Average Error: 0.4 → 0.3
Time: 7.0s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r216883 = 1.0;
        double r216884 = 5.0;
        double r216885 = v;
        double r216886 = r216885 * r216885;
        double r216887 = r216884 * r216886;
        double r216888 = r216883 - r216887;
        double r216889 = atan2(1.0, 0.0);
        double r216890 = t;
        double r216891 = r216889 * r216890;
        double r216892 = 2.0;
        double r216893 = 3.0;
        double r216894 = r216893 * r216886;
        double r216895 = r216883 - r216894;
        double r216896 = r216892 * r216895;
        double r216897 = sqrt(r216896);
        double r216898 = r216891 * r216897;
        double r216899 = r216883 - r216886;
        double r216900 = r216898 * r216899;
        double r216901 = r216888 / r216900;
        return r216901;
}
↓
double f(double v, double t) {
        double r216902 = 1.0;
        double r216903 = 5.0;
        double r216904 = v;
        double r216905 = r216904 * r216904;
        double r216906 = r216903 * r216905;
        double r216907 = r216902 - r216906;
        double r216908 = sqrt(r216907);
        double r216909 = atan2(1.0, 0.0);
        double r216910 = r216908 / r216909;
        double r216911 = t;
        double r216912 = 2.0;
        double r216913 = 3.0;
        double r216914 = r216913 * r216905;
        double r216915 = r216902 - r216914;
        double r216916 = r216912 * r216915;
        double r216917 = sqrt(r216916);
        double r216918 = r216911 * r216917;
        double r216919 = r216910 / r216918;
        double r216920 = r216902 - r216905;
        double r216921 = r216908 / r216920;
        double r216922 = r216919 * r216921;
        return r216922;
}

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. Using strategy rm

  3. Applied associate-*l*0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied times-frac0.4

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

  8. Applied associate-/r*0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))