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Derivation

1. Split input into 2 regimes

2. if (- (/ (+ x 4) y) (* (/ x y) z)) < -1.1806749666350639e-60 or 2.776482164677181e-149 < (- (/ (+ x 4) y) (* (/ x y) z))

   1. Initial program 0.3

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

   if -1.1806749666350639e-60 < (- (/ (+ x 4) y) (* (/ x y) z)) < 2.776482164677181e-149

   1. Initial program 5.9

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

   3. Applied div-inv5.9

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

   4. Applied associate-*l*0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1.1806749666350639 \cdot 10^{-60}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 2.776482164677181 \cdot 10^{-149}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(\frac{1}{y} \cdot z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Runtime

Time bar (total: 18.4s)Debug logProfile

herbie shell --seed 2018355 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))