Average Error: 18.7 → 1.3
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{v}{t1 + u} \cdot \frac{-t1}{t1 - u}\right) \cdot \frac{t1 - u}{t1 + u}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.7

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

  2. Initial simplification1.6

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

  4. Applied *-un-lft-identity1.6

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

  5. Applied associate-/l*1.7

    \[\leadsto \frac{\frac{-t1}{t1 + u}}{\color{blue}{\frac{1}{\frac{v}{t1 + u}}}}\]
  6. Using strategy rm

  7. Applied associate-/r/1.7

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

  8. Applied flip-+18.3

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

  9. Applied associate-/r/19.1

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

  10. Applied times-frac18.6

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

  11. Simplified1.3

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

  12. Final simplification1.3

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))