Average Error: 1.0 → 0.0
Time: 3.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r219971 = 4.0;
        double r219972 = 3.0;
        double r219973 = atan2(1.0, 0.0);
        double r219974 = r219972 * r219973;
        double r219975 = 1.0;
        double r219976 = v;
        double r219977 = r219976 * r219976;
        double r219978 = r219975 - r219977;
        double r219979 = r219974 * r219978;
        double r219980 = 2.0;
        double r219981 = 6.0;
        double r219982 = r219981 * r219977;
        double r219983 = r219980 - r219982;
        double r219984 = sqrt(r219983);
        double r219985 = r219979 * r219984;
        double r219986 = r219971 / r219985;
        return r219986;
}
↓
double f(double v) {
        double r219987 = 1.0;
        double r219988 = 3.0;
        double r219989 = atan2(1.0, 0.0);
        double r219990 = r219988 * r219989;
        double r219991 = 1.0;
        double r219992 = v;
        double r219993 = r219992 * r219992;
        double r219994 = r219991 - r219993;
        double r219995 = r219990 * r219994;
        double r219996 = r219987 / r219995;
        double r219997 = 4.0;
        double r219998 = 2.0;
        double r219999 = 6.0;
        double r220000 = r219999 * r219993;
        double r220001 = r219998 - r220000;
        double r220002 = sqrt(r220001);
        double r220003 = r219997 / r220002;
        double r220004 = r219996 * r220003;
        return r220004;
}

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 *-un-lft-identity1.0

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))