Average Error: 31.6 → 0.3
Time: 4.5s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0203295408193157160769981572912001865916 \lor \neg \left(x \le 0.02935328501821784022429717708746466087177\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0203295408193157160769981572912001865916 \lor \neg \left(x \le 0.02935328501821784022429717708746466087177\right):\\
\;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\end{array}
double f(double x) {
        double r27764 = 1.0;
        double r27765 = x;
        double r27766 = cos(r27765);
        double r27767 = r27764 - r27766;
        double r27768 = r27765 * r27765;
        double r27769 = r27767 / r27768;
        return r27769;
}
↓
double f(double x) {
        double r27770 = x;
        double r27771 = -0.020329540819315716;
        bool r27772 = r27770 <= r27771;
        double r27773 = 0.02935328501821784;
        bool r27774 = r27770 <= r27773;
        double r27775 = !r27774;
        bool r27776 = r27772 || r27775;
        double r27777 = 1.0;
        double r27778 = cos(r27770);
        double r27779 = r27777 - r27778;
        double r27780 = r27779 / r27770;
        double r27781 = 1.0;
        double r27782 = r27781 / r27770;
        double r27783 = r27780 * r27782;
        double r27784 = 4.0;
        double r27785 = pow(r27770, r27784);
        double r27786 = 0.001388888888888889;
        double r27787 = 0.5;
        double r27788 = 0.041666666666666664;
        double r27789 = 2.0;
        double r27790 = pow(r27770, r27789);
        double r27791 = r27788 * r27790;
        double r27792 = r27787 - r27791;
        double r27793 = fma(r27785, r27786, r27792);
        double r27794 = r27776 ? r27783 : r27793;
        return r27794;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.020329540819315716 or 0.02935328501821784 < x

    1. Initial program 1.2

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.3

      \[\leadsto \frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{x \cdot x}\]

    4. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}}\]
    5. Using strategy rm

    6. Applied div-inv0.6

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

    7. Applied associate-*r*0.6

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

    8. Simplified0.5

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x}} \cdot \frac{1}{x}\]

    if -0.020329540819315716 < x < 0.02935328501821784

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0203295408193157160769981572912001865916 \lor \neg \left(x \le 0.02935328501821784022429717708746466087177\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))