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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -2.4100190819319214e+122 or -1.4390835684418586e-161 < (* a1 a2) < 0.0

   1. Initial program 18.2

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

   2. Initial simplification6.8

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

   4. Applied div-inv6.9

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

   5. Applied associate-*l*6.3

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

   if -2.4100190819319214e+122 < (* a1 a2) < -1.4390835684418586e-161

   1. Initial program 3.7

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

   2. Initial simplification15.1

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

   4. Applied div-inv15.2

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

   5. Applied associate-*l*14.4

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

   6. Taylor expanded around inf 3.7

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

   if 0.0 < (* a1 a2) < 3.180610833167153e+101

   1. Initial program 4.5

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

   2. Initial simplification13.3

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

   4. Applied associate-*l/10.5

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

   5. Taylor expanded around 0 4.5

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

   if 3.180610833167153e+101 < (* a1 a2)

   1. Initial program 20.6

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

   2. Initial simplification13.1

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

   4. Applied div-inv13.2

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

   5. Applied associate-*r*15.3

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

4. Final simplification6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.4100190819319214 \cdot 10^{+122}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b1} \cdot \frac{1}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le -1.4390835684418586 \cdot 10^{-161}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 0.0:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b1} \cdot \frac{1}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 3.180610833167153 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(a2 \cdot \frac{a1}{b2}\right)\\ \end{array}\]

Runtime

Time bar (total: 10.5s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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