Average Error: 18.3 → 1.8
Time: 3.3s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{1}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\frac{v}{\sqrt[3]{t1 + u}}}{-1 - \frac{u}{t1}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{1}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\frac{v}{\sqrt[3]{t1 + u}}}{-1 - \frac{u}{t1}}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (*
  (/ 1.0 (* (cbrt (+ t1 u)) (cbrt (+ t1 u))))
  (/ (/ v (cbrt (+ 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 (1.0 / (cbrt(t1 + u) * cbrt(t1 + u))) * ((v / cbrt(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. Using strategy rm

  4. Applied *-un-lft-identity_binary641.3

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

  5. Applied add-cube-cbrt_binary642.0

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

  6. Applied *-un-lft-identity_binary642.0

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

  7. Applied times-frac_binary642.0

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

  8. Applied times-frac_binary641.8

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

  9. Simplified1.8

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

  10. Final simplification1.8

    \[\leadsto \frac{1}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\frac{v}{\sqrt[3]{t1 + u}}}{-1 - \frac{u}{t1}}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))