Derivation

Split input into 2 regimes
if x < -7.435382644996612e+19
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied div-inv0.2
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]
4. Applied associate-*l*0.2
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
if -7.435382644996612e+19 < x
1. Initial program 1.9
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied associate-*l/1.7
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
4. Applied sub-div1.7
  \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
5. Taylor expanded around -inf 1.7
  \[\leadsto \left|\frac{\color{blue}{\left(x + 4\right) - x \cdot z}}{y}\right|\]
Recombined 2 regimes into one program.
Final simplification1.4
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.435382644996612 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(z \cdot \frac{1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \end{array}\]

Runtime

herbie shell --seed 2018252 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

In
Out