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Target

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -1.0545201117055032e+184

   1. Initial program 16.3

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

   2. Initial simplification4.9

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

   if -1.0545201117055032e+184 < (* b1 b2) < -3.8220188897971772e-236

   1. Initial program 4.5

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

   3. Applied div-inv4.5

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

   5. Applied associate-/r*4.5

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

   if -3.8220188897971772e-236 < (* b1 b2) < 2.0259147619971227e-296 or 23619375384618.953 < (* b1 b2)

   1. Initial program 17.1

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

   3. Applied times-frac8.7

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

   if 2.0259147619971227e-296 < (* b1 b2) < 23619375384618.953

   1. Initial program 6.3

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

   3. Applied div-inv6.4

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

   5. Applied associate-*l*6.2

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

4. Final simplification6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.0545201117055032 \cdot 10^{+184}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.8220188897971772 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{1}{b1}}{b2} \cdot \left(a2 \cdot a1\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 2.0259147619971227 \cdot 10^{-296} \lor \neg \left(b1 \cdot b2 \le 23619375384618.953\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{1}{b1 \cdot b2} \cdot a2\right)\\ \end{array}\]

Runtime

Time bar (total: 6.8s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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