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Derivation

1. Split input into 2 regimes

2. if (- (/ (+ x 4) y) (* (/ x y) z)) < 1.052154923565608e+109

   1. Initial program 2.1

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

   3. Applied associate-*l/2.4

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

   4. Applied sub-div2.4

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

   6. Applied div-inv2.4

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

   if 1.052154923565608e+109 < (- (/ (+ x 4) y) (* (/ x y) z))

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.5

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{x + 4}{y} - \frac{x}{y} \cdot z}}\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 1.052154923565608 \cdot 10^{+109}:\\ \;\;\;\;\left|\left(\left(4 + x\right) - x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{4 + x}{y} - \frac{x}{y} \cdot z} \cdot \sqrt{\frac{4 + x}{y} - \frac{x}{y} \cdot z}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

Time bar (total: 9.0s)Debug log