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Derivation

1. Initial program 17.9

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

2. Initial simplification1.5

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

4. Applied div-inv1.6

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

5. Applied div-inv1.6

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

6. Applied times-frac1.5

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

7. Simplified1.3

   \[\leadsto \frac{t1}{t1 + u} \cdot \color{blue}{\frac{-v}{t1 + u}}\]
8. Using strategy rm

9. Applied div-inv1.4

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

11. Applied associate-*l/1.5

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

12. Simplified1.5

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

13. Final simplification1.5

    \[\leadsto \frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018336 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))