Average Error: 17.6 → 1.8
Time: 13.4s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{1}{u + t1}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\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}}}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (*
  (/ v (* (cbrt (- -1.0 (/ u t1))) (cbrt (- -1.0 (/ u t1)))))
  (/ (/ 1.0 (+ u t1)) (cbrt (- -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 / (cbrt(-1.0 - (u / t1)) * cbrt(-1.0 - (u / t1)))) * ((1.0 / (u + t1)) / cbrt(-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. Using strategy rm

  4. Applied add-cube-cbrt_binary64_1131.6

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

  5. Applied div-inv_binary64_751.7

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

  6. Applied times-frac_binary64_841.8

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

  7. Final simplification1.8

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

Reproduce

herbie shell --seed 2021007 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))