Average Error: 1.0 → 0.0
Time: 6.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
\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{4}{\left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}
double f(double v) {
        double r245619 = 4.0;
        double r245620 = 3.0;
        double r245621 = atan2(1.0, 0.0);
        double r245622 = r245620 * r245621;
        double r245623 = 1.0;
        double r245624 = v;
        double r245625 = r245624 * r245624;
        double r245626 = r245623 - r245625;
        double r245627 = r245622 * r245626;
        double r245628 = 2.0;
        double r245629 = 6.0;
        double r245630 = r245629 * r245625;
        double r245631 = r245628 - r245630;
        double r245632 = sqrt(r245631);
        double r245633 = r245627 * r245632;
        double r245634 = r245619 / r245633;
        return r245634;
}
↓
double f(double v) {
        double r245635 = 4.0;
        double r245636 = 3.0;
        double r245637 = atan2(1.0, 0.0);
        double r245638 = r245636 * r245637;
        double r245639 = 1.0;
        double r245640 = v;
        double r245641 = r245640 * r245640;
        double r245642 = r245639 - r245641;
        double r245643 = r245638 * r245642;
        double r245644 = 2.0;
        double r245645 = 6.0;
        double r245646 = r245645 * r245641;
        double r245647 = r245644 - r245646;
        double r245648 = sqrt(r245647);
        double r245649 = cbrt(r245648);
        double r245650 = r245649 * r245649;
        double r245651 = r245643 * r245650;
        double r245652 = r245651 * r245649;
        double r245653 = r245635 / r245652;
        return r245653;
}

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-cube-cbrt1.0

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

  4. Applied associate-*r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))