Average Error: 1.0 → 0.0
Time: 6.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{{\left(\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)}^{3}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\sqrt[3]{{\left(\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)}^{3}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r304285 = 4.0;
        double r304286 = 3.0;
        double r304287 = atan2(1.0, 0.0);
        double r304288 = r304286 * r304287;
        double r304289 = 1.0;
        double r304290 = v;
        double r304291 = r304290 * r304290;
        double r304292 = r304289 - r304291;
        double r304293 = r304288 * r304292;
        double r304294 = 2.0;
        double r304295 = 6.0;
        double r304296 = r304295 * r304291;
        double r304297 = r304294 - r304296;
        double r304298 = sqrt(r304297);
        double r304299 = r304293 * r304298;
        double r304300 = r304285 / r304299;
        return r304300;
}
↓
double f(double v) {
        double r304301 = 1.0;
        double r304302 = 3.0;
        double r304303 = atan2(1.0, 0.0);
        double r304304 = r304302 * r304303;
        double r304305 = 1.0;
        double r304306 = v;
        double r304307 = r304306 * r304306;
        double r304308 = r304305 - r304307;
        double r304309 = r304304 * r304308;
        double r304310 = r304301 / r304309;
        double r304311 = 3.0;
        double r304312 = pow(r304310, r304311);
        double r304313 = cbrt(r304312);
        double r304314 = 4.0;
        double r304315 = 2.0;
        double r304316 = 6.0;
        double r304317 = r304316 * r304307;
        double r304318 = r304315 - r304317;
        double r304319 = sqrt(r304318);
        double r304320 = r304314 / r304319;
        double r304321 = r304313 * r304320;
        return r304321;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity1.0

    \[\leadsto \frac{\color{blue}{1 \cdot 4}}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  4. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.0

    \[\leadsto \frac{1}{\left(3 \cdot \pi\right) \cdot \color{blue}{\sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  7. Applied add-cbrt-cube1.0

    \[\leadsto \frac{1}{\left(3 \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  8. Applied add-cbrt-cube1.6

    \[\leadsto \frac{1}{\left(\color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}} \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  9. Applied cbrt-unprod1.0

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\left(3 \cdot 3\right) \cdot 3\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}} \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  10. Applied cbrt-unprod1.0

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\left(\left(3 \cdot 3\right) \cdot 3\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)}}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  11. Applied add-cbrt-cube1.0

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(\left(\left(3 \cdot 3\right) \cdot 3\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  12. Applied cbrt-undiv0.0

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(\left(\left(3 \cdot 3\right) \cdot 3\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)}}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  13. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)}^{3}}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  14. Final simplification0.0

    \[\leadsto \sqrt[3]{{\left(\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)}^{3}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))