Average Error: 1.0 → 0.0
Time: 19.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left({1}^{3} - {\left(v \cdot v\right)}^{3}\right) \cdot \left(\left(3 \cdot \pi\right) \cdot \left(v \cdot v + 1\right)\right)}}{\frac{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)}} \cdot \left(v \cdot v + 1\right)\]
\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({1}^{3} - {\left(v \cdot v\right)}^{3}\right) \cdot \left(\left(3 \cdot \pi\right) \cdot \left(v \cdot v + 1\right)\right)}}{\frac{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)}} \cdot \left(v \cdot v + 1\right)
double f(double v) {
        double r6864814 = 4.0;
        double r6864815 = 3.0;
        double r6864816 = atan2(1.0, 0.0);
        double r6864817 = r6864815 * r6864816;
        double r6864818 = 1.0;
        double r6864819 = v;
        double r6864820 = r6864819 * r6864819;
        double r6864821 = r6864818 - r6864820;
        double r6864822 = r6864817 * r6864821;
        double r6864823 = 2.0;
        double r6864824 = 6.0;
        double r6864825 = r6864824 * r6864820;
        double r6864826 = r6864823 - r6864825;
        double r6864827 = sqrt(r6864826);
        double r6864828 = r6864822 * r6864827;
        double r6864829 = r6864814 / r6864828;
        return r6864829;
}
↓
double f(double v) {
        double r6864830 = 4.0;
        double r6864831 = 1.0;
        double r6864832 = 3.0;
        double r6864833 = pow(r6864831, r6864832);
        double r6864834 = v;
        double r6864835 = r6864834 * r6864834;
        double r6864836 = pow(r6864835, r6864832);
        double r6864837 = r6864833 - r6864836;
        double r6864838 = 3.0;
        double r6864839 = atan2(1.0, 0.0);
        double r6864840 = r6864838 * r6864839;
        double r6864841 = r6864835 + r6864831;
        double r6864842 = r6864840 * r6864841;
        double r6864843 = r6864837 * r6864842;
        double r6864844 = r6864830 / r6864843;
        double r6864845 = 2.0;
        double r6864846 = 6.0;
        double r6864847 = r6864835 * r6864846;
        double r6864848 = r6864845 - r6864847;
        double r6864849 = sqrt(r6864848);
        double r6864850 = r6864831 * r6864831;
        double r6864851 = r6864835 * r6864841;
        double r6864852 = r6864850 + r6864851;
        double r6864853 = r6864849 / r6864852;
        double r6864854 = r6864844 / r6864853;
        double r6864855 = r6864854 * r6864841;
        return r6864855;
}

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 flip--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  9. Applied flip3--0.0

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

  10. Applied associate-*r/0.0

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

  11. Applied associate-/r/0.0

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

  12. Applied associate-/l*0.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))