Average Error: 18.1 → 1.7
Time: 4.1s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{-t1}{t1 + u} \cdot \frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{v}}{\sqrt[3]{t1 + u}}\]
\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}}
double code(double u, double v, double t1) {
	return ((-t1 * v) / ((t1 + u) * (t1 + u)));
}
↓
double code(double u, double v, double t1) {
	return (((-t1 / (t1 + u)) * ((cbrt(v) * cbrt(v)) / (cbrt((t1 + u)) * cbrt((t1 + u))))) * (cbrt(v) / cbrt((t1 + u))));
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.1

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

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

  6. Applied add-cube-cbrt2.4

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

  7. Applied times-frac2.4

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

  8. Applied associate-*r*1.7

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

  9. Final simplification1.7

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

Reproduce

herbie shell --seed 2020103 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))