r1187780 = r1187779 * r1187779;
        double r1187781 = 2.0;
        double r1187782 = r1187781 * r1187778;
        double r1187783 = r1187782 * r1187782;
        double r1187784 = r1187780 / r1187783;
        double r1187785 = 1.0;
        double r1187786 = r1187783 - r1187785;
        double r1187787 = r1187784 / r1187786;
        return r1187787;
}
↓
double f(double i) {
        double r1187788 = i;
        double r1187789 = 1645.4224050168862;
        bool r1187790 = r1187788 <= r1187789;
        double r1187791 = 0.25;
        double r1187792 = r1187791 * r1187788;
        double r1187793 = r1187792 * r1187788;
        double r1187794 = 4.0;
        double r1187795 = r1187788 * r1187788;
        double r1187796 = r1187794 * r1187795;
        double r1187797 = 1.0;
        double r1187798 = r1187796 - r1187797;
        double r1187799 = r1187793 / r1187798;
        double r1187800 = 0.015625;
        double r1187801 = 1.0;
        double r1187802 = r1187801 / r1187795;
        double r1187803 = r1187802 / r1187795;
        double r1187804 = 0.00390625;
        double r1187805 = 0.0625;
        double r1187806 = fma(r1187803, r1187804, r1187805);
        double r1187807 = fma(r1187800, r1187802, r1187806);
        double r1187808 = r1187790 ? r1187799 : r1187807;
        return r1187808;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1645.4224050168862

    1. Initial program 43.9

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]
    3. Using strategy rm

    4. Applied associate-*l*0.0

      \[\leadsto \frac{\color{blue}{i \cdot \left(i \cdot \frac{1}{4}\right)}}{\left(i \cdot i\right) \cdot 4 - 1.0}\]

    if 1645.4224050168862 < i

    1. Initial program 46.9

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Simplified30.6

      \[\leadsto \color{blue}{\frac{\left(i \cdot i\right) \cdot \frac{1}{4}}{\left(i \cdot i\right) \cdot 4 - 1.0}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \left(\frac{1}{16} + 0.00390625 \cdot \frac{1}{{i}^{4}}\right)}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1645.4224050168862:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot i\right) \cdot i}{4 \cdot \left(i \cdot i\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.015625, \frac{1}{i \cdot i}, \mathsf{fma}\left(\frac{\frac{1}{i \cdot i}}{i \cdot i}, 0.00390625, \frac{1}{16}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))