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Derivation

1. Split input into 2 regimes

2. if x < -2.25551849996464e-47 or 3.1941018314886564e+37 < x

   1. Initial program 0.2

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

   3. Applied div-inv0.4

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

   4. Applied prod-diff0.4

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

   5. Simplified0.2

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

   6. Simplified0.2

      \[\leadsto \left|\left(\frac{4 + x}{y} - \frac{z}{y} \cdot x\right) + \color{blue}{0}\right|\]

   if -2.25551849996464e-47 < x < 3.1941018314886564e+37

   1. Initial program 2.6

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

   3. Applied associate-*l/0.1

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

   4. Applied sub-div0.1

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

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.25551849996464 \cdot 10^{-47} \lor \neg \left(x \le 3.1941018314886564 \cdot 10^{+37}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

Time bar (total: 7.7s)Debug log