Average Error: 0.4 → 0.3
Time: 21.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{t} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}
double f(double v, double t) {
        double r214902 = 1.0;
        double r214903 = 5.0;
        double r214904 = v;
        double r214905 = r214904 * r214904;
        double r214906 = r214903 * r214905;
        double r214907 = r214902 - r214906;
        double r214908 = atan2(1.0, 0.0);
        double r214909 = t;
        double r214910 = r214908 * r214909;
        double r214911 = 2.0;
        double r214912 = 3.0;
        double r214913 = r214912 * r214905;
        double r214914 = r214902 - r214913;
        double r214915 = r214911 * r214914;
        double r214916 = sqrt(r214915);
        double r214917 = r214910 * r214916;
        double r214918 = r214902 - r214905;
        double r214919 = r214917 * r214918;
        double r214920 = r214907 / r214919;
        return r214920;
}
↓
double f(double v, double t) {
        double r214921 = 1.0;
        double r214922 = 5.0;
        double r214923 = v;
        double r214924 = r214923 * r214923;
        double r214925 = r214922 * r214924;
        double r214926 = r214921 - r214925;
        double r214927 = t;
        double r214928 = r214926 / r214927;
        double r214929 = 1.0;
        double r214930 = atan2(1.0, 0.0);
        double r214931 = r214929 / r214930;
        double r214932 = 2.0;
        double r214933 = 3.0;
        double r214934 = r214933 * r214924;
        double r214935 = r214921 - r214934;
        double r214936 = r214932 * r214935;
        double r214937 = sqrt(r214936);
        double r214938 = r214931 / r214937;
        double r214939 = r214928 * r214938;
        double r214940 = r214921 - r214924;
        double r214941 = r214939 / r214940;
        return r214941;
}

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-/r*0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied associate-/r*0.3

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

  9. Applied div-inv0.3

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

  10. Applied times-frac0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))))