Average Error: 31.3 → 0.3
Time: 4.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{elif}\;x \le 0.0246736691289138228:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031288658239448007:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\

\mathbf{elif}\;x \le 0.0246736691289138228:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

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

\end{array}
double f(double x) {
        double r30229 = 1.0;
        double r30230 = x;
        double r30231 = cos(r30230);
        double r30232 = r30229 - r30231;
        double r30233 = r30230 * r30230;
        double r30234 = r30232 / r30233;
        return r30234;
}
↓
double f(double x) {
        double r30235 = x;
        double r30236 = -0.03128865823944801;
        bool r30237 = r30235 <= r30236;
        double r30238 = 1.0;
        double r30239 = r30238 / r30235;
        double r30240 = 1.0;
        double r30241 = cos(r30235);
        double r30242 = r30240 - r30241;
        double r30243 = r30242 / r30235;
        double r30244 = r30239 * r30243;
        double r30245 = 0.024673669128913823;
        bool r30246 = r30235 <= r30245;
        double r30247 = 0.001388888888888889;
        double r30248 = 4.0;
        double r30249 = pow(r30235, r30248);
        double r30250 = r30247 * r30249;
        double r30251 = 0.5;
        double r30252 = r30250 + r30251;
        double r30253 = 0.041666666666666664;
        double r30254 = 2.0;
        double r30255 = pow(r30235, r30254);
        double r30256 = r30253 * r30255;
        double r30257 = r30252 - r30256;
        double r30258 = r30235 / r30242;
        double r30259 = r30238 / r30258;
        double r30260 = r30239 * r30259;
        double r30261 = r30246 ? r30257 : r30260;
        double r30262 = r30237 ? r30244 : r30261;
        return r30262;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03128865823944801

    1. Initial program 0.8

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

    3. Applied *-un-lft-identity0.8

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

    4. Applied times-frac0.5

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

    if -0.03128865823944801 < x < 0.024673669128913823

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    if 0.024673669128913823 < x

    1. Initial program 1.1

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

    3. Applied *-un-lft-identity1.1

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

    4. Applied times-frac0.6

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

    6. Applied clear-num0.6

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \mathbf{elif}\;x \le 0.0246736691289138228:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))