Average Error: 18.4 → 1.5
Time: 2.6s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\sqrt[3]{v} \cdot \frac{\sqrt[3]{v}}{t1 + u}\right) \cdot \frac{\sqrt[3]{v}}{-1 - \frac{u}{t1}}\]

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. Simplified3.3

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

  4. Applied add-cube-cbrt4.0

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

  5. Applied times-frac1.5

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

  6. Simplified1.5

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

  7. Final simplification1.5

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

Reproduce

herbie shell --seed 2020184 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (neg t1) v) (* (+ t1 u) (+ t1 u))))