Average Error: 1.0 → 0.0
Time: 22.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt[3]{\left(\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi} \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}\right) \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\sqrt[3]{\left(\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi} \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}\right) \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r6394662 = 4.0;
        double r6394663 = 3.0;
        double r6394664 = atan2(1.0, 0.0);
        double r6394665 = r6394663 * r6394664;
        double r6394666 = 1.0;
        double r6394667 = v;
        double r6394668 = r6394667 * r6394667;
        double r6394669 = r6394666 - r6394668;
        double r6394670 = r6394665 * r6394669;
        double r6394671 = 2.0;
        double r6394672 = 6.0;
        double r6394673 = r6394672 * r6394668;
        double r6394674 = r6394671 - r6394673;
        double r6394675 = sqrt(r6394674);
        double r6394676 = r6394670 * r6394675;
        double r6394677 = r6394662 / r6394676;
        return r6394677;
}
↓
double f(double v) {
        double r6394678 = 1.3333333333333333;
        double r6394679 = atan2(1.0, 0.0);
        double r6394680 = v;
        double r6394681 = r6394680 * r6394680;
        double r6394682 = r6394681 * r6394679;
        double r6394683 = r6394679 - r6394682;
        double r6394684 = r6394678 / r6394683;
        double r6394685 = r6394684 * r6394684;
        double r6394686 = r6394685 * r6394684;
        double r6394687 = cbrt(r6394686);
        double r6394688 = -6.0;
        double r6394689 = 2.0;
        double r6394690 = fma(r6394688, r6394681, r6394689);
        double r6394691 = sqrt(r6394690);
        double r6394692 = r6394687 / r6394691;
        return r6394692;
}

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 add-cbrt-cube1.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))