Rosa's DopplerBench

Average Error: 18.1 → 1.2

Time: 19.4s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r1096869 = t1;
        double r1096870 = -r1096869;
        double r1096871 = v;
        double r1096872 = r1096870 * r1096871;
        double r1096873 = u;
        double r1096874 = r1096869 + r1096873;
        double r1096875 = r1096874 * r1096874;
        double r1096876 = r1096872 / r1096875;
        return r1096876;
}

↓

double f(double u, double v, double t1) {
        double r1096877 = t1;
        double r1096878 = u;
        double r1096879 = r1096878 + r1096877;
        double r1096880 = -r1096879;
        double r1096881 = r1096877 / r1096880;
        double r1096882 = v;
        double r1096883 = r1096881 * r1096882;
        double r1096884 = r1096883 / r1096879;
        return r1096884;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 18.1

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
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 frac-2neg 1.2

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

8. Simplified 1.2

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

9. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))