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Target

Derivation

1. Split input into 3 regimes

2. if a2 < -3.744170422286937e-227 or 9.808602920791376e-287 < a2 < 2.3271170994576882e-206

   1. Initial program 10.9

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

   2. Initial simplification11.6

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

   4. Applied div-inv11.7

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

   5. Applied associate-*l*11.7

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

   7. Applied associate-*r*11.7

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

   9. Applied pow111.7

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

   10. Applied pow111.7

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

   11. Applied pow-prod-down11.7

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

   12. Simplified10.3

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

   if -3.744170422286937e-227 < a2 < 9.808602920791376e-287

   1. Initial program 9.0

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

   2. Initial simplification11.7

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

   4. Applied div-inv11.7

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

   5. Applied associate-*l*12.1

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

   7. Applied associate-*r*11.7

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

   9. Applied associate-*r/11.7

      \[\leadsto \color{blue}{\frac{a1 \cdot 1}{b2}} \cdot \frac{a2}{b1}\]

   10. Applied frac-times9.0

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

   11. Simplified9.0

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

   if 2.3271170994576882e-206 < a2

   1. Initial program 10.5

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

   2. Initial simplification10.6

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

   4. Applied div-inv10.6

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

   5. Applied associate-*l*11.0

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

   7. Applied associate-*r*10.6

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

4. Final simplification10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \le -3.744170422286937 \cdot 10^{-227}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;a2 \le 9.808602920791376 \cdot 10^{-287}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a2 \le 2.3271170994576882 \cdot 10^{-206}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{1}{b2}\right) \cdot \frac{a2}{b1}\\ \end{array}\]

Runtime

Time bar (total: 12.9s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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