Average Error: 0.4 → 0.1
Time: 1.6m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}
double f(double v, double t) {
        double r10123842 = 1.0;
        double r10123843 = 5.0;
        double r10123844 = v;
        double r10123845 = r10123844 * r10123844;
        double r10123846 = r10123843 * r10123845;
        double r10123847 = r10123842 - r10123846;
        double r10123848 = atan2(1.0, 0.0);
        double r10123849 = t;
        double r10123850 = r10123848 * r10123849;
        double r10123851 = 2.0;
        double r10123852 = 3.0;
        double r10123853 = r10123852 * r10123845;
        double r10123854 = r10123842 - r10123853;
        double r10123855 = r10123851 * r10123854;
        double r10123856 = sqrt(r10123855);
        double r10123857 = r10123850 * r10123856;
        double r10123858 = r10123842 - r10123845;
        double r10123859 = r10123857 * r10123858;
        double r10123860 = r10123847 / r10123859;
        return r10123860;
}
↓
double f(double v, double t) {
        double r10123861 = 1.0;
        double r10123862 = 5.0;
        double r10123863 = v;
        double r10123864 = r10123862 * r10123863;
        double r10123865 = r10123864 * r10123863;
        double r10123866 = r10123861 - r10123865;
        double r10123867 = r10123863 * r10123863;
        double r10123868 = r10123861 - r10123867;
        double r10123869 = r10123866 / r10123868;
        double r10123870 = atan2(1.0, 0.0);
        double r10123871 = r10123869 / r10123870;
        double r10123872 = 8.0;
        double r10123873 = -216.0;
        double r10123874 = r10123867 * r10123867;
        double r10123875 = r10123874 * r10123867;
        double r10123876 = r10123873 * r10123875;
        double r10123877 = r10123872 + r10123876;
        double r10123878 = sqrt(r10123877);
        double r10123879 = r10123871 / r10123878;
        double r10123880 = t;
        double r10123881 = r10123879 / r10123880;
        double r10123882 = 4.0;
        double r10123883 = 2.0;
        double r10123884 = -6.0;
        double r10123885 = r10123867 * r10123884;
        double r10123886 = r10123883 * r10123885;
        double r10123887 = r10123882 - r10123886;
        double r10123888 = r10123885 * r10123885;
        double r10123889 = r10123887 + r10123888;
        double r10123890 = sqrt(r10123889);
        double r10123891 = r10123881 * r10123890;
        return r10123891;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{-6 \cdot \left(v \cdot v\right) + 2}}}\]
  3. Using strategy rm

  4. Applied flip3-+0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{\color{blue}{\frac{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  5. Applied sqrt-div0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \color{blue}{\frac{\sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}}{\sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  6. Applied associate-*r/0.3

    \[\leadsto \frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{\color{blue}{\frac{t \cdot \sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}}{\sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}}}\]

  7. Applied associate-/r/0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}{\pi}}{t \cdot \sqrt{{\left(-6 \cdot \left(v \cdot v\right)\right)}^{3} + {2}^{3}}} \cdot \sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1 - v \cdot \left(5 \cdot v\right)}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t}} \cdot \sqrt{\left(-6 \cdot \left(v \cdot v\right)\right) \cdot \left(-6 \cdot \left(v \cdot v\right)\right) + \left(2 \cdot 2 - \left(-6 \cdot \left(v \cdot v\right)\right) \cdot 2\right)}\]

  9. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{1 - v \cdot v}}{\pi}}{\sqrt{8 + -216 \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right)\right)}}}{t} \cdot \sqrt{\left(4 - 2 \cdot \left(\left(v \cdot v\right) \cdot -6\right)\right) + \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))