Derivation

Split input into 2 regimes
if y < -5.295443062842791e+25
1. Initial program 2.4
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification0.1
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
3. Using strategy rm
4. Applied div-inv0.2
  \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{x \cdot \frac{1}{\frac{y}{z}}}\right|\]
5. Taylor expanded around -inf 0.1
  \[\leadsto \left|\frac{4 + x}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
if -5.295443062842791e+25 < y
1. Initial program 1.2
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification3.9
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
3. Using strategy rm
4. Applied div-inv4.4
  \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{x \cdot \frac{1}{\frac{y}{z}}}\right|\]
5. Taylor expanded around -inf 4.4
  \[\leadsto \left|\frac{4 + x}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
6. Using strategy rm
7. Applied associate-*r/2.4
  \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
Recombined 2 regimes into one program.
Final simplification1.8
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.295443062842791 \cdot 10^{+25}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \end{array}\]

Runtime

herbie shell --seed 2018256 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

In
Out