Average Error: 32.0 → 0.2
Time: 17.6s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03103024640337792919297932314748322824016 \lor \neg \left(x \le 0.03397543731978374531577102857227146159858\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03103024640337792919297932314748322824016 \lor \neg \left(x \le 0.03397543731978374531577102857227146159858\right):\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)\\

\end{array}
double f(double x) {
        double r29762 = 1.0;
        double r29763 = x;
        double r29764 = cos(r29763);
        double r29765 = r29762 - r29764;
        double r29766 = r29763 * r29763;
        double r29767 = r29765 / r29766;
        return r29767;
}
↓
double f(double x) {
        double r29768 = x;
        double r29769 = -0.03103024640337793;
        bool r29770 = r29768 <= r29769;
        double r29771 = 0.033975437319783745;
        bool r29772 = r29768 <= r29771;
        double r29773 = !r29772;
        bool r29774 = r29770 || r29773;
        double r29775 = 1.0;
        double r29776 = cos(r29768);
        double r29777 = r29775 - r29776;
        double r29778 = r29777 / r29768;
        double r29779 = r29778 / r29768;
        double r29780 = -0.041666666666666664;
        double r29781 = r29768 * r29768;
        double r29782 = 0.001388888888888889;
        double r29783 = 4.0;
        double r29784 = pow(r29768, r29783);
        double r29785 = 0.5;
        double r29786 = fma(r29782, r29784, r29785);
        double r29787 = fma(r29780, r29781, r29786);
        double r29788 = expm1(r29787);
        double r29789 = log1p(r29788);
        double r29790 = r29774 ? r29779 : r29789;
        return r29790;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03103024640337793 or 0.033975437319783745 < x

    1. Initial program 1.1

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

    3. Applied associate-/r*0.5

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

    if -0.03103024640337793 < x < 0.033975437319783745

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)}\]
    4. Using strategy rm

    5. Applied log1p-expm1-u0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03103024640337792919297932314748322824016 \lor \neg \left(x \le 0.03397543731978374531577102857227146159858\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))