Rosa's DopplerBench

Average Error: 17.8 → 1.3

Time: 1.9m

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r1748626 = t1;
        double r1748627 = -r1748626;
        double r1748628 = v;
        double r1748629 = r1748627 * r1748628;
        double r1748630 = u;
        double r1748631 = r1748626 + r1748630;
        double r1748632 = r1748631 * r1748631;
        double r1748633 = r1748629 / r1748632;
        return r1748633;
}

↓

double f(double u, double v, double t1) {
        double r1748634 = v;
        double r1748635 = -1.0;
        double r1748636 = u;
        double r1748637 = t1;
        double r1748638 = r1748636 + r1748637;
        double r1748639 = r1748635 / r1748638;
        double r1748640 = r1748639 * r1748637;
        double r1748641 = r1748634 * r1748640;
        double r1748642 = r1748641 / r1748638;
        return r1748642;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 17.8

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

3. Applied times-frac 1.3

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

5. Applied associate-*r/ 1.2

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

7. Applied div-inv 1.3

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

8. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019144 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))