Average Error: 18.2 → 1.5
Time: 9.0s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt[3]{v}}{u + t1}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt[3]{v}}{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
 (*
  (/
   (* (cbrt v) (cbrt v))
   (* (cbrt (- -1.0 (/ u t1))) (cbrt (- -1.0 (/ u t1)))))
  (/ (/ (cbrt v) (+ 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 ((cbrt(v) * cbrt(v)) / (cbrt(-1.0 - (u / t1)) * cbrt(-1.0 - (u / t1)))) * ((cbrt(v) / (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 18.2

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

  2. Simplified1.3

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

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

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

  6. Applied add-cube-cbrt_binary64_1132.1

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

  7. Applied times-frac_binary64_842.1

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

  8. Applied times-frac_binary64_841.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2021043 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))