Average Error: 18.0 → 1.5
Time: 2.6s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

  1. Initial program 18.0

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

  2. Simplified3.6

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

  4. Applied associate-/r*1.8

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

  6. Applied div-inv1.9

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

  8. Applied associate-*l/1.5

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

  9. Simplified1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2020185 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))