Error

Derivation

1. Split input into 2 regimes

2. if x < -3.669195055065622e+140 or 5.1753847487161616e-114 < x

   1. Initial program 0.7

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

   2. Taylor expanded around -inf 7.3

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

   3. Simplified0.7

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

   if -3.669195055065622e+140 < x < 5.1753847487161616e-114

   1. Initial program 2.2

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

   3. Applied div-inv2.2

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

   4. Applied associate-*l*5.0

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

   5. Taylor expanded around inf 0.7

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

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.669195055065622 \cdot 10^{+140} \lor \neg \left(x \le 5.1753847487161616 \cdot 10^{-114}\right):\\ \;\;\;\;\left|(z \cdot \left(-\frac{x}{y}\right) + \left(\frac{x}{y} + \frac{4}{y}\right))_*\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z \cdot x}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

Time bar (total: 10.6s)Debug log