Average Error: 18.2 → 1.3
Time: 2.4s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{-t1}{t1 + u} \cdot v}{t1 + u}\]

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-frac1.4

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

  5. Applied add-cube-cbrt2.1

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

  6. Applied *-un-lft-identity2.1

    \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{\color{blue}{1 \cdot v}}{\left(\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}\right) \cdot \sqrt[3]{t1 + u}}\]

  7. Applied times-frac2.1

    \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{v}{\sqrt[3]{t1 + u}}\right)}\]

  8. Applied associate-*r*2.9

    \[\leadsto \color{blue}{\left(\frac{-t1}{t1 + u} \cdot \frac{1}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\right) \cdot \frac{v}{\sqrt[3]{t1 + u}}}\]

  9. Simplified2.9

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

  11. Applied frac-times2.0

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

  12. Simplified1.3

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

  13. Final simplification1.3

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

Reproduce

herbie shell --seed 2020157 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))