Average Error: 1.0 → 0.0
Time: 10.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {v}^{6}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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}}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {v}^{6}\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 \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r201594 = 4.0;
        double r201595 = 3.0;
        double r201596 = atan2(1.0, 0.0);
        double r201597 = r201595 * r201596;
        double r201598 = 1.0;
        double r201599 = v;
        double r201600 = r201599 * r201599;
        double r201601 = r201598 - r201600;
        double r201602 = r201597 * r201601;
        double r201603 = 2.0;
        double r201604 = 6.0;
        double r201605 = r201604 * r201600;
        double r201606 = r201603 - r201605;
        double r201607 = sqrt(r201606);
        double r201608 = r201602 * r201607;
        double r201609 = r201594 / r201608;
        return r201609;
}
↓
double f(double v) {
        double r201610 = 4.0;
        double r201611 = sqrt(r201610);
        double r201612 = 3.0;
        double r201613 = atan2(1.0, 0.0);
        double r201614 = r201612 * r201613;
        double r201615 = 1.0;
        double r201616 = 3.0;
        double r201617 = pow(r201615, r201616);
        double r201618 = v;
        double r201619 = 6.0;
        double r201620 = pow(r201618, r201619);
        double r201621 = r201617 - r201620;
        double r201622 = r201614 * r201621;
        double r201623 = r201615 * r201615;
        double r201624 = r201618 * r201618;
        double r201625 = r201624 * r201624;
        double r201626 = r201615 * r201624;
        double r201627 = r201625 + r201626;
        double r201628 = r201623 + r201627;
        double r201629 = r201622 / r201628;
        double r201630 = r201611 / r201629;
        double r201631 = 2.0;
        double r201632 = 6.0;
        double r201633 = r201632 * r201624;
        double r201634 = r201631 - r201633;
        double r201635 = sqrt(r201634);
        double r201636 = r201611 / r201635;
        double r201637 = r201630 * r201636;
        return r201637;
}

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 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 flip3--0.0

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

  7. Applied associate-*r/0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

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)))))))