Error

Derivation

1. Initial program 18.4

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
2. Using strategy rm

3. Applied *-un-lft-identity18.4

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

4. Applied *-un-lft-identity18.4

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

5. Applied distribute-lft-out18.4

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

6. Applied associate-*r*18.4

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

7. Applied times-frac1.3

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

8. Simplified1.3

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

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

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

11. Applied distribute-lft-neg-in1.3

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

12. Applied associate-/l*1.5

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

13. Applied associate-*l/1.5

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

14. Simplified1.5

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

15. Taylor expanded around 0 1.4

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

16. Final simplification1.4

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))