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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0986873089252876 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{1}{\frac{\frac{y}{z}}{x}}\right|\\ \mathbf{elif}\;x \le 5.707205910123564 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -1.0986873089252876e+41
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Initial simplification0.1
  \[\leadsto \left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\]
3. Using strategy rm
4. Applied clear-num0.2
  \[\leadsto \left|\frac{4 + x}{y} - \color{blue}{\frac{1}{\frac{\frac{y}{z}}{x}}}\right|\]
if -1.0986873089252876e+41 < x < 5.707205910123564e+21
1. Initial program 2.4
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around 0 0.2
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
if 5.707205910123564e+21 < x
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
Recombined 3 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0986873089252876 \cdot 10^{+41}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{1}{\frac{\frac{y}{z}}{x}}\right|\\ \mathbf{elif}\;x \le 5.707205910123564 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - z \cdot \frac{x}{y}\right|\\ \end{array}\]

Runtime

herbie shell --seed 2018251 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Error

Try it out

Derivation

`if x < -1.0986873089252876e+41`

`if -1.0986873089252876e+41 < x < 5.707205910123564e+21`

`if 5.707205910123564e+21 < x`

Runtime

In
Out