Average Error: 18.1 → 1.3
Time: 5.2s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{u + t1}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{u + t1}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (* (/ v (- -1.0 (/ u t1))) (/ 1.0 (+ 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 (v / (-1.0 - (u / t1))) * (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 18.1

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

  2. Simplified1.5

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

  4. Applied clear-num_binary642.0

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

  6. Applied associate-/r/_binary641.8

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

  7. Applied add-sqr-sqrt_binary641.8

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

  8. Applied times-frac_binary641.4

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

  9. Simplified1.3

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

  10. Simplified1.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2020268 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))