Average Error: 1.0 → 0.0
Time: 18.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\frac{\left(\sqrt{2 - \left(v \cdot 6\right) \cdot v} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)\right)}{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot 1\right) + 1 \cdot 1}}\]
\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{4}{\frac{\left(\sqrt{2 - \left(v \cdot 6\right) \cdot v} \cdot 3\right) \cdot \left(\pi \cdot \left(1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)\right)}{\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot 1\right) + 1 \cdot 1}}
double f(double v) {
        double r11109322 = 4.0;
        double r11109323 = 3.0;
        double r11109324 = atan2(1.0, 0.0);
        double r11109325 = r11109323 * r11109324;
        double r11109326 = 1.0;
        double r11109327 = v;
        double r11109328 = r11109327 * r11109327;
        double r11109329 = r11109326 - r11109328;
        double r11109330 = r11109325 * r11109329;
        double r11109331 = 2.0;
        double r11109332 = 6.0;
        double r11109333 = r11109332 * r11109328;
        double r11109334 = r11109331 - r11109333;
        double r11109335 = sqrt(r11109334);
        double r11109336 = r11109330 * r11109335;
        double r11109337 = r11109322 / r11109336;
        return r11109337;
}
↓
double f(double v) {
        double r11109338 = 4.0;
        double r11109339 = 2.0;
        double r11109340 = v;
        double r11109341 = 6.0;
        double r11109342 = r11109340 * r11109341;
        double r11109343 = r11109342 * r11109340;
        double r11109344 = r11109339 - r11109343;
        double r11109345 = sqrt(r11109344);
        double r11109346 = 3.0;
        double r11109347 = r11109345 * r11109346;
        double r11109348 = atan2(1.0, 0.0);
        double r11109349 = 1.0;
        double r11109350 = r11109349 * r11109349;
        double r11109351 = r11109349 * r11109350;
        double r11109352 = r11109340 * r11109340;
        double r11109353 = r11109352 * r11109352;
        double r11109354 = r11109353 * r11109352;
        double r11109355 = r11109351 - r11109354;
        double r11109356 = r11109348 * r11109355;
        double r11109357 = r11109347 * r11109356;
        double r11109358 = r11109352 * r11109349;
        double r11109359 = r11109353 + r11109358;
        double r11109360 = r11109359 + r11109350;
        double r11109361 = r11109357 / r11109360;
        double r11109362 = r11109338 / r11109361;
        return r11109362;
}

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. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +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)))))))