Average Error: 18.1 → 1.5
Time: 45.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{t1 + u} \cdot \left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \sqrt[3]{\frac{-t1}{t1 + u}}\right)\right)\]

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 18.1

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

  2. Initial simplification1.5

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

  4. Applied div-inv1.6

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

  5. Applied div-inv1.6

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

  6. Applied times-frac1.4

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

  7. Simplified1.3

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

  9. Applied add-cube-cbrt1.5

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

  10. Final simplification1.5

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

Runtime

Time bar (total: 45.4s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))