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Target

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -7.627138752121106e+55

   1. Initial program 18.1

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

   2. Initial simplification13.5

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

   4. Applied div-inv13.6

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

   5. Applied associate-*l*11.9

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

   7. Applied associate-*l/11.8

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

   8. Simplified11.8

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

   if -7.627138752121106e+55 < (* a1 a2) < -5.106090969630976e-119 or -0.0 < (* a1 a2) < 6.416315954955407e+192

   1. Initial program 4.1

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

   2. Initial simplification13.5

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

   4. Applied div-inv13.6

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

   5. Applied associate-*l*13.6

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

   7. Applied associate-*l/13.5

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

   8. Simplified13.5

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

   9. Taylor expanded around -inf 4.1

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

   if -5.106090969630976e-119 < (* a1 a2) < -0.0 or 6.416315954955407e+192 < (* a1 a2)

   1. Initial program 18.2

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

   2. Initial simplification6.2

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

   4. Applied div-inv6.2

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

   5. Applied associate-*l*6.2

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

   7. Applied associate-*r*6.2

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

   9. Applied pow16.2

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

   10. Applied pow16.2

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

   11. Applied pow-prod-down6.2

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

   12. Simplified6.3

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

4. Final simplification6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -7.627138752121106 \cdot 10^{+55}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -5.106090969630976 \cdot 10^{-119}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.416315954955407 \cdot 10^{+192}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Runtime

Time bar (total: 10.0s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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