Average Error: 18.2 → 1.5
Time: 4.5s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(-\frac{v}{\frac{t1 + u}{t1}}\right) \cdot \frac{1}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\left(-\frac{v}{\frac{t1 + u}{t1}}\right) \cdot \frac{1}{t1 + u}
double f(double u, double v, double t1) {
        double r30674 = t1;
        double r30675 = -r30674;
        double r30676 = v;
        double r30677 = r30675 * r30676;
        double r30678 = u;
        double r30679 = r30674 + r30678;
        double r30680 = r30679 * r30679;
        double r30681 = r30677 / r30680;
        return r30681;
}
↓
double f(double u, double v, double t1) {
        double r30682 = v;
        double r30683 = t1;
        double r30684 = u;
        double r30685 = r30683 + r30684;
        double r30686 = r30685 / r30683;
        double r30687 = r30682 / r30686;
        double r30688 = -r30687;
        double r30689 = 1.0;
        double r30690 = r30689 / r30685;
        double r30691 = r30688 * r30690;
        return r30691;
}

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-frac1.5

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

  5. Applied *-un-lft-identity1.5

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

  6. Applied distribute-lft-neg-in1.5

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

  7. Applied associate-/l*1.7

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

  9. Applied div-inv1.8

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

  10. Applied associate-*r*1.6

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

  11. Simplified1.5

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

  12. Final simplification1.5

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))