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

↓

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

\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 r19042 = 1.0;
        double r19043 = x;
        double r19044 = cos(r19043);
        double r19045 = r19042 - r19044;
        double r19046 = r19043 * r19043;
        double r19047 = r19045 / r19046;
        return r19047;
}

↓

double f(double x) {
        double r19048 = x;
        double r19049 = -0.032962631270408896;
        bool r19050 = r19048 <= r19049;
        double r19051 = 1.0;
        double r19052 = cos(r19048);
        double r19053 = r19051 - r19052;
        double r19054 = cbrt(r19053);
        double r19055 = r19054 * r19054;
        double r19056 = r19055 / r19048;
        double r19057 = r19054 / r19048;
        double r19058 = r19056 * r19057;
        double r19059 = 0.02432033747417449;
        bool r19060 = r19048 <= r19059;
        double r19061 = r19048 * r19048;
        double r19062 = -0.041666666666666664;
        double r19063 = 0.001388888888888889;
        double r19064 = 4.0;
        double r19065 = pow(r19048, r19064);
        double r19066 = 0.5;
        double r19067 = fma(r19063, r19065, r19066);
        double r19068 = fma(r19061, r19062, r19067);
        double r19069 = 3.0;
        double r19070 = pow(r19051, r19069);
        double r19071 = pow(r19052, r19069);
        double r19072 = r19070 - r19071;
        double r19073 = log(r19072);
        double r19074 = exp(r19073);
        double r19075 = r19051 + r19052;
        double r19076 = r19052 * r19075;
        double r19077 = fma(r19051, r19051, r19076);
        double r19078 = r19061 * r19077;
        double r19079 = r19074 / r19078;
        double r19080 = r19060 ? r19068 : r19079;
        double r19081 = r19050 ? r19058 : r19080;
        return r19081;
}

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