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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -8.972689661583338e+284 or -5.6379784274601786e-219 < (* a1 a2) < 2.401871222154145e-300

   1. Initial program 21.8

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

   2. Initial simplification3.9

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

   4. Applied associate-*r/8.8

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

   6. Applied associate-/l*3.9

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

   if -8.972689661583338e+284 < (* a1 a2) < -5.6379784274601786e-219

   1. Initial program 5.2

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

   2. Initial simplification14.4

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

   4. Applied associate-*r/11.9

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

   6. Applied div-inv11.9

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

   7. Applied associate-*l*11.1

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

   9. Applied associate-*l/11.0

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

   10. Applied associate-*r/5.2

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

   11. Applied associate-/l/5.2

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

   12. Simplified5.2

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

   if 2.401871222154145e-300 < (* a1 a2) < 1.1971016819116577e+198

   1. Initial program 5.3

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

   2. Initial simplification13.5

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

   4. Applied associate-*r/10.5

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

   6. Applied div-inv10.5

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

   7. Applied associate-*l*11.0

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

   8. Taylor expanded around -inf 4.4

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

   10. Applied div-inv4.5

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

   if 1.1971016819116577e+198 < (* a1 a2)

   1. Initial program 34.8

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

   2. Initial simplification9.4

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

   4. Applied div-inv9.4

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

   5. Applied associate-*l*9.4

      \[\leadsto \color{blue}{a1 \cdot \left(\frac{1}{b2} \cdot \frac{a2}{b1}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -8.972689661583338 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -5.6379784274601786 \cdot 10^{-219}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.401871222154145 \cdot 10^{-300}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.1971016819116577 \cdot 10^{+198}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1 \cdot a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b1} \cdot \frac{1}{b2}\right)\\ \end{array}\]

Runtime

Time bar (total: 6.3s)Debug logProfile

herbie shell --seed 2018297 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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