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Target

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -3.06922008117946e+92 or -2.656558819306604e-164 < (* b1 b2) < 4.452037822227978e-180

   1. Initial program 18.4

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

   3. Applied times-frac8.8

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

   if -3.06922008117946e+92 < (* b1 b2) < -2.656558819306604e-164

   1. Initial program 3.3

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

   3. Applied times-frac16.6

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

   4. Taylor expanded around 0 3.3

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

   if 4.452037822227978e-180 < (* b1 b2) < 1.625112373520668e+281

   1. Initial program 4.8

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

   3. Applied associate-/l*4.2

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

   if 1.625112373520668e+281 < (* b1 b2)

   1. Initial program 21.2

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

   3. Applied times-frac2.8

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

   5. Applied associate-*l/3.5

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

4. Final simplification5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.06922008117946 \cdot 10^{+92}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le -2.656558819306604 \cdot 10^{-164}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 4.452037822227978 \cdot 10^{-180}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.625112373520668 \cdot 10^{+281}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2} \cdot a1}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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

Details

Time bar (total: 5.4s)Debug log