Average Error: 1.0 → 0.0
Time: 23.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
\frac{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r157341 = 4.0;
        double r157342 = 3.0;
        double r157343 = atan2(1.0, 0.0);
        double r157344 = r157342 * r157343;
        double r157345 = 1.0;
        double r157346 = v;
        double r157347 = r157346 * r157346;
        double r157348 = r157345 - r157347;
        double r157349 = r157344 * r157348;
        double r157350 = 2.0;
        double r157351 = 6.0;
        double r157352 = r157351 * r157347;
        double r157353 = r157350 - r157352;
        double r157354 = sqrt(r157353);
        double r157355 = r157349 * r157354;
        double r157356 = r157341 / r157355;
        return r157356;
}
↓
double f(double v) {
        double r157357 = 1.0;
        double r157358 = v;
        double r157359 = r157358 * r157358;
        double r157360 = 4.0;
        double r157361 = pow(r157358, r157360);
        double r157362 = fma(r157357, r157359, r157361);
        double r157363 = 3.0;
        double r157364 = pow(r157362, r157363);
        double r157365 = 6.0;
        double r157366 = pow(r157357, r157365);
        double r157367 = r157364 + r157366;
        double r157368 = 4.0;
        double r157369 = pow(r157357, r157363);
        double r157370 = pow(r157358, r157365);
        double r157371 = r157369 - r157370;
        double r157372 = r157368 / r157371;
        double r157373 = 3.0;
        double r157374 = atan2(1.0, 0.0);
        double r157375 = r157373 * r157374;
        double r157376 = r157372 / r157375;
        double r157377 = r157367 * r157376;
        double r157378 = fma(r157358, r157358, r157357);
        double r157379 = r157357 * r157357;
        double r157380 = -r157379;
        double r157381 = fma(r157359, r157378, r157380);
        double r157382 = pow(r157357, r157360);
        double r157383 = fma(r157362, r157381, r157382);
        double r157384 = 2.0;
        double r157385 = 6.0;
        double r157386 = r157385 * r157359;
        double r157387 = r157384 - r157386;
        double r157388 = sqrt(r157387);
        double r157389 = r157383 * r157388;
        double r157390 = r157377 / r157389;
        return r157390;
}

Error

Bits error versus v

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

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

  9. Applied flip3-+0.0

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

  10. Applied frac-times0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))