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Derivation

1. Initial program 18.0

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]

2. Initial simplification1.5

   \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}\]
3. Using strategy rm

4. Applied neg-mul-11.5

   \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}}\]

5. Applied *-un-lft-identity1.5

   \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{1 \cdot \left(t1 + u\right)}}{-1 \cdot t1}}\]

6. Applied times-frac1.5

   \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{1}{-1} \cdot \frac{t1 + u}{t1}}}\]

7. Applied div-inv1.6

   \[\leadsto \frac{\color{blue}{v \cdot \frac{1}{t1 + u}}}{\frac{1}{-1} \cdot \frac{t1 + u}{t1}}\]

8. Applied times-frac3.3

   \[\leadsto \color{blue}{\frac{v}{\frac{1}{-1}} \cdot \frac{\frac{1}{t1 + u}}{\frac{t1 + u}{t1}}}\]

9. Simplified3.3

   \[\leadsto \color{blue}{\left(-v\right)} \cdot \frac{\frac{1}{t1 + u}}{\frac{t1 + u}{t1}}\]

10. Simplified3.3

    \[\leadsto \left(-v\right) \cdot \color{blue}{\frac{\frac{t1}{t1 + u}}{t1 + u}}\]
11. Using strategy rm

12. Applied div-inv3.3

    \[\leadsto \left(-v\right) \cdot \color{blue}{\left(\frac{t1}{t1 + u} \cdot \frac{1}{t1 + u}\right)}\]

13. Applied associate-*r*1.3

    \[\leadsto \color{blue}{\left(\left(-v\right) \cdot \frac{t1}{t1 + u}\right) \cdot \frac{1}{t1 + u}}\]
14. Using strategy rm

15. Applied clear-num1.5

    \[\leadsto \left(\left(-v\right) \cdot \color{blue}{\frac{1}{\frac{t1 + u}{t1}}}\right) \cdot \frac{1}{t1 + u}\]

16. Final simplification1.5

    \[\leadsto \left(\frac{1}{\frac{t1 + u}{t1}} \cdot v\right) \cdot \frac{-1}{t1 + u}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))