Rosa's DopplerBench

Average Error: 18.2 → 1.2

Time: 10.1s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r422072 = t1;
        double r422073 = -r422072;
        double r422074 = v;
        double r422075 = r422073 * r422074;
        double r422076 = u;
        double r422077 = r422072 + r422076;
        double r422078 = r422077 * r422077;
        double r422079 = r422075 / r422078;
        return r422079;
}

↓

double f(double u, double v, double t1) {
        double r422080 = t1;
        double r422081 = u;
        double r422082 = r422081 + r422080;
        double r422083 = r422080 / r422082;
        double r422084 = v;
        double r422085 = r422083 * r422084;
        double r422086 = r422085 / r422082;
        double r422087 = -r422086;
        return r422087;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

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-frac 1.4

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

5. Applied div-inv 1.5

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

7. Applied distribute-lft-neg-out 1.5

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

8. Applied distribute-lft-neg-out 1.5

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

9. Simplified 1.2

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

10. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))