Rosa's DopplerBench

Average Error: 18.2 → 1.6

Time: 3.1s

Precision: binary64

Math

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

↓

\[\left(\frac{-t1}{t1 + u} \cdot \frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{v}}{\sqrt[3]{t1 + u}}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 18.2

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

3. Applied times-frac 1.2

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

5. Applied add-cube-cbrt 1.9

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

6. Applied add-cube-cbrt 2.1

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

7. Applied times-frac 2.1

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

8. Applied associate-*r* 1.6

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

9. Final simplification 1.6

   \[\leadsto \left(\frac{-t1}{t1 + u} \cdot \frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{v}}{\sqrt[3]{t1 + u}}\]

Reproduce

herbie shell --seed 2020173 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))