Rosa's DopplerBench

Average Error: 18.4 → 1.5

Time: 5.4s

Precision: binary64

Math

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

↓

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

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 18.4

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

3. Applied times-frac 1.3

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

5. Applied neg-mul-1 1.3

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

6. Applied associate-/l* 1.5

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

7. Taylor expanded around 0 1.5

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

8. Final simplification 1.5

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

Reproduce

herbie shell --seed 2020162 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))