Average Error: 1.0 → 0.0
Time: 5.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \frac{\mathsf{fma}\left(v, v, 1\right) \cdot \sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \frac{\mathsf{fma}\left(v, v, 1\right) \cdot \sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r244450 = 4.0;
        double r244451 = 3.0;
        double r244452 = atan2(1.0, 0.0);
        double r244453 = r244451 * r244452;
        double r244454 = 1.0;
        double r244455 = v;
        double r244456 = r244455 * r244455;
        double r244457 = r244454 - r244456;
        double r244458 = r244453 * r244457;
        double r244459 = 2.0;
        double r244460 = 6.0;
        double r244461 = r244460 * r244456;
        double r244462 = r244459 - r244461;
        double r244463 = sqrt(r244462);
        double r244464 = r244458 * r244463;
        double r244465 = r244450 / r244464;
        return r244465;
}
↓
double f(double v) {
        double r244466 = 4.0;
        double r244467 = sqrt(r244466);
        double r244468 = 3.0;
        double r244469 = atan2(1.0, 0.0);
        double r244470 = r244468 * r244469;
        double r244471 = 1.0;
        double r244472 = r244471 * r244471;
        double r244473 = v;
        double r244474 = r244473 * r244473;
        double r244475 = r244474 * r244474;
        double r244476 = r244472 - r244475;
        double r244477 = r244470 * r244476;
        double r244478 = r244467 / r244477;
        double r244479 = fma(r244473, r244473, r244471);
        double r244480 = r244479 * r244467;
        double r244481 = 2.0;
        double r244482 = 6.0;
        double r244483 = r244482 * r244474;
        double r244484 = r244481 - r244483;
        double r244485 = sqrt(r244484);
        double r244486 = r244480 / r244485;
        double r244487 = r244478 * r244486;
        return r244487;
}

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 add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-*r/0.0

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

  8. Applied associate-/r/0.0

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

  9. Applied associate-*l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))