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Derivation

1. Split input into 2 regimes

2. if x < -2.7659169745111363e+18 or 85558716.64139932 < x

   1. Initial program 0.1

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

   2. Taylor expanded around 0 7.9

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

   3. Simplified0.1

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

   5. Applied div-inv0.2

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

   6. Applied associate-/r*0.1

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

   if -2.7659169745111363e+18 < x < 85558716.64139932

   1. Initial program 2.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   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 simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.7659169745111363 \cdot 10^{+18} \lor \neg \left(x \le 85558716.64139932\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{\frac{x}{y}}{\frac{1}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \end{array}\]

Runtime

Time bar (total: 14.2s)Debug logProfile

herbie shell --seed 2018286 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))