Average Error: 18.1 → 1.5
Time: 2.0s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}
double code(double u, double v, double t1) {
	return ((-t1 * v) / ((t1 + u) * (t1 + u)));
}
↓
double code(double u, double v, double t1) {
	return ((-t1 / (t1 + u)) / ((t1 + u) / v));
}

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 18.1

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

  3. Applied *-commutative18.1

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

  4. Applied times-frac1.2

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

  6. Applied clear-num1.6

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

  7. Applied associate-*l/1.5

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

  8. Simplified1.5

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

  9. Final simplification1.5

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

Reproduce

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