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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + 4\right) - x \cdot z \le -2.5088017987598225 \cdot 10^{+144}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;\left(x + 4\right) - x \cdot z \le 4.6375526016656766 \cdot 10^{+148}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Derivation

Split input into 3 regimes
if (- (+ x 4) (* x z)) < -2.5088017987598225e+144
1. Initial program 0.8
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied div-inv0.9
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]
4. Applied associate-*l*0.8
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
5. Applied simplify0.8
  \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
if -2.5088017987598225e+144 < (- (+ x 4) (* x z)) < 4.6375526016656766e+148
1. Initial program 2.2
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
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
  \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
if 4.6375526016656766e+148 < (- (+ x 4) (* x z))
1. Initial program 0.9
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed 2018214 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Error

Try it out

Derivation

`if (- (+ x 4) (* x z)) < -2.5088017987598225e+144`

`if -2.5088017987598225e+144 < (- (+ x 4) (* x z)) < 4.6375526016656766e+148`

`if 4.6375526016656766e+148 < (- (+ x 4) (* x z))`

Runtime

In
Out