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Target

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -3.9286586837245714e+101 or -2.3659078828417293e-224 < (* a1 a2) < 8.060373702016246e-150

   1. Initial program 15.7

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

   2. Initial simplification6.1

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

   if -3.9286586837245714e+101 < (* a1 a2) < -2.3659078828417293e-224 or 8.060373702016246e-150 < (* a1 a2) < 1.4601818134507145e+206

   1. Initial program 3.8

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

   2. Initial simplification14.6

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

   4. Applied associate-*r/11.4

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

   5. Taylor expanded around inf 3.8

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

   if 1.4601818134507145e+206 < (* a1 a2)

   1. Initial program 34.1

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

   2. Initial simplification10.5

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

   4. Applied associate-*r/18.5

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

   6. Applied clear-num18.6

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{\frac{a1}{b2} \cdot a2}}}\]
   7. Using strategy rm

   8. Applied associate-/r/18.6

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

   10. Applied pow118.6

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

   11. Applied pow118.6

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

   12. Applied pow-prod-down18.6

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

   13. Simplified10.7

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

4. Final simplification5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.9286586837245714 \cdot 10^{+101}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.3659078828417293 \cdot 10^{-224}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 8.060373702016246 \cdot 10^{-150}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4601818134507145 \cdot 10^{+206}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array}\]

Runtime

Time bar (total: 8.7s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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