Average Error: 31.9 → 0.2
Time: 19.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\
\;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \frac{1}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\

\end{array}
double f(double x) {
        double r868457 = 1.0;
        double r868458 = x;
        double r868459 = cos(r868458);
        double r868460 = r868457 - r868459;
        double r868461 = r868458 * r868458;
        double r868462 = r868460 / r868461;
        return r868462;
}
↓
double f(double x) {
        double r868463 = x;
        double r868464 = -0.03115379972147734;
        bool r868465 = r868463 <= r868464;
        double r868466 = 1.0;
        double r868467 = cos(r868463);
        double r868468 = r868466 - r868467;
        double r868469 = r868468 / r868463;
        double r868470 = r868469 / r868463;
        double r868471 = 0.027390056951785625;
        bool r868472 = r868463 <= r868471;
        double r868473 = r868463 * r868463;
        double r868474 = 0.001388888888888889;
        double r868475 = r868473 * r868474;
        double r868476 = -0.041666666666666664;
        double r868477 = 0.5;
        double r868478 = fma(r868476, r868473, r868477);
        double r868479 = fma(r868473, r868475, r868478);
        double r868480 = r868472 ? r868479 : r868470;
        double r868481 = r868465 ? r868470 : r868480;
        return r868481;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03115379972147734 or 0.027390056951785625 < x

    1. Initial program 1.0

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

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

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

    6. Applied associate-*r/0.5

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

    7. Simplified0.5

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

    if -0.03115379972147734 < x < 0.027390056951785625

    1. Initial program 62.4

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))