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Derivation

1. Split input into 2 regimes

2. if (fabs (- (/ (+ x 4) y) (* (/ x y) z))) < 14884601944.499592

   1. Initial program 3.8

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

   3. Applied div-inv3.9

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

   4. Applied associate-*l*0.1

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

   5. Applied simplify0.1

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

   if 14884601944.499592 < (fabs (- (/ (+ x 4) y) (* (/ x y) z)))

   1. Initial program 0.1

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

   2. Taylor expanded around 0 5.1

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

   3. Applied simplify0.1

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

   5. Applied add-cube-cbrt0.9

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

Runtime

Time bar (total: 32.5s)Debug logProfile

herbie shell --seed 2018170 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))