Average Error: 1.0 → 0.0
Time: 12.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3 \cdot \pi}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)} \cdot \left(\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)\]
\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}{3 \cdot \pi}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} - {v}^{6}\right)} \cdot \left(\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)
double f(double v) {
        double r255675 = 4.0;
        double r255676 = 3.0;
        double r255677 = atan2(1.0, 0.0);
        double r255678 = r255676 * r255677;
        double r255679 = 1.0;
        double r255680 = v;
        double r255681 = r255680 * r255680;
        double r255682 = r255679 - r255681;
        double r255683 = r255678 * r255682;
        double r255684 = 2.0;
        double r255685 = 6.0;
        double r255686 = r255685 * r255681;
        double r255687 = r255684 - r255686;
        double r255688 = sqrt(r255687);
        double r255689 = r255683 * r255688;
        double r255690 = r255675 / r255689;
        return r255690;
}
↓
double f(double v) {
        double r255691 = 4.0;
        double r255692 = 3.0;
        double r255693 = atan2(1.0, 0.0);
        double r255694 = r255692 * r255693;
        double r255695 = r255691 / r255694;
        double r255696 = 2.0;
        double r255697 = 3.0;
        double r255698 = pow(r255696, r255697);
        double r255699 = 6.0;
        double r255700 = v;
        double r255701 = r255700 * r255700;
        double r255702 = r255699 * r255701;
        double r255703 = pow(r255702, r255697);
        double r255704 = r255698 - r255703;
        double r255705 = sqrt(r255704);
        double r255706 = 1.0;
        double r255707 = pow(r255706, r255697);
        double r255708 = 6.0;
        double r255709 = pow(r255700, r255708);
        double r255710 = r255707 - r255709;
        double r255711 = r255705 * r255710;
        double r255712 = r255695 / r255711;
        double r255713 = r255706 * r255706;
        double r255714 = r255701 * r255701;
        double r255715 = r255706 * r255701;
        double r255716 = r255714 + r255715;
        double r255717 = r255713 + r255716;
        double r255718 = r255696 * r255696;
        double r255719 = r255702 * r255702;
        double r255720 = r255696 * r255702;
        double r255721 = r255719 + r255720;
        double r255722 = r255718 + r255721;
        double r255723 = sqrt(r255722);
        double r255724 = r255717 * r255723;
        double r255725 = r255712 * r255724;
        return r255725;
}

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 flip3--1.0

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

  4. Applied sqrt-div1.0

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

  5. Applied flip3--1.0

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

  6. Applied associate-*r/1.0

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

  7. Applied frac-times1.0

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

  8. Applied associate-/r/1.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))