Average Error: 18.1 → 1.5
Time: 3.3s
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 times-frac1.3

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

  5. Applied associate-*r/1.3

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

  6. Simplified1.2

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

  8. Applied clear-num1.6

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

  10. Applied distribute-lft-neg-out1.6

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

  11. Applied distribute-frac-neg1.6

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

  12. Simplified1.5

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

  13. Final simplification1.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))