Rosa's DopplerBench

Average Error: 18.3 → 1.2

Time: 19.1s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r749592 = t1;
        double r749593 = -r749592;
        double r749594 = v;
        double r749595 = r749593 * r749594;
        double r749596 = u;
        double r749597 = r749592 + r749596;
        double r749598 = r749597 * r749597;
        double r749599 = r749595 / r749598;
        return r749599;
}

↓

double f(double u, double v, double t1) {
        double r749600 = t1;
        double r749601 = u;
        double r749602 = r749600 + r749601;
        double r749603 = r749600 / r749602;
        double r749604 = v;
        double r749605 = r749603 * r749604;
        double r749606 = r749605 / r749602;
        double r749607 = -r749606;
        return r749607;
}

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 1.3

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

5. Applied div-inv 1.4

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

6. Applied associate-*r* 1.3

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

8. Applied associate-*r/ 1.2

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

9. Simplified 1.2

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

10. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))