Average Error: 1.0 → 0.0
Time: 4.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}}{\frac{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]
\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)}}
↓
\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}}{\frac{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}
double f(double v) {
        double r259311 = 4.0;
        double r259312 = 3.0;
        double r259313 = atan2(1.0, 0.0);
        double r259314 = r259312 * r259313;
        double r259315 = 1.0;
        double r259316 = v;
        double r259317 = r259316 * r259316;
        double r259318 = r259315 - r259317;
        double r259319 = r259314 * r259318;
        double r259320 = 2.0;
        double r259321 = 6.0;
        double r259322 = r259321 * r259317;
        double r259323 = r259320 - r259322;
        double r259324 = sqrt(r259323);
        double r259325 = r259319 * r259324;
        double r259326 = r259311 / r259325;
        return r259326;
}
↓
double f(double v) {
        double r259327 = 4.0;
        double r259328 = 3.0;
        double r259329 = atan2(1.0, 0.0);
        double r259330 = r259328 * r259329;
        double r259331 = 1.0;
        double r259332 = 3.0;
        double r259333 = pow(r259331, r259332);
        double r259334 = v;
        double r259335 = r259334 * r259334;
        double r259336 = pow(r259335, r259332);
        double r259337 = r259333 - r259336;
        double r259338 = r259330 * r259337;
        double r259339 = r259327 / r259338;
        double r259340 = 2.0;
        double r259341 = 6.0;
        double r259342 = r259341 * r259335;
        double r259343 = r259340 - r259342;
        double r259344 = sqrt(r259343);
        double r259345 = r259331 * r259331;
        double r259346 = r259335 * r259335;
        double r259347 = r259331 * r259335;
        double r259348 = r259346 + r259347;
        double r259349 = r259345 + r259348;
        double r259350 = r259344 / r259349;
        double r259351 = r259339 / r259350;
        return r259351;
}

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 associate-/r*0.0

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

  5. Applied flip3--0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied associate-/r/0.0

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

  8. Applied associate-/l*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019346 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))