Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.1

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r177176 = 1.0;
        double r177177 = 5.0;
        double r177178 = v;
        double r177179 = r177178 * r177178;
        double r177180 = r177177 * r177179;
        double r177181 = r177176 - r177180;
        double r177182 = atan2(1.0, 0.0);
        double r177183 = t;
        double r177184 = r177182 * r177183;
        double r177185 = 2.0;
        double r177186 = 3.0;
        double r177187 = r177186 * r177179;
        double r177188 = r177176 - r177187;
        double r177189 = r177185 * r177188;
        double r177190 = sqrt(r177189);
        double r177191 = r177184 * r177190;
        double r177192 = r177176 - r177179;
        double r177193 = r177191 * r177192;
        double r177194 = r177181 / r177193;
        return r177194;
}

↓

double f(double v, double t) {
        double r177195 = 1.0;
        double r177196 = atan2(1.0, 0.0);
        double r177197 = r177195 / r177196;
        double r177198 = 2.0;
        double r177199 = 3.0;
        double r177200 = pow(r177195, r177199);
        double r177201 = 3.0;
        double r177202 = v;
        double r177203 = r177202 * r177202;
        double r177204 = r177201 * r177203;
        double r177205 = pow(r177204, r177199);
        double r177206 = r177200 - r177205;
        double r177207 = r177198 * r177206;
        double r177208 = sqrt(r177207);
        double r177209 = r177197 / r177208;
        double r177210 = t;
        double r177211 = r177209 / r177210;
        double r177212 = 6.0;
        double r177213 = pow(r177195, r177212);
        double r177214 = 2.0;
        double r177215 = pow(r177202, r177214);
        double r177216 = pow(r177215, r177212);
        double r177217 = r177213 - r177216;
        double r177218 = r177211 / r177217;
        double r177219 = 5.0;
        double r177220 = r177219 * r177202;
        double r177221 = r177217 * r177210;
        double r177222 = r177221 * r177208;
        double r177223 = r177220 / r177222;
        double r177224 = r177202 / r177196;
        double r177225 = r177223 * r177224;
        double r177226 = r177218 - r177225;
        double r177227 = r177195 * r177195;
        double r177228 = r177204 * r177204;
        double r177229 = r177195 * r177204;
        double r177230 = r177228 + r177229;
        double r177231 = r177227 + r177230;
        double r177232 = sqrt(r177231);
        double r177233 = pow(r177203, r177199);
        double r177234 = r177200 + r177233;
        double r177235 = r177232 * r177234;
        double r177236 = r177226 * r177235;
        double r177237 = r177203 * r177203;
        double r177238 = r177195 * r177203;
        double r177239 = r177237 + r177238;
        double r177240 = r177227 + r177239;
        double r177241 = r177236 * r177240;
        return r177241;
}

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

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

4. Applied associate-*r/ 0.4

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

5. Applied associate-/r/ 0.4

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

7. Applied associate-*l* 0.5

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

9. Applied flip-- 0.5

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

10. Applied flip3-- 0.5

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

11. Applied associate-*r/ 0.5

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

12. Applied sqrt-div 0.5

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

13. Applied associate-*r/ 0.5

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

14. Applied associate-*r/ 0.5

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

15. Applied frac-times 0.5

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

16. Applied associate-/r/ 0.5

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

17. Simplified 0.3

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

19. Applied div-sub 0.3

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

20. Applied div-sub 0.3

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

21. Simplified 0.1

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

22. Simplified 0.1

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

23. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020027 
(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)))))