Average Error: 32.0 → 0.2
Time: 15.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2}\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

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

\end{array}
double f(double x) {
        double r22573 = 1.0;
        double r22574 = x;
        double r22575 = cos(r22574);
        double r22576 = r22573 - r22575;
        double r22577 = r22574 * r22574;
        double r22578 = r22576 / r22577;
        return r22578;
}
↓
double f(double x) {
        double r22579 = x;
        double r22580 = -0.029468940558385545;
        bool r22581 = r22579 <= r22580;
        double r22582 = 0.029870589015516296;
        bool r22583 = r22579 <= r22582;
        double r22584 = !r22583;
        bool r22585 = r22581 || r22584;
        double r22586 = 1.0;
        double r22587 = cos(r22579);
        double r22588 = r22586 - r22587;
        double r22589 = r22588 / r22579;
        double r22590 = r22589 / r22579;
        double r22591 = 2.0;
        double r22592 = pow(r22579, r22591);
        double r22593 = -0.041666666666666664;
        double r22594 = 4.0;
        double r22595 = pow(r22579, r22594);
        double r22596 = 0.001388888888888889;
        double r22597 = 0.5;
        double r22598 = fma(r22595, r22596, r22597);
        double r22599 = fma(r22592, r22593, r22598);
        double r22600 = r22585 ? r22590 : r22599;
        return r22600;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.029468940558385545 or 0.029870589015516296 < x

    1. Initial program 1.0

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

    3. Applied associate-/r*0.5

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

    if -0.029468940558385545 < x < 0.029870589015516296

    1. Initial program 62.2

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

    3. Applied associate-/r*61.3

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02946894055838554515869276428929879330099 \lor \neg \left(x \le 0.02987058901551629605530813194036454660818\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))