Average Error: 18.4 → 1.1
Time: 20.6s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{v}{t1 + u} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\left(\frac{v}{t1 + u} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}\right) \cdot \frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}
double f(double u, double v, double t1) {
        double r1122145 = t1;
        double r1122146 = -r1122145;
        double r1122147 = v;
        double r1122148 = r1122146 * r1122147;
        double r1122149 = u;
        double r1122150 = r1122145 + r1122149;
        double r1122151 = r1122150 * r1122150;
        double r1122152 = r1122148 / r1122151;
        return r1122152;
}
↓
double f(double u, double v, double t1) {
        double r1122153 = v;
        double r1122154 = t1;
        double r1122155 = u;
        double r1122156 = r1122154 + r1122155;
        double r1122157 = r1122153 / r1122156;
        double r1122158 = -r1122154;
        double r1122159 = cbrt(r1122158);
        double r1122160 = cbrt(r1122156);
        double r1122161 = r1122159 / r1122160;
        double r1122162 = r1122157 * r1122161;
        double r1122163 = r1122159 * r1122159;
        double r1122164 = r1122160 * r1122160;
        double r1122165 = r1122163 / r1122164;
        double r1122166 = r1122162 * r1122165;
        return r1122166;
}

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 add-cube-cbrt2.1

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

  6. Applied add-cube-cbrt1.6

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

  7. Applied times-frac1.6

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

  8. Applied associate-*l*1.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))