Average Error: 1.0 → 0.0
Time: 13.0s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r5782778 = 4.0;
        double r5782779 = 3.0;
        double r5782780 = atan2(1.0, 0.0);
        double r5782781 = r5782779 * r5782780;
        double r5782782 = 1.0;
        double r5782783 = v;
        double r5782784 = r5782783 * r5782783;
        double r5782785 = r5782782 - r5782784;
        double r5782786 = r5782781 * r5782785;
        double r5782787 = 2.0;
        double r5782788 = 6.0;
        double r5782789 = r5782788 * r5782784;
        double r5782790 = r5782787 - r5782789;
        double r5782791 = sqrt(r5782790);
        double r5782792 = r5782786 * r5782791;
        double r5782793 = r5782778 / r5782792;
        return r5782793;
}
↓
double f(double v) {
        double r5782794 = 4.0;
        double r5782795 = atan2(1.0, 0.0);
        double r5782796 = 3.0;
        double r5782797 = r5782795 * r5782796;
        double r5782798 = 1.0;
        double r5782799 = v;
        double r5782800 = r5782799 * r5782799;
        double r5782801 = r5782798 - r5782800;
        double r5782802 = r5782797 * r5782801;
        double r5782803 = r5782794 / r5782802;
        double r5782804 = 2.0;
        double r5782805 = 6.0;
        double r5782806 = r5782805 * r5782800;
        double r5782807 = r5782804 - r5782806;
        double r5782808 = sqrt(r5782807);
        double r5782809 = r5782803 / r5782808;
        return r5782809;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))