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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.563448987301963 \cdot 10^{+89}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 3.620984115508776 \cdot 10^{+83}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -3.563448987301963e+89
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
if -3.563448987301963e+89 < x < 3.620984115508776e+83
1. Initial program 2.0
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification3.5
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
3. Taylor expanded around -inf 0.5
  \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
if 3.620984115508776e+83 < x
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification0.1
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
Recombined 3 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.563448987301963 \cdot 10^{+89}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 3.620984115508776 \cdot 10^{+83}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	1.5	0.4	0.0	1.5	76.9%

herbie shell --seed 2018285 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Error

Try it out

Derivation

`if x < -3.563448987301963e+89`

`if -3.563448987301963e+89 < x < 3.620984115508776e+83`

`if 3.620984115508776e+83 < x`

Runtime

In
Out