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Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -inf.0

   1. Initial program 60.9

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

   2. Initial simplification6.7

      \[\leadsto \frac{a1}{b2} \cdot \frac{a2}{b1}\]

   if -inf.0 < (* a1 a2) < -3.1565697845360545e-228

   1. Initial program 5.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

   2. Initial simplification13.3

      \[\leadsto \frac{a1}{b2} \cdot \frac{a2}{b1}\]
   3. Using strategy rm

   4. Applied associate-*l/11.2

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b1}}{b2}}\]
   5. Using strategy rm

   6. Applied associate-*r/5.2

      \[\leadsto \frac{\color{blue}{\frac{a1 \cdot a2}{b1}}}{b2}\]

   7. Applied associate-/l/5.2

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b2 \cdot b1}}\]

   if -3.1565697845360545e-228 < (* a1 a2) < 6.338080416842529e-249 or 4.8764018383057663e+219 < (* a1 a2)

   1. Initial program 19.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

   2. Initial simplification5.1

      \[\leadsto \frac{a1}{b2} \cdot \frac{a2}{b1}\]
   3. Using strategy rm

   4. Applied div-inv5.1

      \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b2}\right)} \cdot \frac{a2}{b1}\]

   5. Applied associate-*l*5.1

      \[\leadsto \color{blue}{a1 \cdot \left(\frac{1}{b2} \cdot \frac{a2}{b1}\right)}\]
   6. Using strategy rm

   7. Applied pow15.1

      \[\leadsto a1 \cdot \left(\frac{1}{b2} \cdot \color{blue}{{\left(\frac{a2}{b1}\right)}^{1}}\right)\]

   8. Applied pow15.1

      \[\leadsto a1 \cdot \left(\color{blue}{{\left(\frac{1}{b2}\right)}^{1}} \cdot {\left(\frac{a2}{b1}\right)}^{1}\right)\]

   9. Applied pow-prod-down5.1

      \[\leadsto a1 \cdot \color{blue}{{\left(\frac{1}{b2} \cdot \frac{a2}{b1}\right)}^{1}}\]

   10. Simplified5.3

       \[\leadsto a1 \cdot {\color{blue}{\left(\frac{\frac{a2}{b2}}{b1}\right)}}^{1}\]

   if 6.338080416842529e-249 < (* a1 a2) < 4.8764018383057663e+219

   1. Initial program 4.3

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

   2. Initial simplification13.7

      \[\leadsto \frac{a1}{b2} \cdot \frac{a2}{b1}\]
   3. Using strategy rm

   4. Applied associate-*l/11.3

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b1}}{b2}}\]
   5. Using strategy rm

   6. Applied div-inv11.3

      \[\leadsto \frac{a1 \cdot \color{blue}{\left(a2 \cdot \frac{1}{b1}\right)}}{b2}\]

   7. Applied associate-*r*4.3

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}}{b2}\]
3. Recombined 4 regimes into one program.

4. Final simplification5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.1565697845360545 \cdot 10^{-228}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.338080416842529 \cdot 10^{-249}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 4.8764018383057663 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{1}{b1} \cdot \left(a1 \cdot a2\right)}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \end{array}\]

Runtime

Time bar (total: 11.4s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))