Derivation

Split input into 2 regimes
if x < -5.678729188719287e+48 or 1.3622734763920842e-98 < x
1. Initial program 0.5
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied div-inv0.5
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]
4. Applied associate-*l*0.4
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
5. Applied simplify0.4
  \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
if -5.678729188719287e+48 < x < 1.3622734763920842e-98
1. Initial program 2.8
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied associate-*l/0.2
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
4. Applied sub-div0.2
  \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
Recombined 2 regimes into one program.
Applied simplify0.3
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -5.678729188719287 \cdot 10^{+48} \lor \neg \left(x \le 1.3622734763920842 \cdot 10^{-98}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - z \cdot x}{y}\right|\\ \end{array}}\]

Runtime

herbie shell --seed 2018206 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

In
Out