Average Error: 1.0 → 0.0
Time: 13.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\left(\pi \cdot \pi\right) \cdot \left(\pi - \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \pi\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(\left(\pi \cdot \left(v \cdot v\right)\right) \cdot \left(\pi \cdot \left(v \cdot v\right)\right) + \left(\pi \cdot \left(v \cdot v\right)\right) \cdot \pi\right) + \pi \cdot \pi\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\frac{4}{3}}{\left(\pi \cdot \pi\right) \cdot \left(\pi - \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \pi\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(\left(\pi \cdot \left(v \cdot v\right)\right) \cdot \left(\pi \cdot \left(v \cdot v\right)\right) + \left(\pi \cdot \left(v \cdot v\right)\right) \cdot \pi\right) + \pi \cdot \pi\right)}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r5856307 = 4.0;
        double r5856308 = 3.0;
        double r5856309 = atan2(1.0, 0.0);
        double r5856310 = r5856308 * r5856309;
        double r5856311 = 1.0;
        double r5856312 = v;
        double r5856313 = r5856312 * r5856312;
        double r5856314 = r5856311 - r5856313;
        double r5856315 = r5856310 * r5856314;
        double r5856316 = 2.0;
        double r5856317 = 6.0;
        double r5856318 = r5856317 * r5856313;
        double r5856319 = r5856316 - r5856318;
        double r5856320 = sqrt(r5856319);
        double r5856321 = r5856315 * r5856320;
        double r5856322 = r5856307 / r5856321;
        return r5856322;
}
↓
double f(double v) {
        double r5856323 = 1.3333333333333333;
        double r5856324 = atan2(1.0, 0.0);
        double r5856325 = r5856324 * r5856324;
        double r5856326 = v;
        double r5856327 = r5856326 * r5856326;
        double r5856328 = r5856327 * r5856327;
        double r5856329 = r5856328 * r5856324;
        double r5856330 = r5856329 * r5856327;
        double r5856331 = r5856324 - r5856330;
        double r5856332 = r5856325 * r5856331;
        double r5856333 = r5856323 / r5856332;
        double r5856334 = r5856324 * r5856327;
        double r5856335 = r5856334 * r5856334;
        double r5856336 = r5856334 * r5856324;
        double r5856337 = r5856335 + r5856336;
        double r5856338 = r5856337 + r5856325;
        double r5856339 = r5856333 * r5856338;
        double r5856340 = -6.0;
        double r5856341 = 2.0;
        double r5856342 = fma(r5856340, r5856327, r5856341);
        double r5856343 = sqrt(r5856342);
        double r5856344 = r5856339 / r5856343;
        return r5856344;
}

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. Simplified0.0

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

  4. Applied flip3--0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))