Average Error: 17.6 → 1.5
Time: 4.9s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}
double f(double u, double v, double t1) {
        double r34907 = t1;
        double r34908 = -r34907;
        double r34909 = v;
        double r34910 = r34908 * r34909;
        double r34911 = u;
        double r34912 = r34907 + r34911;
        double r34913 = r34912 * r34912;
        double r34914 = r34910 / r34913;
        return r34914;
}
↓
double f(double u, double v, double t1) {
        double r34915 = t1;
        double r34916 = -r34915;
        double r34917 = u;
        double r34918 = r34915 + r34917;
        double r34919 = r34916 / r34918;
        double r34920 = v;
        double r34921 = r34918 / r34920;
        double r34922 = r34919 / r34921;
        return r34922;
}

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 17.6

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

  3. Applied times-frac1.2

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

  5. Applied clear-num1.6

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

  7. Applied *-un-lft-identity1.6

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

  8. Applied *-un-lft-identity1.6

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

  9. Applied times-frac1.6

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

  10. Applied associate-*l*1.6

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

  11. Simplified1.5

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

  12. Final simplification1.5

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))