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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.033320062920463786:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0329960873997033124:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.033320062920463786:\\
\;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\\

\mathbf{elif}\;x \le 0.0329960873997033124:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

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

\end{array}

double f(double x) {
        double r35764 = 1.0;
        double r35765 = x;
        double r35766 = cos(r35765);
        double r35767 = r35764 - r35766;
        double r35768 = r35765 * r35765;
        double r35769 = r35767 / r35768;
        return r35769;
}

↓

double f(double x) {
        double r35770 = x;
        double r35771 = -0.033320062920463786;
        bool r35772 = r35770 <= r35771;
        double r35773 = 1.0;
        double r35774 = 3.0;
        double r35775 = pow(r35773, r35774);
        double r35776 = cos(r35770);
        double r35777 = pow(r35776, r35774);
        double r35778 = log1p(r35777);
        double r35779 = expm1(r35778);
        double r35780 = r35775 - r35779;
        double r35781 = r35780 / r35770;
        double r35782 = r35770 * r35773;
        double r35783 = r35770 * r35776;
        double r35784 = r35773 + r35776;
        double r35785 = r35783 * r35784;
        double r35786 = fma(r35773, r35782, r35785);
        double r35787 = r35781 / r35786;
        double r35788 = 0.03299608739970331;
        bool r35789 = r35770 <= r35788;
        double r35790 = 4.0;
        double r35791 = pow(r35770, r35790);
        double r35792 = 0.001388888888888889;
        double r35793 = 0.5;
        double r35794 = 0.041666666666666664;
        double r35795 = 2.0;
        double r35796 = pow(r35770, r35795);
        double r35797 = r35794 * r35796;
        double r35798 = r35793 - r35797;
        double r35799 = fma(r35791, r35792, r35798);
        double r35800 = r35773 - r35776;
        double r35801 = log(r35800);
        double r35802 = exp(r35801);
        double r35803 = r35770 * r35770;
        double r35804 = r35802 / r35803;
        double r35805 = r35789 ? r35799 : r35804;
        double r35806 = r35772 ? r35787 : r35805;
        return r35806;
}

Derivation

Split input into 3 regimes
if x < -0.033320062920463786
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-frac0.5
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
5. Using strategy rm
6. Applied div-inv0.6
  \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(1 - \cos x\right) \cdot \frac{1}{x}\right)}\]
7. Using strategy rm
8. Applied flip3--0.6
  \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}} \cdot \frac{1}{x}\right)\]
9. Applied associate-*l/0.6
  \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\left({1}^{3} - {\left(\cos x\right)}^{3}\right) \cdot \frac{1}{x}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\]
10. Applied frac-times0.6
  \[\leadsto \color{blue}{\frac{1 \cdot \left(\left({1}^{3} - {\left(\cos x\right)}^{3}\right) \cdot \frac{1}{x}\right)}{x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]
11. Simplified0.5
  \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x}}}{x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}\]
12. Simplified0.5
  \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x}}{\color{blue}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}}\]
13. Using strategy rm
14. Applied expm1-log1p-u0.5
  \[\leadsto \frac{\frac{{1}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\]
if -0.033320062920463786 < x < 0.03299608739970331
1. Initial program 62.2
  \[\frac{1 - \cos x}{x \cdot x}\]
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
  \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
if 0.03299608739970331 < x
1. Initial program 1.0
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied add-exp-log1.0
  \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x \cdot x}\]
Recombined 3 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033320062920463786:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0329960873997033124:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

Error

Derivation

`if x < -0.033320062920463786`

`if -0.033320062920463786 < x < 0.03299608739970331`

`if 0.03299608739970331 < x`

Reproduce