Error

Try it out

Derivation

1. Split input into 2 regimes

2. if x < -1.305821992521901e-26 or 5.613510658928969e-09 < x

   1. Initial program 0.1

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified0.1

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

   4. Taylor expanded around inf 7.7

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

   5. Simplified0.2

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

   if -1.305821992521901e-26 < x < 5.613510658928969e-09

   1. Initial program 2.8

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

   2. Taylor expanded around 0 2.8

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

   3. Simplified2.8

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

   4. Taylor expanded around inf 0.1

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

   5. Simplified5.2

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

   7. Applied *-un-lft-identity5.2

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

   8. Applied *-un-lft-identity5.2

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

   9. Applied *-un-lft-identity5.2

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

   10. Applied distribute-lft-out5.2

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

   11. Applied distribute-lft-out--5.2

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

   12. Simplified2.8

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

   13. Taylor expanded around 0 0.1

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

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.305821992521901 \cdot 10^{-26} \lor \neg \left(x \le 5.613510658928969 \cdot 10^{-09}\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - \frac{x \cdot z}{y}\right)\right|\\ \end{array}\]

Reproduce

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

Details

Time bar (total: 10.1s)Debug log