Average Error: 17.9 → 1.3
Time: 3.0s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{t1}{t1 + u} \cdot \frac{-v}{t1 + u}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

  1. Initial program 17.9

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

  2. Simplified18.2

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

  4. Applied *-un-lft-identity18.2

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

  5. Applied times-frac11.4

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

  6. Applied associate-*r*1.3

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

  7. Simplified1.3

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

  8. Final simplification1.3

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

Reproduce

herbie shell --seed 2020184 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))