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Derivation

1. Split input into 2 regimes

2. if x < -1.6436486530612315e+33 or 5.5078194604740656e-111 < x

   1. Initial program 0.5

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

   3. Applied *-un-lft-identity0.5

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

   4. Applied associate-/l*0.6

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{y}{x + 4}}} - \frac{x}{y} \cdot z\right|\]
   5. Using strategy rm

   6. Applied add-cube-cbrt1.3

      \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}} \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}} - \frac{x}{y} \cdot z\right|\]

   7. Applied prod-diff1.3

      \[\leadsto \left|\color{blue}{(\left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}} \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) + \left(-z \cdot \frac{x}{y}\right))_* + (\left(-z\right) \cdot \left(\frac{x}{y}\right) + \left(z \cdot \frac{x}{y}\right))_*}\right|\]

   8. Simplified0.5

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

   9. Simplified0.5

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

   if -1.6436486530612315e+33 < x < 5.5078194604740656e-111

   1. Initial program 2.4

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

   3. Applied *-un-lft-identity2.4

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

   4. Applied associate-/l*2.5

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{y}{x + 4}}} - \frac{x}{y} \cdot z\right|\]
   5. Using strategy rm

   6. Applied add-cube-cbrt3.5

      \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}} \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}} - \frac{x}{y} \cdot z\right|\]

   7. Applied prod-diff3.5

      \[\leadsto \left|\color{blue}{(\left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}} \cdot \sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\frac{y}{x + 4}}}\right) + \left(-z \cdot \frac{x}{y}\right))_* + (\left(-z\right) \cdot \left(\frac{x}{y}\right) + \left(z \cdot \frac{x}{y}\right))_*}\right|\]

   8. Simplified2.6

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

   9. Simplified2.6

      \[\leadsto \left|\left(\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right) + \color{blue}{0}\right|\]
   10. Using strategy rm

   11. Applied associate-/r/5.4

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

   12. Applied add-sqr-sqrt32.9

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

   13. Applied prod-diff32.9

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

   14. Simplified5.4

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

   15. Simplified5.0

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

   16. Taylor expanded around inf 0.1

       \[\leadsto \left|\left(\left(\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right) + 0\right) + 0\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6436486530612315 \cdot 10^{+33} \lor \neg \left(x \le 5.5078194604740656 \cdot 10^{-111}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))

Details

Time bar (total: 18.1s)Debug log