Average Error: 18.4 → 1.2
Time: 20.0s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{\sqrt[3]{t1} \cdot \sqrt[3]{t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\frac{t1 + u}{\frac{-\sqrt[3]{t1}}{\frac{\sqrt[3]{t1 + u}}{v}}}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{\sqrt[3]{t1} \cdot \sqrt[3]{t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\frac{t1 + u}{\frac{-\sqrt[3]{t1}}{\frac{\sqrt[3]{t1 + u}}{v}}}}
double f(double u, double v, double t1) {
        double r27823 = t1;
        double r27824 = -r27823;
        double r27825 = v;
        double r27826 = r27824 * r27825;
        double r27827 = u;
        double r27828 = r27823 + r27827;
        double r27829 = r27828 * r27828;
        double r27830 = r27826 / r27829;
        return r27830;
}
↓
double f(double u, double v, double t1) {
        double r27831 = t1;
        double r27832 = cbrt(r27831);
        double r27833 = r27832 * r27832;
        double r27834 = u;
        double r27835 = r27831 + r27834;
        double r27836 = cbrt(r27835);
        double r27837 = r27836 * r27836;
        double r27838 = r27833 / r27837;
        double r27839 = -r27832;
        double r27840 = v;
        double r27841 = r27836 / r27840;
        double r27842 = r27839 / r27841;
        double r27843 = r27835 / r27842;
        double r27844 = r27838 / r27843;
        return r27844;
}

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 clear-num1.7

    \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}}\]
  6. Using strategy rm

  7. Applied associate-*l/1.6

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

  8. Simplified1.4

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

  10. Applied *-un-lft-identity1.4

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

  11. Applied add-cube-cbrt2.1

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

  12. Applied times-frac2.1

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

  13. Applied add-cube-cbrt1.8

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

  14. Applied distribute-rgt-neg-in1.8

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

  15. Applied times-frac1.0

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

  16. Applied associate-/l*1.2

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

  17. Final simplification1.2

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

Reproduce

herbie shell --seed 2019322 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))