Average Error: 17.0 → 1.6
Time: 4.8s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{-1 - \frac{u}{t1}}\\ \frac{v}{t_1 \cdot t_1} \cdot \frac{\frac{1}{u + t1}}{t_1} \end{array} \]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\begin{array}{l}
t_1 := \sqrt[3]{-1 - \frac{u}{t1}}\\
\frac{v}{t_1 \cdot t_1} \cdot \frac{\frac{1}{u + t1}}{t_1}
\end{array}

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (cbrt (- -1.0 (/ u t1)))))
   (* (/ v (* t_1 t_1)) (/ (/ 1.0 (+ u t1)) t_1))))
double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}
↓
double code(double u, double v, double t1) {
	double t_1 = cbrt(-1.0 - (u / t1));
	return (v / (t_1 * t_1)) * ((1.0 / (u + t1)) / t_1);
}

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 17.0

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

  2. Simplified1.5

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

  3. Applied add-cube-cbrt_binary641.7

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

  4. Applied div-inv_binary641.8

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

  5. Applied times-frac_binary641.6

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

  6. Final simplification1.6

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

Reproduce

herbie shell --seed 2022088 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))