Error

Target

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -3.31206064810173e+224

   1. Initial program 36.4

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

   3. Applied times-frac11.0

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

   if -3.31206064810173e+224 < (* a1 a2) < -3.366555438131595e-289 or 4.675538401688919e-137 < (* a1 a2) < 6.301409825747376e+130

   1. Initial program 4.2

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

   3. Applied associate-/r*4.1

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

   if -3.366555438131595e-289 < (* a1 a2) < 4.675538401688919e-137

   1. Initial program 13.9

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

   3. Applied associate-/l*7.8

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

   5. Applied *-un-lft-identity7.8

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

   6. Applied times-frac5.2

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

   7. Applied *-un-lft-identity5.2

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

   8. Applied times-frac7.6

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

   9. Simplified7.6

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

   11. Applied frac-times5.2

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

   12. Simplified5.2

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

   if 6.301409825747376e+130 < (* a1 a2)

   1. Initial program 24.1

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

   3. Applied associate-/l*16.2

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

   5. Applied *-un-lft-identity16.2

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

   6. Applied times-frac12.6

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

   7. Applied *-un-lft-identity12.6

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

   8. Applied times-frac16.7

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

   9. Simplified16.7

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

   11. Applied div-inv16.7

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

   12. Applied associate-/r*15.8

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

   14. Applied frac-times9.1

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

   15. Simplified9.1

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

   16. Simplified9.1

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

4. Final simplification5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.31206064810173 \cdot 10^{+224}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -3.366555438131595 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 4.675538401688919 \cdot 10^{-137}:\\ \;\;\;\;\frac{a1}{\frac{b2}{a2} \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 6.301409825747376 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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