Average Error: 18.2 → 2.2
Time: 16.9s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{1}{\frac{-1 - \frac{u}{t1}}{v} \cdot \left(u + t1\right)}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{1}{\frac{-1 - \frac{u}{t1}}{v} \cdot \left(u + t1\right)}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (/ 1.0 (* (/ (- -1.0 (/ u t1)) v) (+ 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 1.0 / (((-1.0 - (u / t1)) / v) * (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 18.2

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

  2. Simplified1.5

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

  4. Applied *-un-lft-identity_binary641.5

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

  5. Applied associate-/l*_binary642.1

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

  6. Simplified2.1

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

  8. Applied associate-/r/_binary642.2

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

  9. Final simplification2.2

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

Reproduce

herbie shell --seed 2020231 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))