\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\right):\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

\end{array}

double f(double x, double y, double z) {
        double r26564 = x;
        double r26565 = 4.0;
        double r26566 = r26564 + r26565;
        double r26567 = y;
        double r26568 = r26566 / r26567;
        double r26569 = r26564 / r26567;
        double r26570 = z;
        double r26571 = r26569 * r26570;
        double r26572 = r26568 - r26571;
        double r26573 = fabs(r26572);
        return r26573;
}

↓

double f(double x, double y, double z) {
        double r26574 = x;
        double r26575 = -1.384698452851939e-17;
        bool r26576 = r26574 <= r26575;
        double r26577 = 4.1763185447825497e-35;
        bool r26578 = r26574 <= r26577;
        double r26579 = !r26578;
        bool r26580 = r26576 || r26579;
        double r26581 = 4.0;
        double r26582 = y;
        double r26583 = r26581 / r26582;
        double r26584 = r26574 / r26582;
        double r26585 = r26583 + r26584;
        double r26586 = z;
        double r26587 = r26586 / r26582;
        double r26588 = r26574 * r26587;
        double r26589 = r26585 - r26588;
        double r26590 = fabs(r26589);
        double r26591 = r26574 + r26581;
        double r26592 = r26574 * r26586;
        double r26593 = r26591 - r26592;
        double r26594 = r26593 / r26582;
        double r26595 = fabs(r26594);
        double r26596 = r26580 ? r26590 : r26595;
        return r26596;
}

Derivation

Split input into 2 regimes
if x < -1.384698452851939e-17 or 4.1763185447825497e-35 < x
1. Initial program 0.2
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around 0 0.2
  \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]
3. Simplified0.2
  \[\leadsto \left|\color{blue}{\left(\frac{4}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]
4. Using strategy rm
5. Applied div-inv0.2
  \[\leadsto \left|\left(\frac{4}{y} + \frac{x}{y}\right) - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]
6. Applied associate-*l*0.3
  \[\leadsto \left|\left(\frac{4}{y} + \frac{x}{y}\right) - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
7. Simplified0.3
  \[\leadsto \left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \color{blue}{\frac{z}{y}}\right|\]
if -1.384698452851939e-17 < x < 4.1763185447825497e-35
1. Initial program 2.8
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied associate-*l/0.1
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
4. Applied sub-div0.1
  \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Try it out

Derivation

`if x < -1.384698452851939e-17 or 4.1763185447825497e-35 < x`

`if -1.384698452851939e-17 < x < 4.1763185447825497e-35`

Reproduce

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