Average Error: 1.0 → 0.0
Time: 13.1s
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 r6676825 = 4.0;
        double r6676826 = 3.0;
        double r6676827 = atan2(1.0, 0.0);
        double r6676828 = r6676826 * r6676827;
        double r6676829 = 1.0;
        double r6676830 = v;
        double r6676831 = r6676830 * r6676830;
        double r6676832 = r6676829 - r6676831;
        double r6676833 = r6676828 * r6676832;
        double r6676834 = 2.0;
        double r6676835 = 6.0;
        double r6676836 = r6676835 * r6676831;
        double r6676837 = r6676834 - r6676836;
        double r6676838 = sqrt(r6676837);
        double r6676839 = r6676833 * r6676838;
        double r6676840 = r6676825 / r6676839;
        return r6676840;
}
↓
double f(double v) {
        double r6676841 = 4.0;
        double r6676842 = atan2(1.0, 0.0);
        double r6676843 = 3.0;
        double r6676844 = r6676842 * r6676843;
        double r6676845 = 1.0;
        double r6676846 = v;
        double r6676847 = r6676846 * r6676846;
        double r6676848 = r6676845 - r6676847;
        double r6676849 = r6676844 * r6676848;
        double r6676850 = r6676841 / r6676849;
        double r6676851 = 2.0;
        double r6676852 = 6.0;
        double r6676853 = r6676852 * r6676847;
        double r6676854 = r6676851 - r6676853;
        double r6676855 = sqrt(r6676854);
        double r6676856 = r6676850 / r6676855;
        return r6676856;
}

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 2019171 +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)))))))