Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 12.0

Time: 1.1m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\frac{\frac{\left|\sqrt[3]{2}\right|}{{\left(\sqrt[3]{t}\right)}^{3}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{\tan k}\]

C

double f(double t, double l, double k) {
        double r185147 = 2.0;
        double r185148 = t;
        double r185149 = 3.0;
        double r185150 = pow(r185148, r185149);
        double r185151 = l;
        double r185152 = r185151 * r185151;
        double r185153 = r185150 / r185152;
        double r185154 = k;
        double r185155 = sin(r185154);
        double r185156 = r185153 * r185155;
        double r185157 = tan(r185154);
        double r185158 = r185156 * r185157;
        double r185159 = 1.0;
        double r185160 = r185154 / r185148;
        double r185161 = pow(r185160, r185147);
        double r185162 = r185159 + r185161;
        double r185163 = r185162 + r185159;
        double r185164 = r185158 * r185163;
        double r185165 = r185147 / r185164;
        return r185165;
}

↓

double f(double t, double l, double k) {
        double r185166 = 2.0;
        double r185167 = cbrt(r185166);
        double r185168 = fabs(r185167);
        double r185169 = t;
        double r185170 = cbrt(r185169);
        double r185171 = 3.0;
        double r185172 = pow(r185170, r185171);
        double r185173 = r185168 / r185172;
        double r185174 = 2.0;
        double r185175 = 1.0;
        double r185176 = k;
        double r185177 = r185176 / r185169;
        double r185178 = pow(r185177, r185166);
        double r185179 = fma(r185174, r185175, r185178);
        double r185180 = sqrt(r185179);
        double r185181 = sqrt(r185166);
        double r185182 = l;
        double r185183 = r185172 / r185182;
        double r185184 = sin(r185176);
        double r185185 = r185183 * r185184;
        double r185186 = r185181 / r185185;
        double r185187 = cbrt(r185186);
        double r185188 = r185187 * r185187;
        double r185189 = r185180 / r185188;
        double r185190 = r185173 / r185189;
        double r185191 = sqrt(r185167);
        double r185192 = r185191 / r185183;
        double r185193 = r185180 / r185187;
        double r185194 = r185192 / r185193;
        double r185195 = tan(r185176);
        double r185196 = r185194 / r185195;
        double r185197 = r185190 * r185196;
        return r185197;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.3

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

2. Simplified 32.3

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 32.5

   \[\leadsto \frac{\frac{\frac{2}{\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

5. Applied unpow-prod-down 32.5

   \[\leadsto \frac{\frac{\frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

6. Applied times-frac 25.5

   \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

7. Applied associate-*l* 23.7

   \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]
8. Using strategy rm

9. Applied *-un-lft-identity 23.7

   \[\leadsto \frac{\frac{\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\color{blue}{1 \cdot \ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

10. Applied unpow-prod-down 23.7

    \[\leadsto \frac{\frac{\frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{1 \cdot \ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

11. Applied times-frac 18.4

    \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{1} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

12. Simplified 18.4

    \[\leadsto \frac{\frac{\frac{2}{\left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]
13. Using strategy rm

14. Applied add-sqr-sqrt 18.4

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

15. Applied times-frac 18.2

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}} \cdot \frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

16. Applied associate-/l* 16.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{\tan k}\]
17. Using strategy rm

18. Applied *-un-lft-identity 16.4

    \[\leadsto \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}{\color{blue}{1 \cdot \tan k}}\]

19. Applied add-cube-cbrt 16.4

    \[\leadsto \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\left(\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}\right) \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}}{1 \cdot \tan k}\]

20. Applied add-sqr-sqrt 16.4

    \[\leadsto \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}{\left(\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}\right) \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{1 \cdot \tan k}\]

21. Applied times-frac 16.4

    \[\leadsto \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}} \cdot \frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}}{1 \cdot \tan k}\]

22. Applied add-cube-cbrt 16.5

    \[\leadsto \frac{\frac{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}} \cdot \frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{1 \cdot \tan k}\]

23. Applied sqrt-prod 16.5

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \sqrt{\sqrt[3]{2}}}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}} \cdot \frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{1 \cdot \tan k}\]

24. Applied times-frac 16.2

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}} \cdot \frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{1 \cdot \tan k}\]

25. Applied times-frac 13.9

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}}{1 \cdot \tan k}\]

26. Applied times-frac 12.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{1} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{\tan k}}\]

27. Simplified 12.0

    \[\leadsto \color{blue}{\frac{\frac{\left|\sqrt[3]{2}\right|}{{\left(\sqrt[3]{t}\right)}^{3}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{\tan k}\]

28. Final simplification 12.0

    \[\leadsto \frac{\frac{\left|\sqrt[3]{2}\right|}{{\left(\sqrt[3]{t}\right)}^{3}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}} \cdot \sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{2}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}}}{\frac{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\sqrt[3]{\frac{\sqrt{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}}}{\tan k}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))