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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 8.863395261203298 \cdot 10^{+109}:\\ \;\;\;\;\left|(z \cdot \left(\frac{-x}{y}\right) + \left(\frac{x}{y} + \frac{4}{y}\right))_*\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Derivation

Split input into 2 regimes
if z < 8.863395261203298e+109
1. Initial program 0.9
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around inf 2.2
  \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}}\right|\]
3. Simplified0.9
  \[\leadsto \left|\color{blue}{(z \cdot \left(-\frac{x}{y}\right) + \left(\frac{x}{y} + \frac{4}{y}\right))_*}\right|\]
if 8.863395261203298e+109 < z
1. Initial program 6.4
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification8.9
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
Recombined 2 regimes into one program.
Final simplification1.9
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le 8.863395261203298 \cdot 10^{+109}:\\ \;\;\;\;\left|(z \cdot \left(\frac{-x}{y}\right) + \left(\frac{x}{y} + \frac{4}{y}\right))_*\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	1.6	1.9	0.0	1.6	-21.2%

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Error

Derivation

`if z < 8.863395261203298e+109`

`if 8.863395261203298e+109 < z`

Runtime