Rosa's DopplerBench

Average Error: 17.9 → 1.3

Time: 37.9s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r1225600 = t1;
        double r1225601 = -r1225600;
        double r1225602 = v;
        double r1225603 = r1225601 * r1225602;
        double r1225604 = u;
        double r1225605 = r1225600 + r1225604;
        double r1225606 = r1225605 * r1225605;
        double r1225607 = r1225603 / r1225606;
        return r1225607;
}

↓

double f(double u, double v, double t1) {
        double r1225608 = t1;
        double r1225609 = -r1225608;
        double r1225610 = u;
        double r1225611 = r1225608 + r1225610;
        double r1225612 = r1225609 / r1225611;
        double r1225613 = v;
        double r1225614 = r1225612 * r1225613;
        double r1225615 = 1.0;
        double r1225616 = r1225615 / r1225611;
        double r1225617 = r1225614 * r1225616;
        return r1225617;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 17.9

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

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

6. Applied associate-*r* 1.3

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

7. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))