Average Error: 18.6 → 1.4
Time: 3.0s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[-1 \cdot \frac{\frac{v}{t1 + u}}{1 + \frac{u}{t1}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
-1 \cdot \frac{\frac{v}{t1 + u}}{1 + \frac{u}{t1}}
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 * ((v / (t1 + u)) / (1.0 + (u / t1))));
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.6

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

  3. Applied times-frac1.4

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

  5. Applied neg-mul-11.4

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

  6. Applied associate-/l*1.5

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

  7. Taylor expanded around 0 1.5

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

  9. Applied div-inv1.5

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

  10. Applied associate-*l*1.5

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

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2020102 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))