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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03305631426271166128927347926946822553873:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot x}}{x}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001388888888888870076776527184847509488463, {x}^{5}, x \cdot 0.5\right) - {x}^{3} \cdot 0.04166666666666668517038374375260900706053}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.03305631426271166128927347926946822553873:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot x}}{x}\\

\mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.001388888888888870076776527184847509488463, {x}^{5}, x \cdot 0.5\right) - {x}^{3} \cdot 0.04166666666666668517038374375260900706053}{x}\\

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

\end{array}

double f(double x) {
        double r24137 = 1.0;
        double r24138 = x;
        double r24139 = cos(r24138);
        double r24140 = r24137 - r24139;
        double r24141 = r24138 * r24138;
        double r24142 = r24140 / r24141;
        return r24142;
}

↓

double f(double x) {
        double r24143 = x;
        double r24144 = -0.03305631426271166;
        bool r24145 = r24143 <= r24144;
        double r24146 = 1.0;
        double r24147 = 3.0;
        double r24148 = pow(r24146, r24147);
        double r24149 = cos(r24143);
        double r24150 = pow(r24149, r24147);
        double r24151 = r24148 - r24150;
        double r24152 = r24146 + r24149;
        double r24153 = r24146 * r24146;
        double r24154 = fma(r24149, r24152, r24153);
        double r24155 = r24154 * r24143;
        double r24156 = r24151 / r24155;
        double r24157 = r24156 / r24143;
        double r24158 = 0.033302914266201536;
        bool r24159 = r24143 <= r24158;
        double r24160 = 0.00138888888888887;
        double r24161 = 5.0;
        double r24162 = pow(r24143, r24161);
        double r24163 = 0.5;
        double r24164 = r24143 * r24163;
        double r24165 = fma(r24160, r24162, r24164);
        double r24166 = pow(r24143, r24147);
        double r24167 = 0.041666666666666685;
        double r24168 = r24166 * r24167;
        double r24169 = r24165 - r24168;
        double r24170 = r24169 / r24143;
        double r24171 = r24146 - r24149;
        double r24172 = log(r24171);
        double r24173 = exp(r24172);
        double r24174 = r24143 * r24143;
        double r24175 = r24173 / r24174;
        double r24176 = r24159 ? r24170 : r24175;
        double r24177 = r24145 ? r24157 : r24176;
        return r24177;
}

Derivation

Split input into 3 regimes
if x < -0.03305631426271166
1. Initial program 1.3
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied associate-/r*0.5
  \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
4. Using strategy rm
5. Applied flip3--0.5
  \[\leadsto \frac{\frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x}}{x}\]
6. Applied associate-/l/0.5
  \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}}{x}\]
7. Simplified0.5
  \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot x}}}{x}\]
if -0.03305631426271166 < x < 0.033302914266201536
1. Initial program 62.3
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied associate-/r*61.3
  \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
4. Using strategy rm
5. Applied flip3--61.3
  \[\leadsto \frac{\frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x}}{x}\]
6. Applied associate-/l/61.3
  \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}}{x}\]
7. Simplified61.3
  \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot x}}}{x}\]
8. Taylor expanded around 0 0.0
  \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 0.001388888888888870076776527184847509488463 \cdot {x}^{5}\right) - 0.04166666666666668517038374375260900706053 \cdot {x}^{3}}}{x}\]
9. Simplified0.0
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.001388888888888870076776527184847509488463, {x}^{5}, 0.5 \cdot x\right) - 0.04166666666666668517038374375260900706053 \cdot {x}^{3}}}{x}\]
if 0.033302914266201536 < x
1. Initial program 1.1
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied add-exp-log1.1
  \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x \cdot x}\]
Recombined 3 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03305631426271166128927347926946822553873:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot x}}{x}\\ \mathbf{elif}\;x \le 0.03330291426620153594218010084659908898175:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.001388888888888870076776527184847509488463, {x}^{5}, x \cdot 0.5\right) - {x}^{3} \cdot 0.04166666666666668517038374375260900706053}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

Error

Derivation

`if x < -0.03305631426271166`

`if -0.03305631426271166 < x < 0.033302914266201536`

`if 0.033302914266201536 < x`

Reproduce