Rosa's DopplerBench

Average Error: 17.8 → 1.8

Time: 3.5s

Precision: binary64

Math

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

↓

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

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 17.8

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

3. Applied times-frac 1.4

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

5. Applied clear-num 1.8

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

6. Final simplification 1.8

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

Reproduce

herbie shell --seed 2020171 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))