Average Error: 17.5 → 1.3
Time: 2.9m
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{t1 + u} \cdot \frac{\frac{-1}{t1 + u}}{\frac{1}{t1}}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied div-inv1.5

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

  5. Applied div-inv1.5

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

  6. Applied times-frac1.4

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

  7. Simplified1.2

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

  9. Applied clear-num1.3

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

  11. Applied div-inv1.4

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

  12. Applied associate-/r*1.3

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

  13. Final simplification1.3

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))