Average Error: 18.4 → 1.5
Time: 10.8s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \frac{-t1}{t1 + u}\right) \cdot \frac{\sqrt[3]{v}}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\left(\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \frac{-t1}{t1 + u}\right) \cdot \frac{\sqrt[3]{v}}{t1 + u}
double f(double u, double v, double t1) {
        double r37856 = t1;
        double r37857 = -r37856;
        double r37858 = v;
        double r37859 = r37857 * r37858;
        double r37860 = u;
        double r37861 = r37856 + r37860;
        double r37862 = r37861 * r37861;
        double r37863 = r37859 / r37862;
        return r37863;
}
↓
double f(double u, double v, double t1) {
        double r37864 = v;
        double r37865 = cbrt(r37864);
        double r37866 = r37865 * r37865;
        double r37867 = t1;
        double r37868 = -r37867;
        double r37869 = u;
        double r37870 = r37867 + r37869;
        double r37871 = r37868 / r37870;
        double r37872 = r37866 * r37871;
        double r37873 = r37865 / r37870;
        double r37874 = r37872 * r37873;
        return r37874;
}

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 18.4

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

  3. Applied times-frac1.3

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

  5. Applied *-un-lft-identity1.3

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

  6. Applied add-cube-cbrt2.0

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

  7. Applied times-frac2.0

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

  8. Applied associate-*r*1.5

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

  9. Simplified1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))