Average Error: 18.2 → 1.3
Time: 5.5s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{-1 - \frac{u}{t1}}}{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))) (+ 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))) / (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. Simplified1.6

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

  4. Applied clear-num_binary64_771.9

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

  6. Applied *-un-lft-identity_binary64_781.9

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

  7. Applied div-inv_binary64_752.0

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

  8. Applied add-sqr-sqrt_binary64_1002.0

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

  9. Applied times-frac_binary64_841.8

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

  10. Applied times-frac_binary64_841.5

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

  11. Simplified1.5

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

  12. Simplified1.4

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

  14. Applied associate-*l/_binary64_211.3

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

  15. Simplified1.3

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

  16. Final simplification1.3

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

Reproduce

herbie shell --seed 2020355 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))