Rosa's DopplerBench

Average Error: 18.3 → 1.3

Time: 30.6s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r1102463 = t1;
        double r1102464 = -r1102463;
        double r1102465 = v;
        double r1102466 = r1102464 * r1102465;
        double r1102467 = u;
        double r1102468 = r1102463 + r1102467;
        double r1102469 = r1102468 * r1102468;
        double r1102470 = r1102466 / r1102469;
        return r1102470;
}

↓

double f(double u, double v, double t1) {
        double r1102471 = t1;
        double r1102472 = u;
        double r1102473 = r1102472 + r1102471;
        double r1102474 = r1102471 / r1102473;
        double r1102475 = v;
        double r1102476 = r1102474 * r1102475;
        double r1102477 = -1.0;
        double r1102478 = r1102477 / r1102473;
        double r1102479 = r1102476 * r1102478;
        return r1102479;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 18.3

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

3. Applied times-frac 1.1

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

5. Applied div-inv 1.3

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

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

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))