Rosa's DopplerBench

Average Error: 17.3 → 1.3

Time: 13.9s

Precision: 64

Math

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

↓

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

C

double f(double u, double v, double t1) {
        double r19221 = t1;
        double r19222 = -r19221;
        double r19223 = v;
        double r19224 = r19222 * r19223;
        double r19225 = u;
        double r19226 = r19221 + r19225;
        double r19227 = r19226 * r19226;
        double r19228 = r19224 / r19227;
        return r19228;
}

↓

double f(double u, double v, double t1) {
        double r19229 = v;
        double r19230 = u;
        double r19231 = t1;
        double r19232 = r19230 + r19231;
        double r19233 = r19229 / r19232;
        double r19234 = -r19231;
        double r19235 = r19233 * r19234;
        double r19236 = r19235 / r19232;
        return r19236;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

1. Initial program 17.3

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

3. Applied associate-/r* 10.7

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

4. Simplified 1.2

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

6. Applied div-inv 1.3

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

7. Applied associate-*l* 1.4

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

8. Simplified 1.3

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

9. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))