Average Error: 18.2 → 1.5
Time: 3.6s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{-\sqrt[3]{t1} \cdot \sqrt[3]{t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \left(\frac{\sqrt[3]{t1}}{\sqrt[3]{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \sqrt[3]{\sqrt[3]{t1 + u}}} \cdot \frac{v}{t1 + u}\right)\]

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-frac1.5

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

  5. Applied add-cube-cbrt2.2

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

  6. Applied add-cube-cbrt1.8

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

  7. Applied distribute-lft-neg-in1.8

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

  8. Applied times-frac1.8

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

  9. Applied associate-*l*1.2

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

  11. Applied add-cube-cbrt1.4

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

  12. Applied cbrt-prod1.5

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

  13. Final simplification1.5

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

Reproduce

herbie shell --seed 2020140 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))