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Target

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -2.06541239684983e+256

   1. Initial program 17.7

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

   3. Applied associate-/r*7.9

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

   if -2.06541239684983e+256 < (* b1 b2) < -1.794287396877486e-178

   1. Initial program 4.6

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

   3. Applied associate-/l*4.5

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

   5. Applied div-inv4.6

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

   6. Applied associate-/r*4.6

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

   8. Applied div-inv4.6

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

   if -1.794287396877486e-178 < (* b1 b2) < 2.200863960063993e-305

   1. Initial program 35.8

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

   2. Initial simplification10.0

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

   if 2.200863960063993e-305 < (* b1 b2) < 3.641633386252542e+119

   1. Initial program 4.6

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

   3. Applied associate-/l*5.5

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

   5. Applied div-inv5.6

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

   6. Applied associate-/r*5.3

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

   if 3.641633386252542e+119 < (* b1 b2)

   1. Initial program 11.5

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

   3. Applied associate-/l*12.4

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

   5. Applied div-inv12.4

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

   6. Applied associate-/r*12.1

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

   8. Applied *-un-lft-identity12.1

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

   9. Applied div-inv12.1

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

   10. Applied times-frac12.0

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

   11. Simplified12.0

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

   12. Simplified7.6

       \[\leadsto a1 \cdot \color{blue}{\frac{\frac{a2}{b2}}{b1}}\]
3. Recombined 5 regimes into one program.

4. Final simplification6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -2.06541239684983 \cdot 10^{+256}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -1.794287396877486 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{1}{a2}} \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 2.200863960063993 \cdot 10^{-305}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.641633386252542 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{b1} \cdot a1\\ \end{array}\]

Runtime

Time bar (total: 9.3s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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