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Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -4.329455158949287e-189

   1. Initial program 9.0

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

   3. Applied associate-/r*9.5

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

   if -4.329455158949287e-189 < (* a1 a2) < 1.0265589641605737e-245

   1. Initial program 14.2

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

   3. Applied associate-/r*15.3

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

   5. Applied clear-num15.6

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

   7. Applied associate-/r*8.5

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

   if 1.0265589641605737e-245 < (* a1 a2) < 6.654002524214254e+203

   1. Initial program 4.6

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

   3. Applied associate-/r*4.6

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

   5. Applied clear-num4.8

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

   7. Applied associate-/r/4.6

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

   8. Applied associate-/l*4.5

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

   if 6.654002524214254e+203 < (* a1 a2)

   1. Initial program 34.0

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

   3. Applied associate-/r*33.6

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

   5. Applied clear-num33.7

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

   7. Applied associate-/r*17.6

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

   9. Applied associate-/r/17.7

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

   10. Applied associate-/l*9.5

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

   11. Simplified9.3

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

4. Final simplification7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -4.329455158949287 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.0265589641605737 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{b1}{a1}}{a2}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.654002524214254 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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

Details

Time bar (total: 5.4s)Debug log