Average Error: 31.7 → 0.5
Time: 15.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03296263127040889584495886310833157040179:\\ \;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\ \mathbf{elif}\;x \le 0.02432033747417448876770862398188910447061:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03296263127040889584495886310833157040179:\\
\;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\

\mathbf{elif}\;x \le 0.02432033747417448876770862398188910447061:\\
\;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\end{array}
double f(double x) {
        double r18285 = 1.0;
        double r18286 = x;
        double r18287 = cos(r18286);
        double r18288 = r18285 - r18287;
        double r18289 = r18286 * r18286;
        double r18290 = r18288 / r18289;
        return r18290;
}
↓
double f(double x) {
        double r18291 = x;
        double r18292 = -0.032962631270408896;
        bool r18293 = r18291 <= r18292;
        double r18294 = 1.0;
        double r18295 = cos(r18291);
        double r18296 = r18294 - r18295;
        double r18297 = cbrt(r18296);
        double r18298 = r18297 * r18297;
        double r18299 = r18298 / r18291;
        double r18300 = r18297 / r18291;
        double r18301 = r18299 * r18300;
        double r18302 = 0.02432033747417449;
        bool r18303 = r18291 <= r18302;
        double r18304 = 0.001388888888888889;
        double r18305 = 4.0;
        double r18306 = pow(r18291, r18305);
        double r18307 = r18304 * r18306;
        double r18308 = 0.5;
        double r18309 = r18307 + r18308;
        double r18310 = 0.041666666666666664;
        double r18311 = 2.0;
        double r18312 = pow(r18291, r18311);
        double r18313 = r18310 * r18312;
        double r18314 = r18309 - r18313;
        double r18315 = 3.0;
        double r18316 = pow(r18294, r18315);
        double r18317 = pow(r18295, r18315);
        double r18318 = r18316 - r18317;
        double r18319 = log(r18318);
        double r18320 = exp(r18319);
        double r18321 = r18291 * r18291;
        double r18322 = r18294 + r18295;
        double r18323 = r18295 * r18322;
        double r18324 = r18294 * r18294;
        double r18325 = r18323 + r18324;
        double r18326 = r18321 * r18325;
        double r18327 = r18320 / r18326;
        double r18328 = r18303 ? r18314 : r18327;
        double r18329 = r18293 ? r18301 : r18328;
        return r18329;
}

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 < -0.032962631270408896

    1. Initial program 1.2

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

    3. Applied add-cube-cbrt1.6

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

    4. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}}\]

    if -0.032962631270408896 < x < 0.02432033747417449

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.0

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

    if 0.02432033747417449 < x

    1. Initial program 1.1

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

    3. Applied add-exp-log1.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x \cdot x}\]
    4. Using strategy rm

    5. Applied flip3--1.2

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

    6. Applied log-div1.2

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

    7. Applied exp-diff1.2

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

    8. Applied associate-/l/1.2

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

    9. Simplified1.2

      \[\leadsto \frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03296263127040889584495886310833157040179:\\ \;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\ \mathbf{elif}\;x \le 0.02432033747417448876770862398188910447061:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))