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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.033994064032605062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \mathbf{elif}\;x \le 0.032573444318334172:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.033994064032605062:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\

\mathbf{elif}\;x \le 0.032573444318334172:\\
\;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\

\end{array}

double f(double x) {
        double r31546 = 1.0;
        double r31547 = x;
        double r31548 = cos(r31547);
        double r31549 = r31546 - r31548;
        double r31550 = r31547 * r31547;
        double r31551 = r31549 / r31550;
        return r31551;
}

↓

double f(double x) {
        double r31552 = x;
        double r31553 = -0.03399406403260506;
        bool r31554 = r31552 <= r31553;
        double r31555 = 1.0;
        double r31556 = r31555 / r31552;
        double r31557 = 1.0;
        double r31558 = cos(r31552);
        double r31559 = r31557 - r31558;
        double r31560 = exp(r31559);
        double r31561 = log(r31560);
        double r31562 = r31561 / r31552;
        double r31563 = r31556 * r31562;
        double r31564 = 0.03257344431833417;
        bool r31565 = r31552 <= r31564;
        double r31566 = 4.0;
        double r31567 = pow(r31552, r31566);
        double r31568 = 0.001388888888888889;
        double r31569 = 0.5;
        double r31570 = 0.041666666666666664;
        double r31571 = 2.0;
        double r31572 = pow(r31552, r31571);
        double r31573 = r31570 * r31572;
        double r31574 = r31569 - r31573;
        double r31575 = fma(r31567, r31568, r31574);
        double r31576 = r31552 / r31559;
        double r31577 = r31555 / r31576;
        double r31578 = r31556 * r31577;
        double r31579 = r31565 ? r31575 : r31578;
        double r31580 = r31554 ? r31563 : r31579;
        return r31580;
}

Derivation

Split input into 3 regimes
if x < -0.03399406403260506
1. Initial program 1.2
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied *-un-lft-identity1.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
4. Applied times-frac0.4
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
5. Using strategy rm
6. Applied add-log-exp0.5
  \[\leadsto \frac{1}{x} \cdot \frac{1 - \color{blue}{\log \left(e^{\cos x}\right)}}{x}\]
7. Applied add-log-exp0.5
  \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}{x}\]
8. Applied diff-log0.6
  \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}{x}\]
9. Simplified0.5
  \[\leadsto \frac{1}{x} \cdot \frac{\log \color{blue}{\left(e^{1 - \cos x}\right)}}{x}\]
if -0.03399406403260506 < x < 0.03257344431833417
1. Initial program 62.1
  \[\frac{1 - \cos x}{x \cdot x}\]
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 0.03257344431833417 < x
1. Initial program 0.9
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied *-un-lft-identity0.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
4. Applied times-frac0.4
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
5. Using strategy rm
6. Applied clear-num0.5
  \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{1 - \cos x}}}\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033994064032605062:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{1 - \cos x}\right)}{x}\\ \mathbf{elif}\;x \le 0.032573444318334172:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \end{array}\]

Error

Derivation

`if x < -0.03399406403260506`

`if -0.03399406403260506 < x < 0.03257344431833417`

`if 0.03257344431833417 < x`

Reproduce