Average Error: 0.4 → 0.1
Time: 6.4m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(v, \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right), \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{2}}}{t - v \cdot t}\right)\]
\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)}
↓
\sqrt{1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\frac{1}{\mathsf{fma}\left(v, \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right), \left(\sqrt{1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\pi}}{\sqrt{2}}}{t - v \cdot t}\right)
double f(double v, double t) {
        double r168032003 = 1.0;
        double r168032004 = 5.0;
        double r168032005 = v;
        double r168032006 = r168032005 * r168032005;
        double r168032007 = r168032004 * r168032006;
        double r168032008 = r168032003 - r168032007;
        double r168032009 = atan2(1.0, 0.0);
        double r168032010 = t;
        double r168032011 = r168032009 * r168032010;
        double r168032012 = 2.0;
        double r168032013 = 3.0;
        double r168032014 = r168032013 * r168032006;
        double r168032015 = r168032003 - r168032014;
        double r168032016 = r168032012 * r168032015;
        double r168032017 = sqrt(r168032016);
        double r168032018 = r168032011 * r168032017;
        double r168032019 = r168032003 - r168032006;
        double r168032020 = r168032018 * r168032019;
        double r168032021 = r168032008 / r168032020;
        return r168032021;
}
↓
double f(double v, double t) {
        double r168032022 = 1.0;
        double r168032023 = 3.0;
        double r168032024 = v;
        double r168032025 = r168032024 * r168032024;
        double r168032026 = r168032023 * r168032025;
        double r168032027 = r168032026 * r168032026;
        double r168032028 = r168032027 + r168032026;
        double r168032029 = r168032022 + r168032028;
        double r168032030 = sqrt(r168032029);
        double r168032031 = r168032026 * r168032027;
        double r168032032 = r168032022 - r168032031;
        double r168032033 = sqrt(r168032032);
        double r168032034 = fma(r168032024, r168032033, r168032033);
        double r168032035 = r168032022 / r168032034;
        double r168032036 = -5.0;
        double r168032037 = fma(r168032036, r168032025, r168032022);
        double r168032038 = atan2(1.0, 0.0);
        double r168032039 = r168032037 / r168032038;
        double r168032040 = 2.0;
        double r168032041 = sqrt(r168032040);
        double r168032042 = r168032039 / r168032041;
        double r168032043 = t;
        double r168032044 = r168032024 * r168032043;
        double r168032045 = r168032043 - r168032044;
        double r168032046 = r168032042 / r168032045;
        double r168032047 = r168032035 * r168032046;
        double r168032048 = r168032030 * r168032047;
        return r168032048;
}

Error

Bits error versus v

Bits error versus t

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 flip3--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied sqrt-div0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied associate-*l/0.4

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

  8. Applied associate-/r/0.4

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

  9. Simplified0.3

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

  11. Applied sqrt-prod0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied difference-of-squares0.3

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

  14. Applied *-un-lft-identity0.3

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

  15. Applied times-frac0.3

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

  16. Applied times-frac0.3

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

  17. Simplified0.3

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(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)))))