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Derivation

1. Initial program 18.1

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

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-frac1.4

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

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. Applied associate-*r*1.4

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

11. Final simplification1.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))