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Derivation

1. Initial program 17.5

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

2. Initial simplification1.4

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

4. Applied add-cube-cbrt2.2

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

5. Applied add-cube-cbrt1.7

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

6. Applied times-frac1.8

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

7. Applied *-un-lft-identity1.8

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

8. Applied times-frac1.0

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

9. Simplified1.0

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

10. Final simplification1.0

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018216 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))