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Derivation

1. Split input into 3 regimes

2. if x < -9749406881751794.0

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   if -9749406881751794.0 < x < 653141293.4019859

   1. Initial program 2.5

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

   2. Taylor expanded around -inf 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]
   3. Using strategy rm

   4. Applied flip-+0.1

      \[\leadsto \left|\frac{\color{blue}{\frac{x \cdot x - 4 \cdot 4}{x - 4}}}{y} - \frac{x \cdot z}{y}\right|\]

   5. Applied associate-/l/0.1

      \[\leadsto \left|\color{blue}{\frac{x \cdot x - 4 \cdot 4}{y \cdot \left(x - 4\right)}} - \frac{x \cdot z}{y}\right|\]

   6. Simplified0.1

      \[\leadsto \left|\frac{\color{blue}{-16 + x \cdot x}}{y \cdot \left(x - 4\right)} - \frac{x \cdot z}{y}\right|\]

   if 653141293.4019859 < x

   1. Initial program 0.1

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

   2. Taylor expanded around 0 0.1

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right)} - \frac{x}{y} \cdot z\right|\]

   3. Simplified0.1

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + \frac{4}{y}\right)} - \frac{x}{y} \cdot z\right|\]
3. Recombined 3 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9749406881751794.0:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \le 653141293.4019859:\\ \;\;\;\;\left|\frac{-16 + x \cdot x}{y \cdot \left(x - 4\right)} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

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