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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -1.1598356346117476e+168

   1. Initial program 26.8

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

   3. Applied associate-/r*26.7

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

   5. Applied associate-/l*16.4

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

   if -1.1598356346117476e+168 < (* a1 a2) < -5.904730415373209e-181

   1. Initial program 4.2

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

   3. Applied associate-/r*3.8

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

   4. Taylor expanded around inf 4.2

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

   if -5.904730415373209e-181 < (* a1 a2) < 3.7488137802570405e-238 or 1.5550221396521212e+204 < (* a1 a2)

   1. Initial program 18.8

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

   3. Applied times-frac5.2

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

   if 3.7488137802570405e-238 < (* a1 a2) < 1.5550221396521212e+204

   1. Initial program 4.3

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

   3. Applied associate-/r*4.7

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

   4. Taylor expanded around inf 4.3

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

   6. Applied associate-/r*4.6

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

4. Final simplification5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.1598356346117476 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -5.904730415373209 \cdot 10^{-181}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.7488137802570405 \cdot 10^{-238}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.5550221396521212 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Runtime

Time bar (total: 6.7s)Debug logProfile

herbie shell --seed 2018351 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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