Average Error: 18.3 → 1.3
Time: 8.0s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
↓
\[\frac{-\frac{v}{t1 + u}}{1 + \frac{u}{t1}} \]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{-\frac{v}{t1 + u}}{1 + \frac{u}{t1}}

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1) :precision binary64 (/ (- (/ v (+ t1 u))) (+ 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 / (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 18.3

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

  2. Simplified1.3

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

  3. Applied div-inv_binary641.4

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

  4. Applied frac-2neg_binary641.4

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

  5. Simplified1.3

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

  6. Simplified1.3

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

  7. Final simplification1.3

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

Reproduce

herbie shell --seed 2022125 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))