Average Error: 1.0 → 0.0
Time: 23.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r5021301 = 4.0;
        double r5021302 = 3.0;
        double r5021303 = atan2(1.0, 0.0);
        double r5021304 = r5021302 * r5021303;
        double r5021305 = 1.0;
        double r5021306 = v;
        double r5021307 = r5021306 * r5021306;
        double r5021308 = r5021305 - r5021307;
        double r5021309 = r5021304 * r5021308;
        double r5021310 = 2.0;
        double r5021311 = 6.0;
        double r5021312 = r5021311 * r5021307;
        double r5021313 = r5021310 - r5021312;
        double r5021314 = sqrt(r5021313);
        double r5021315 = r5021309 * r5021314;
        double r5021316 = r5021301 / r5021315;
        return r5021316;
}
↓
double f(double v) {
        double r5021317 = 1.0;
        double r5021318 = r5021317 * r5021317;
        double r5021319 = v;
        double r5021320 = r5021319 * r5021319;
        double r5021321 = r5021320 * r5021317;
        double r5021322 = r5021320 * r5021320;
        double r5021323 = r5021321 + r5021322;
        double r5021324 = r5021318 + r5021323;
        double r5021325 = 4.0;
        double r5021326 = 3.0;
        double r5021327 = atan2(1.0, 0.0);
        double r5021328 = r5021326 * r5021327;
        double r5021329 = r5021325 / r5021328;
        double r5021330 = r5021317 * r5021318;
        double r5021331 = r5021320 * r5021322;
        double r5021332 = r5021330 - r5021331;
        double r5021333 = r5021329 / r5021332;
        double r5021334 = 2.0;
        double r5021335 = 6.0;
        double r5021336 = r5021335 * r5021320;
        double r5021337 = r5021334 - r5021336;
        double r5021338 = sqrt(r5021337);
        double r5021339 = r5021333 / r5021338;
        double r5021340 = r5021324 * r5021339;
        return r5021340;
}

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 flip3--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))