Average Error: 18.7 → 1.3
Time: 6.5s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{-t1}{u + t1} \cdot v\right) \cdot \frac{1}{t1 + u}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

  1. Initial program 18.7

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

  3. Applied associate-/r*11.6

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

  4. Simplified1.2

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

  6. Applied div-inv1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2020182 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))