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

Average Error: 0.4 → 0.1

Time: 24.5s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r181122 = 1.0;
        double r181123 = 5.0;
        double r181124 = v;
        double r181125 = r181124 * r181124;
        double r181126 = r181123 * r181125;
        double r181127 = r181122 - r181126;
        double r181128 = atan2(1.0, 0.0);
        double r181129 = t;
        double r181130 = r181128 * r181129;
        double r181131 = 2.0;
        double r181132 = 3.0;
        double r181133 = r181132 * r181125;
        double r181134 = r181122 - r181133;
        double r181135 = r181131 * r181134;
        double r181136 = sqrt(r181135);
        double r181137 = r181130 * r181136;
        double r181138 = r181122 - r181125;
        double r181139 = r181137 * r181138;
        double r181140 = r181127 / r181139;
        return r181140;
}

↓

double f(double v, double t) {
        double r181141 = 1.0;
        double r181142 = 5.0;
        double r181143 = v;
        double r181144 = r181143 * r181143;
        double r181145 = r181142 * r181144;
        double r181146 = r181141 - r181145;
        double r181147 = atan2(1.0, 0.0);
        double r181148 = r181146 / r181147;
        double r181149 = 6.0;
        double r181150 = pow(r181141, r181149);
        double r181151 = 3.0;
        double r181152 = r181151 * r181144;
        double r181153 = pow(r181152, r181149);
        double r181154 = r181150 - r181153;
        double r181155 = 2.0;
        double r181156 = r181154 * r181155;
        double r181157 = sqrt(r181156);
        double r181158 = r181148 / r181157;
        double r181159 = t;
        double r181160 = r181158 / r181159;
        double r181161 = 3.0;
        double r181162 = pow(r181141, r181161);
        double r181163 = pow(r181152, r181161);
        double r181164 = r181162 + r181163;
        double r181165 = sqrt(r181164);
        double r181166 = r181160 * r181165;
        double r181167 = pow(r181143, r181149);
        double r181168 = r181162 - r181167;
        double r181169 = r181166 / r181168;
        double r181170 = r181141 * r181141;
        double r181171 = r181152 * r181152;
        double r181172 = r181141 * r181152;
        double r181173 = r181171 + r181172;
        double r181174 = r181170 + r181173;
        double r181175 = sqrt(r181174);
        double r181176 = r181144 * r181144;
        double r181177 = r181141 * r181144;
        double r181178 = r181176 + r181177;
        double r181179 = r181170 + r181178;
        double r181180 = r181175 * r181179;
        double r181181 = r181169 * r181180;
        return r181181;
}

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 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 flip3-- 0.4

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

6. Applied flip3-- 0.4

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

7. Applied associate-*r/ 0.4

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

8. Applied sqrt-div 0.4

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

9. Applied associate-*r/ 0.4

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

10. Applied associate-*r/ 0.4

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

11. Applied frac-times 0.4

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

12. Applied associate-/r/ 0.4

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

13. Simplified 0.4

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

15. Applied flip-- 0.4

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

16. Applied associate-*r/ 0.4

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

17. Applied sqrt-div 0.4

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

18. Applied associate-*r/ 0.4

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

19. Applied associate-*r/ 0.4

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

20. Applied associate-/r/ 0.4

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

21. Simplified 0.3

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

23. Applied associate-/r* 0.1

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

24. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019306 +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)))))