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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 simplification1.3

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

4. Applied *-un-lft-identity1.3

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

5. Applied add-cube-cbrt2.1

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

6. Applied times-frac2.0

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

7. Applied div-inv2.1

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

8. Applied times-frac2.2

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

9. Simplified2.2

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

10. Simplified2.2

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

11. Final simplification2.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018219 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))