Average Error: 0.4 → 0.1
Time: 32.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{4 + \left(2 \cdot \left(6 \cdot \left(v \cdot v\right)\right) + \left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{8 - \left(216 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}}{\pi}}{t}\]
\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{4 + \left(2 \cdot \left(6 \cdot \left(v \cdot v\right)\right) + \left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{8 - \left(216 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}}{\pi}}{t}
double f(double v, double t) {
        double r6626819 = 1.0;
        double r6626820 = 5.0;
        double r6626821 = v;
        double r6626822 = r6626821 * r6626821;
        double r6626823 = r6626820 * r6626822;
        double r6626824 = r6626819 - r6626823;
        double r6626825 = atan2(1.0, 0.0);
        double r6626826 = t;
        double r6626827 = r6626825 * r6626826;
        double r6626828 = 2.0;
        double r6626829 = 3.0;
        double r6626830 = r6626829 * r6626822;
        double r6626831 = r6626819 - r6626830;
        double r6626832 = r6626828 * r6626831;
        double r6626833 = sqrt(r6626832);
        double r6626834 = r6626827 * r6626833;
        double r6626835 = r6626819 - r6626822;
        double r6626836 = r6626834 * r6626835;
        double r6626837 = r6626824 / r6626836;
        return r6626837;
}
↓
double f(double v, double t) {
        double r6626838 = 4.0;
        double r6626839 = 2.0;
        double r6626840 = 6.0;
        double r6626841 = v;
        double r6626842 = r6626841 * r6626841;
        double r6626843 = r6626840 * r6626842;
        double r6626844 = r6626839 * r6626843;
        double r6626845 = r6626843 * r6626843;
        double r6626846 = r6626844 + r6626845;
        double r6626847 = r6626838 + r6626846;
        double r6626848 = sqrt(r6626847);
        double r6626849 = -5.0;
        double r6626850 = 1.0;
        double r6626851 = fma(r6626842, r6626849, r6626850);
        double r6626852 = 8.0;
        double r6626853 = 216.0;
        double r6626854 = r6626842 * r6626842;
        double r6626855 = r6626853 * r6626854;
        double r6626856 = r6626855 * r6626842;
        double r6626857 = r6626852 - r6626856;
        double r6626858 = sqrt(r6626857);
        double r6626859 = r6626850 - r6626842;
        double r6626860 = r6626858 * r6626859;
        double r6626861 = r6626851 / r6626860;
        double r6626862 = atan2(1.0, 0.0);
        double r6626863 = r6626861 / r6626862;
        double r6626864 = t;
        double r6626865 = r6626863 / r6626864;
        double r6626866 = r6626848 * r6626865;
        return r6626866;
}

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. Simplified0.3

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

  4. Applied associate-/r*0.3

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

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

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{\color{blue}{1 \cdot t}}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}\]

  7. Applied div-inv0.3

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right) \cdot \frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}}{1 \cdot t}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}\]

  8. Applied times-frac0.3

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{1} \cdot \frac{\frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}{t}}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}\]

  9. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{1}}{\frac{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}{\frac{\frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}{t}}}}\]

  10. Simplified0.4

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

  12. Applied flip3--0.4

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

  13. Applied sqrt-div0.4

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

  14. Applied associate-*l/0.4

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

  15. Applied associate-*l/0.4

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

  16. Applied associate-/r/0.4

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

  17. Simplified0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2019141 +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)))))