Average Error: 18.1 → 1.3
Time: 2.7s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1}
double code(double u, double v, double t1) {
	return (((double) (((double) -(t1)) * v)) / ((double) (((double) (t1 + u)) * ((double) (t1 + u)))));
}
↓
double code(double u, double v, double t1) {
	return ((v / ((double) (-1.0 - (u / t1)))) / ((double) (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 Error: 18.1 bits

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

  2. SimplifiedError: 3.3 bits

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

  4. Applied *-un-lft-identityError: 3.3 bits

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

  5. Applied times-fracError: 1.5 bits

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

  7. Applied associate-*l/Error: 1.3 bits

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

  8. SimplifiedError: 1.3 bits

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

  9. Final simplificationError: 1.3 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))