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Target

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -4.334952147301378e+177 or -1.051155546863002e-97 < (* a1 a2) < 1.0279751256132542e-190

   1. Initial program 15.6

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

   3. Applied associate-/l*9.4

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

   5. Applied div-inv9.4

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

   6. Applied associate-/r*9.9

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

   8. Applied div-inv9.9

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

   9. Applied div-inv10.2

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

   10. Applied times-frac9.7

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

   11. Simplified9.7

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

   12. Simplified6.6

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

   if -4.334952147301378e+177 < (* a1 a2) < -1.051155546863002e-97 or 1.0279751256132542e-190 < (* a1 a2) < 3.384269109639199e+237

   1. Initial program 4.3

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

   3. Applied associate-/l*12.3

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

   5. Applied associate-/r/11.4

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

   6. Taylor expanded around 0 4.3

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

   if 3.384269109639199e+237 < (* a1 a2)

   1. Initial program 41.4

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

   3. Applied associate-/l*20.7

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

   5. Applied associate-/r/20.7

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

   7. Applied associate-/r*7.6

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

4. Final simplification5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.334952147301378 \cdot 10^{+177}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.051155546863002 \cdot 10^{-97}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0279751256132542 \cdot 10^{-190}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.384269109639199 \cdot 10^{+237}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{b2} \cdot a2\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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

Details

Time bar (total: 5.3s)Debug log