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Derivation

1. Split input into 3 regimes

2. if x < -1.6436486530612315e+33

   1. Initial program 0.1

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

   3. Applied clear-num0.3

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

   4. Taylor expanded around -inf 9.9

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

   5. Simplified0.1

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

   if -1.6436486530612315e+33 < x < 4.530076266194952e+60

   1. Initial program 2.2

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

   3. Applied associate-*l/0.2

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

   4. Applied sub-div0.2

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

   if 4.530076266194952e+60 < x

   1. Initial program 0.1

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

   3. Applied clear-num0.3

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

   4. Taylor expanded around -inf 10.0

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

   5. Simplified0.2

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

   7. Applied div-inv0.2

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

   8. Applied associate-/r*0.2

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{\frac{\frac{x}{y}}{\frac{1}{z}}}\right|\]
3. Recombined 3 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6436486530612315 \cdot 10^{+33}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \le 4.530076266194952 \cdot 10^{+60}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{\frac{x}{y}}{\frac{1}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

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