Average Error: 31.1 → 0.3
Time: 15.5s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\
\;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\

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

\end{array}
double f(double x) {
        double r21034 = 1.0;
        double r21035 = x;
        double r21036 = cos(r21035);
        double r21037 = r21034 - r21036;
        double r21038 = r21035 * r21035;
        double r21039 = r21037 / r21038;
        return r21039;
}
↓
double f(double x) {
        double r21040 = x;
        double r21041 = -0.028173748830909656;
        bool r21042 = r21040 <= r21041;
        double r21043 = 0.030278781168554474;
        bool r21044 = r21040 <= r21043;
        double r21045 = !r21044;
        bool r21046 = r21042 || r21045;
        double r21047 = 1.0;
        double r21048 = r21047 / r21040;
        double r21049 = 1.0;
        double r21050 = 3.0;
        double r21051 = pow(r21049, r21050);
        double r21052 = cos(r21040);
        double r21053 = pow(r21052, r21050);
        double r21054 = r21051 - r21053;
        double r21055 = r21049 + r21052;
        double r21056 = r21052 * r21055;
        double r21057 = r21049 * r21049;
        double r21058 = r21056 + r21057;
        double r21059 = r21058 * r21040;
        double r21060 = r21054 / r21059;
        double r21061 = r21048 * r21060;
        double r21062 = 0.001388888888888889;
        double r21063 = 4.0;
        double r21064 = pow(r21040, r21063);
        double r21065 = r21062 * r21064;
        double r21066 = 0.5;
        double r21067 = r21065 + r21066;
        double r21068 = 0.041666666666666664;
        double r21069 = 2.0;
        double r21070 = pow(r21040, r21069);
        double r21071 = r21068 * r21070;
        double r21072 = r21067 - r21071;
        double r21073 = r21046 ? r21061 : r21072;
        return r21073;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.028173748830909656 or 0.030278781168554474 < 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 flip3--0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x}\]

    7. Applied associate-/l/0.5

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

    8. Simplified0.5

      \[\leadsto \frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}}\]

    if -0.028173748830909656 < x < 0.030278781168554474

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02817374883090965551057927029887650860474 \lor \neg \left(x \le 0.03027878116855447360178388294116302859038\right):\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))