\[\frac{x - \sin x}{x - \tan x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02610929689909074:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.03142629441362508:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

double f(double x) {
        double r1139552 = x;
        double r1139553 = sin(r1139552);
        double r1139554 = r1139552 - r1139553;
        double r1139555 = tan(r1139552);
        double r1139556 = r1139552 - r1139555;
        double r1139557 = r1139554 / r1139556;
        return r1139557;
}

↓

double f(double x) {
        double r1139558 = x;
        double r1139559 = -0.02610929689909074;
        bool r1139560 = r1139558 <= r1139559;
        double r1139561 = tan(r1139558);
        double r1139562 = r1139558 - r1139561;
        double r1139563 = r1139558 / r1139562;
        double r1139564 = sin(r1139558);
        double r1139565 = r1139564 / r1139562;
        double r1139566 = r1139563 - r1139565;
        double r1139567 = 0.03142629441362508;
        bool r1139568 = r1139558 <= r1139567;
        double r1139569 = r1139558 * r1139558;
        double r1139570 = 0.225;
        double r1139571 = r1139569 * r1139570;
        double r1139572 = 0.5;
        double r1139573 = 0.009642857142857142;
        double r1139574 = r1139569 * r1139569;
        double r1139575 = r1139573 * r1139574;
        double r1139576 = r1139572 + r1139575;
        double r1139577 = r1139571 - r1139576;
        double r1139578 = r1139568 ? r1139577 : r1139566;
        double r1139579 = r1139560 ? r1139566 : r1139578;
        return r1139579;
}

\frac{x - \sin x}{x - \tan x}

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.02610929689909074:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 0.03142629441362508:\\
\;\;\;\;\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\end{array}

Derivation

Split input into 2 regimes
if x < -0.02610929689909074 or 0.03142629441362508 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied div-sub0.1
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
if -0.02610929689909074 < x < 0.03142629441362508
1. Initial program 62.7
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800} + \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02610929689909074:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.03142629441362508:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Error

Derivation

`if x < -0.02610929689909074 or 0.03142629441362508 < x`

`if -0.02610929689909074 < x < 0.03142629441362508`

Reproduce