Average Error: 1.0 → 0.0
Time: 9.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\frac{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - {v}^{4}\right)}{1 + v \cdot v}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\frac{4}{\frac{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - {v}^{4}\right)}{1 + v \cdot v}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r329584 = 4.0;
        double r329585 = 3.0;
        double r329586 = atan2(1.0, 0.0);
        double r329587 = r329585 * r329586;
        double r329588 = 1.0;
        double r329589 = v;
        double r329590 = r329589 * r329589;
        double r329591 = r329588 - r329590;
        double r329592 = r329587 * r329591;
        double r329593 = 2.0;
        double r329594 = 6.0;
        double r329595 = r329594 * r329590;
        double r329596 = r329593 - r329595;
        double r329597 = sqrt(r329596);
        double r329598 = r329592 * r329597;
        double r329599 = r329584 / r329598;
        return r329599;
}
↓
double f(double v) {
        double r329600 = 4.0;
        double r329601 = 3.0;
        double r329602 = atan2(1.0, 0.0);
        double r329603 = r329601 * r329602;
        double r329604 = 1.0;
        double r329605 = r329604 * r329604;
        double r329606 = v;
        double r329607 = 4.0;
        double r329608 = pow(r329606, r329607);
        double r329609 = r329605 - r329608;
        double r329610 = r329603 * r329609;
        double r329611 = r329606 * r329606;
        double r329612 = r329604 + r329611;
        double r329613 = r329610 / r329612;
        double r329614 = r329600 / r329613;
        double r329615 = 2.0;
        double r329616 = 6.0;
        double r329617 = r329616 * r329611;
        double r329618 = r329615 - r329617;
        double r329619 = sqrt(r329618);
        double r329620 = r329614 / r329619;
        return r329620;
}

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 associate-/r*0.0

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

  5. Applied flip--0.0

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

  6. Applied associate-*r/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))