Rosa's DopplerBench

Average Error: 18.1 → 1.5

Time: 3.9s

Precision: 64

Math

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

↓

\[-1 \cdot \frac{\frac{t1}{t1 + u}}{\frac{t1 + u}{v}}\]

C

double code(double u, double v, double t1) {
	return ((-t1 * v) / ((t1 + u) * (t1 + u)));
}

↓

double code(double u, double v, double t1) {
	return (-1.0 * ((t1 / (t1 + u)) / ((t1 + u) / v)));
}

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 1.2

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

5. Applied clear-num 1.6

   \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 1.6

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

8. Applied neg-mul-1 1.6

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

9. Applied times-frac 1.6

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

10. Applied associate-*l* 1.6

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

11. Simplified 1.5

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

12. Final simplification 1.5

    \[\leadsto -1 \cdot \frac{\frac{t1}{t1 + u}}{\frac{t1 + u}{v}}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))