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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 simplification1.6

   \[\leadsto \frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}\]
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 *-un-lft-identity1.6

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

6. Applied add-cube-cbrt2.3

   \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}\right) \cdot \sqrt[3]{-t1}}}{1 \cdot \left(t1 + u\right)}}{\left(t1 + u\right) \cdot \frac{1}{v}}\]

7. Applied times-frac2.2

   \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{1} \cdot \frac{\sqrt[3]{-t1}}{t1 + u}}}{\left(t1 + u\right) \cdot \frac{1}{v}}\]

8. Applied times-frac1.0

   \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{1}}{t1 + u} \cdot \frac{\frac{\sqrt[3]{-t1}}{t1 + u}}{\frac{1}{v}}}\]

9. Simplified1.0

   \[\leadsto \color{blue}{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{t1 + u}} \cdot \frac{\frac{\sqrt[3]{-t1}}{t1 + u}}{\frac{1}{v}}\]

10. Simplified1.6

    \[\leadsto \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{t1 + u} \cdot \color{blue}{\left(\sqrt[3]{-t1} \cdot \frac{v}{u + t1}\right)}\]

11. Final simplification1.6

    \[\leadsto \left(\sqrt[3]{-t1} \cdot \frac{v}{t1 + u}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{t1 + u}\]

Runtime

Time bar (total: 27.0s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))