Derivation

Split input into 2 regimes
if x < -265661134690759.5 or 1.0379090337825838e-250 < x
1. Initial program 1.0
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around 0 5.2
  \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{z \cdot x}{y}}\right|\]
3. Applied simplify1.0
  \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot \left(-z\right) + \left(\frac{4}{y} + \frac{x}{y}\right)\right|}\]
4. Taylor expanded around 0 5.2
  \[\leadsto \left|\color{blue}{-1 \cdot \frac{z \cdot x}{y}} + \left(\frac{4}{y} + \frac{x}{y}\right)\right|\]
5. Applied simplify1.1
  \[\leadsto \color{blue}{\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{\frac{y}{x}}\right|}\]
if -265661134690759.5 < x < 1.0379090337825838e-250
1. Initial program 2.6
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied associate-*l/0.1
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
Recombined 2 regimes into one program.
Applied simplify0.7
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -265661134690759.5 \lor \neg \left(x \le 1.0379090337825838 \cdot 10^{-250}\right):\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}}\]

Runtime

herbie shell --seed 2018201 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

In
Out