Average Error: 46.4 → 0.0
Time: 2.3s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 219.764984409217874:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\ \end{array}\]
\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 219.764984409217874:\\
\;\;\;\;\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(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\

\end{array}
double f(double i) {
        double r111761 = i;
        double r111762 = r111761 * r111761;
        double r111763 = r111762 * r111762;
        double r111764 = 2.0;
        double r111765 = r111764 * r111761;
        double r111766 = r111765 * r111765;
        double r111767 = r111763 / r111766;
        double r111768 = 1.0;
        double r111769 = r111766 - r111768;
        double r111770 = r111767 / r111769;
        return r111770;
}
↓
double f(double i) {
        double r111771 = i;
        double r111772 = 219.76498440921787;
        bool r111773 = r111771 <= r111772;
        double r111774 = r111771 * r111771;
        double r111775 = 2.0;
        double r111776 = 1.0;
        double r111777 = sqrt(r111776);
        double r111778 = fma(r111775, r111771, r111777);
        double r111779 = r111775 * r111771;
        double r111780 = r111779 - r111777;
        double r111781 = r111778 * r111780;
        double r111782 = r111775 * r111775;
        double r111783 = r111781 * r111782;
        double r111784 = r111774 / r111783;
        double r111785 = 4.0;
        double r111786 = pow(r111777, r111785);
        double r111787 = pow(r111771, r111785);
        double r111788 = r111786 / r111787;
        double r111789 = 0.00390625;
        double r111790 = 0.0625;
        double r111791 = 0.015625;
        double r111792 = r111791 * r111776;
        double r111793 = 2.0;
        double r111794 = pow(r111771, r111793);
        double r111795 = r111792 / r111794;
        double r111796 = r111790 + r111795;
        double r111797 = fma(r111788, r111789, r111796);
        double r111798 = r111773 ? r111784 : r111797;
        return r111798;
}

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 219.76498440921787

    1. Initial program 44.4

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

    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 219.76498440921787 < i

    1. Initial program 48.4

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

    2. Simplified32.4

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

    4. Applied add-sqr-sqrt32.4

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

    5. Applied difference-of-squares32.4

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

    6. Simplified32.4

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

    7. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}} + \left(0.015625 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{{i}^{2}} + 0.0625\right)}\]

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 219.764984409217874:\\ \;\;\;\;\frac{i \cdot i}{\left(\mathsf{fma}\left(2, i, \sqrt{1}\right) \cdot \left(2 \cdot i - \sqrt{1}\right)\right) \cdot \left(2 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sqrt{1}\right)}^{4}}{{i}^{4}}, 0.00390625, 0.0625 + \frac{0.015625 \cdot 1}{{i}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))