Average Error: 17.6 → 1.4
Time: 9.5s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
↓
\[\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}} \]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1) :precision binary64 (/ (/ v (+ 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 / (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 17.6

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

  2. Simplified1.4

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

  3. Applied add-cube-cbrt_binary641.6

    \[\leadsto \frac{\frac{v}{t1 + u}}{-1 - \color{blue}{\left(\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}}} \]

  4. Applied cancel-sign-sub-inv_binary641.6

    \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}}} \]

  5. Applied *-un-lft-identity_binary641.6

    \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \left(-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}\right)}} \]

  6. Applied *-un-lft-identity_binary641.6

    \[\leadsto \frac{\frac{v}{\color{blue}{1 \cdot \left(t1 + u\right)}}}{1 \cdot \left(-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}\right)} \]

  7. Applied *-un-lft-identity_binary641.6

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot v}}{1 \cdot \left(t1 + u\right)}}{1 \cdot \left(-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}\right)} \]

  8. Applied times-frac_binary641.6

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{v}{t1 + u}}}{1 \cdot \left(-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}\right)} \]

  9. Applied times-frac_binary641.6

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{v}{t1 + u}}{-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}}} \]

  10. Simplified1.6

    \[\leadsto \color{blue}{1} \cdot \frac{\frac{v}{t1 + u}}{-1 + \left(-\sqrt[3]{\frac{u}{t1}} \cdot \sqrt[3]{\frac{u}{t1}}\right) \cdot \sqrt[3]{\frac{u}{t1}}} \]

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2022005 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))