Average Error: 1.0 → 0.0
Time: 6.2s
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 r201229 = 4.0;
        double r201230 = 3.0;
        double r201231 = atan2(1.0, 0.0);
        double r201232 = r201230 * r201231;
        double r201233 = 1.0;
        double r201234 = v;
        double r201235 = r201234 * r201234;
        double r201236 = r201233 - r201235;
        double r201237 = r201232 * r201236;
        double r201238 = 2.0;
        double r201239 = 6.0;
        double r201240 = r201239 * r201235;
        double r201241 = r201238 - r201240;
        double r201242 = sqrt(r201241);
        double r201243 = r201237 * r201242;
        double r201244 = r201229 / r201243;
        return r201244;
}
↓
double f(double v) {
        double r201245 = 1.0;
        double r201246 = 3.0;
        double r201247 = atan2(1.0, 0.0);
        double r201248 = r201246 * r201247;
        double r201249 = 1.0;
        double r201250 = v;
        double r201251 = r201250 * r201250;
        double r201252 = r201249 - r201251;
        double r201253 = r201248 * r201252;
        double r201254 = r201245 / r201253;
        double r201255 = 3.0;
        double r201256 = pow(r201254, r201255);
        double r201257 = cbrt(r201256);
        double r201258 = 4.0;
        double r201259 = 2.0;
        double r201260 = 6.0;
        double r201261 = r201260 * r201251;
        double r201262 = r201259 - r201261;
        double r201263 = sqrt(r201262);
        double r201264 = r201258 / r201263;
        double r201265 = r201257 * r201264;
        return r201265;
}

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 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))