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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -2.972839290677902e+300

   1. Initial program 56.8

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

   3. Applied associate-/l*20.7

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

   if -2.972839290677902e+300 < (* a1 a2) < -9.3988679746644e-313 or 1.1556955786713302e-161 < (* a1 a2) < 2.2514602872102896e+178

   1. Initial program 5.1

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

   3. Applied times-frac13.4

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

   5. Applied associate-*r/10.3

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

   6. Taylor expanded around inf 4.4

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

   if -9.3988679746644e-313 < (* a1 a2) < 1.1556955786713302e-161

   1. Initial program 15.1

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

   3. Applied times-frac4.4

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

   5. Applied associate-*r/9.2

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

   7. Applied associate-/l*4.5

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

   9. Applied associate-/r/4.5

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

   if 2.2514602872102896e+178 < (* a1 a2)

   1. Initial program 30.0

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

   3. Applied times-frac9.9

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

   5. Applied associate-*r/14.4

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

   7. Applied associate-/l*9.7

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

4. Final simplification5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -2.972839290677902 \cdot 10^{+300}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -9.3988679746644 \cdot 10^{-313}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.1556955786713302 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{b2} \cdot a2\\ \mathbf{elif}\;a1 \cdot a2 \le 2.2514602872102896 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Runtime

Time bar (total: 8.5s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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