Average Error: 18.2 → 1.3
Time: 5.2s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{\frac{t1 + u}{-t1}}}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{\frac{t1 + u}{-t1}}}{t1 + u}
double f(double u, double v, double t1) {
        double r35214 = t1;
        double r35215 = -r35214;
        double r35216 = v;
        double r35217 = r35215 * r35216;
        double r35218 = u;
        double r35219 = r35214 + r35218;
        double r35220 = r35219 * r35219;
        double r35221 = r35217 / r35220;
        return r35221;
}
↓
double f(double u, double v, double t1) {
        double r35222 = v;
        double r35223 = t1;
        double r35224 = u;
        double r35225 = r35223 + r35224;
        double r35226 = -r35223;
        double r35227 = r35225 / r35226;
        double r35228 = r35222 / r35227;
        double r35229 = r35228 / r35225;
        return r35229;
}

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-frac1.3

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

  5. Applied clear-num1.4

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

  7. Applied associate-*r/1.4

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

  8. Simplified1.3

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

  9. Final simplification1.3

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))