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

↓

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

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

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

\end{array}

double f(double x) {
        double r19217 = 1.0;
        double r19218 = x;
        double r19219 = cos(r19218);
        double r19220 = r19217 - r19219;
        double r19221 = r19218 * r19218;
        double r19222 = r19220 / r19221;
        return r19222;
}

↓

double f(double x) {
        double r19223 = x;
        double r19224 = -0.032962631270408896;
        bool r19225 = r19223 <= r19224;
        double r19226 = 1.0;
        double r19227 = cos(r19223);
        double r19228 = r19226 - r19227;
        double r19229 = cbrt(r19228);
        double r19230 = r19229 * r19229;
        double r19231 = r19230 / r19223;
        double r19232 = r19229 / r19223;
        double r19233 = r19231 * r19232;
        double r19234 = 0.02432033747417449;
        bool r19235 = r19223 <= r19234;
        double r19236 = r19223 * r19223;
        double r19237 = -0.041666666666666664;
        double r19238 = 0.001388888888888889;
        double r19239 = 4.0;
        double r19240 = pow(r19223, r19239);
        double r19241 = 0.5;
        double r19242 = fma(r19238, r19240, r19241);
        double r19243 = fma(r19236, r19237, r19242);
        double r19244 = 3.0;
        double r19245 = pow(r19226, r19244);
        double r19246 = pow(r19227, r19244);
        double r19247 = r19245 - r19246;
        double r19248 = log(r19247);
        double r19249 = exp(r19248);
        double r19250 = r19226 + r19227;
        double r19251 = r19227 * r19250;
        double r19252 = fma(r19226, r19226, r19251);
        double r19253 = r19236 * r19252;
        double r19254 = r19249 / r19253;
        double r19255 = r19235 ? r19243 : r19254;
        double r19256 = r19225 ? r19233 : r19255;
        return r19256;
}

Derivation

Split input into 3 regimes
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
  \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)}\]
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
  \[\leadsto \frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}}\]
Recombined 3 regimes into one program.
Final simplification0.5
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03296263127040889584495886310833157040179:\\ \;\;\;\;\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\\ \mathbf{elif}\;x \le 0.02432033747417448876770862398188910447061:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(1, 1, \cos x \cdot \left(1 + \cos x\right)\right)}\\ \end{array}\]

Error

Derivation

`if x < -0.032962631270408896`

`if -0.032962631270408896 < x < 0.02432033747417449`

`if 0.02432033747417449 < x`

Reproduce