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Target

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -4.836605524658094e+127

   1. Initial program 23.4

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

   2. Initial simplification10.8

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

   if -4.836605524658094e+127 < (* a1 a2) < -4.111691167559186e-274 or 5.524363222538348e-297 < (* a1 a2) < 3.273019957727645e+151

   1. Initial program 4.3

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

   2. Initial simplification13.2

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

   4. Applied associate-*l/10.7

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

   6. Applied associate-/l*13.5

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

   7. Taylor expanded around inf 4.3

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

   if -4.111691167559186e-274 < (* a1 a2) < 5.524363222538348e-297 or 3.273019957727645e+151 < (* a1 a2)

   1. Initial program 21.5

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

   2. Initial simplification5.7

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

   4. Applied associate-*l/10.3

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

   6. Applied associate-/l*5.6

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

4. Final simplification5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.836605524658094 \cdot 10^{+127}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.111691167559186 \cdot 10^{-274}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.524363222538348 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 3.273019957727645 \cdot 10^{+151}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\]

Runtime

Time bar (total: 11.8s)Debug logProfile

herbie shell --seed 2018339 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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