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Target

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -6.022692576073051e+271 or 3.689553141953114e+57 < (* a1 a2)

   1. Initial program 24.6

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

   2. Initial simplification12.8

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

   if -6.022692576073051e+271 < (* a1 a2) < -1.2478065568477178e-163 or 1.9916657381995613e-279 < (* a1 a2) < 3.689553141953114e+57

   1. Initial program 4.4

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

   3. Applied times-frac13.6

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

   4. Taylor expanded around inf 4.4

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

   if -1.2478065568477178e-163 < (* a1 a2) < 1.9916657381995613e-279

   1. Initial program 14.9

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

   3. Applied associate-/l*7.9

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

   5. Applied associate-/l*4.7

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

4. Final simplification6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.022692576073051 \cdot 10^{+271}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.2478065568477178 \cdot 10^{-163}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.9916657381995613 \cdot 10^{-279}:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.689553141953114 \cdot 10^{+57}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Runtime

Time bar (total: 7.9s)Debug logProfile

herbie shell --seed 2018336 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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