Rosa's DopplerBench

Average Error: 18.3 → 1.4

Time: 42.1s

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 r1459470 = t1;
        double r1459471 = -r1459470;
        double r1459472 = v;
        double r1459473 = r1459471 * r1459472;
        double r1459474 = u;
        double r1459475 = r1459470 + r1459474;
        double r1459476 = r1459475 * r1459475;
        double r1459477 = r1459473 / r1459476;
        return r1459477;
}

↓

double f(double u, double v, double t1) {
        double r1459478 = t1;
        double r1459479 = u;
        double r1459480 = r1459478 + r1459479;
        double r1459481 = r1459478 / r1459480;
        double r1459482 = v;
        double r1459483 = r1459481 * r1459482;
        double r1459484 = -1.0;
        double r1459485 = r1459484 / r1459480;
        double r1459486 = r1459483 * r1459485;
        return r1459486;
}

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 1.2

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
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* 1.4

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

7. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019138 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))