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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -inf.0

   1. Initial program 61.0

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

   2. Initial simplification4.8

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

   4. Applied associate-*l/18.5

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

   if -inf.0 < (* a1 a2) < -1.4942147701964681e-186 or 3.285868405129512e-263 < (* a1 a2) < 1.4090779083505787e+208

   1. Initial program 5.0

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

   2. Initial simplification13.7

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

   4. Applied associate-*r/11.1

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

   5. Taylor expanded around 0 4.9

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

   if -1.4942147701964681e-186 < (* a1 a2) < 3.285868405129512e-263

   1. Initial program 14.5

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

   2. Initial simplification3.8

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

   4. Applied div-inv3.8

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

   5. Applied associate-*l*3.4

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

   if 1.4090779083505787e+208 < (* a1 a2)

   1. Initial program 34.2

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

   2. Initial simplification10.5

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

   4. Applied div-inv10.5

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

   5. Applied associate-*r*18.6

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

   7. Applied associate-*l/35.5

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

   8. Applied associate-*l/34.3

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

   9. Simplified17.5

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

4. Final simplification5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -1.4942147701964681 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.285868405129512 \cdot 10^{-263}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b1} \cdot \frac{1}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.4090779083505787 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{\frac{b1}{a1}}}{b2}\\ \end{array}\]

Runtime

Time bar (total: 8.6s)Debug logProfile

herbie shell --seed 2018355 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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