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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -5.74129741148558 \cdot 10^{-14}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \le 1.5179704607986919 \cdot 10^{-58}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -5.74129741148558e-14 or 1.5179704607986919e-58 < x
1. Initial program 0.2
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Applied taylor 0.2
  \[\leadsto \left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]
3. Taylor expanded around 0 0.2
  
  \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]
4. Applied simplify 0.3
  \[\leadsto \color{blue}{\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|}\]
if -5.74129741148558e-14 < x < 1.5179704607986919e-58
1. Initial program 3.3
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
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-div 0.1
  \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Error

Derivation

`if x < -5.74129741148558e-14 or 1.5179704607986919e-58 < x`

`if -5.74129741148558e-14 < x < 1.5179704607986919e-58`

Runtime