Average Error: 1.0 → 0.0
Time: 6.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}\]
\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}{e^{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}
double f(double v) {
        double r322880 = 4.0;
        double r322881 = 3.0;
        double r322882 = atan2(1.0, 0.0);
        double r322883 = r322881 * r322882;
        double r322884 = 1.0;
        double r322885 = v;
        double r322886 = r322885 * r322885;
        double r322887 = r322884 - r322886;
        double r322888 = r322883 * r322887;
        double r322889 = 2.0;
        double r322890 = 6.0;
        double r322891 = r322890 * r322886;
        double r322892 = r322889 - r322891;
        double r322893 = sqrt(r322892);
        double r322894 = r322888 * r322893;
        double r322895 = r322880 / r322894;
        return r322895;
}
↓
double f(double v) {
        double r322896 = 4.0;
        double r322897 = 3.0;
        double r322898 = atan2(1.0, 0.0);
        double r322899 = r322897 * r322898;
        double r322900 = 1.0;
        double r322901 = v;
        double r322902 = r322901 * r322901;
        double r322903 = r322900 - r322902;
        double r322904 = r322899 * r322903;
        double r322905 = 2.0;
        double r322906 = 6.0;
        double r322907 = r322906 * r322902;
        double r322908 = r322905 - r322907;
        double r322909 = sqrt(r322908);
        double r322910 = r322904 * r322909;
        double r322911 = log(r322910);
        double r322912 = exp(r322911);
        double r322913 = r322896 / r322912;
        return r322913;
}

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-exp-log1.0

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

  4. Applied add-exp-log1.0

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

  5. Applied add-exp-log1.0

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

  6. Applied add-exp-log1.0

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

  7. Applied prod-exp1.0

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

  8. Applied prod-exp1.0

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

  9. Applied prod-exp0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))