Average Error: 18.2 → 1.4
Time: 3.6s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\left(-t1\right) \cdot \frac{v}{t1 + u}\right) \cdot \frac{1}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\left(\left(-t1\right) \cdot \frac{v}{t1 + u}\right) \cdot \frac{1}{t1 + u}
double f(double u, double v, double t1) {
        double r19905 = t1;
        double r19906 = -r19905;
        double r19907 = v;
        double r19908 = r19906 * r19907;
        double r19909 = u;
        double r19910 = r19905 + r19909;
        double r19911 = r19910 * r19910;
        double r19912 = r19908 / r19911;
        return r19912;
}
↓
double f(double u, double v, double t1) {
        double r19913 = t1;
        double r19914 = -r19913;
        double r19915 = v;
        double r19916 = u;
        double r19917 = r19913 + r19916;
        double r19918 = r19915 / r19917;
        double r19919 = r19914 * r19918;
        double r19920 = 1.0;
        double r19921 = r19920 / r19917;
        double r19922 = r19919 * r19921;
        return r19922;
}

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 18.2

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

  3. Applied times-frac1.4

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

  5. Applied div-inv1.5

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

  6. Applied associate-*r*1.3

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

  7. Simplified1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))