Average Error: 1.0 → 0.0
Time: 17.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot 1 + \left(-3 \cdot \left({v}^{2} \cdot \pi\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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}{\left(3 \cdot \pi\right) \cdot 1 + \left(-3 \cdot \left({v}^{2} \cdot \pi\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r128800 = 4.0;
        double r128801 = 3.0;
        double r128802 = atan2(1.0, 0.0);
        double r128803 = r128801 * r128802;
        double r128804 = 1.0;
        double r128805 = v;
        double r128806 = r128805 * r128805;
        double r128807 = r128804 - r128806;
        double r128808 = r128803 * r128807;
        double r128809 = 2.0;
        double r128810 = 6.0;
        double r128811 = r128810 * r128806;
        double r128812 = r128809 - r128811;
        double r128813 = sqrt(r128812);
        double r128814 = r128808 * r128813;
        double r128815 = r128800 / r128814;
        return r128815;
}
↓
double f(double v) {
        double r128816 = 4.0;
        double r128817 = 3.0;
        double r128818 = atan2(1.0, 0.0);
        double r128819 = r128817 * r128818;
        double r128820 = 1.0;
        double r128821 = r128819 * r128820;
        double r128822 = v;
        double r128823 = 2.0;
        double r128824 = pow(r128822, r128823);
        double r128825 = r128824 * r128818;
        double r128826 = r128817 * r128825;
        double r128827 = -r128826;
        double r128828 = r128821 + r128827;
        double r128829 = r128816 / r128828;
        double r128830 = 2.0;
        double r128831 = 6.0;
        double r128832 = r128822 * r128822;
        double r128833 = r128831 * r128832;
        double r128834 = r128830 - r128833;
        double r128835 = sqrt(r128834);
        double r128836 = r128829 / r128835;
        return r128836;
}

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 sub-neg0.0

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

  6. Applied distribute-lft-in0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))