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Target

Derivation

1. Split input into 5 regimes

2. if (* a1 a2) < -2.447653400202506e+210

   1. Initial program 35.3

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

   2. Initial simplification8.9

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

   if -2.447653400202506e+210 < (* a1 a2) < -641902519989.8164

   1. Initial program 5.6

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

   2. Initial simplification15.6

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

   4. Applied associate-*r/11.5

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

   6. Applied clear-num11.7

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{\frac{a1}{b2} \cdot a2}}}\]
   7. Using strategy rm

   8. Applied div-inv11.7

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

   9. Taylor expanded around 0 5.6

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

   if -641902519989.8164 < (* a1 a2) < 4.951301047265912e-294

   1. Initial program 10.5

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

   2. Initial simplification7.8

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

   4. Applied div-inv7.9

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

   5. Applied associate-*l*7.5

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

   if 4.951301047265912e-294 < (* a1 a2) < 1.0177928102460338e+247

   1. Initial program 4.9

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

   2. Initial simplification13.4

      \[\leadsto \frac{a1}{b2} \cdot \frac{a2}{b1}\]
   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 5.0

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

   if 1.0177928102460338e+247 < (* a1 a2)

   1. Initial program 42.5

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

   2. Initial simplification7.1

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

   4. Applied associate-*r/16.3

      \[\leadsto \color{blue}{\frac{\frac{a1}{b2} \cdot a2}{b1}}\]
3. Recombined 5 regimes into one program.

4. Final simplification6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.447653400202506 \cdot 10^{+210}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -641902519989.8164:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 4.951301047265912 \cdot 10^{-294}:\\ \;\;\;\;a1 \cdot \left(\frac{a2}{b1} \cdot \frac{1}{b2}\right)\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0177928102460338 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2} \cdot a2}{b1}\\ \end{array}\]

Runtime

Time bar (total: 9.1s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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