Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 0.5

Time: 3.7m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r9193929 = 2.0;
        double r9193930 = t;
        double r9193931 = 3.0;
        double r9193932 = pow(r9193930, r9193931);
        double r9193933 = l;
        double r9193934 = r9193933 * r9193933;
        double r9193935 = r9193932 / r9193934;
        double r9193936 = k;
        double r9193937 = sin(r9193936);
        double r9193938 = r9193935 * r9193937;
        double r9193939 = tan(r9193936);
        double r9193940 = r9193938 * r9193939;
        double r9193941 = 1.0;
        double r9193942 = r9193936 / r9193930;
        double r9193943 = pow(r9193942, r9193929);
        double r9193944 = r9193941 + r9193943;
        double r9193945 = r9193944 - r9193941;
        double r9193946 = r9193940 * r9193945;
        double r9193947 = r9193929 / r9193946;
        return r9193947;
}

↓

double f(double t, double l, double k) {
        double r9193948 = 2.0;
        double r9193949 = 1.0;
        double r9193950 = t;
        double r9193951 = 1.0;
        double r9193952 = pow(r9193950, r9193951);
        double r9193953 = r9193949 / r9193952;
        double r9193954 = pow(r9193953, r9193951);
        double r9193955 = k;
        double r9193956 = 2.0;
        double r9193957 = r9193948 / r9193956;
        double r9193958 = pow(r9193955, r9193957);
        double r9193959 = r9193949 / r9193958;
        double r9193960 = pow(r9193959, r9193951);
        double r9193961 = sin(r9193955);
        double r9193962 = l;
        double r9193963 = r9193961 / r9193962;
        double r9193964 = r9193960 / r9193963;
        double r9193965 = r9193954 * r9193964;
        double r9193966 = cos(r9193955);
        double r9193967 = r9193963 / r9193966;
        double r9193968 = r9193967 / r9193960;
        double r9193969 = r9193949 / r9193968;
        double r9193970 = r9193965 * r9193969;
        double r9193971 = r9193948 * r9193970;
        return r9193971;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 48.1

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

2. Simplified 41.0

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

3. Taylor expanded around inf 22.5

   \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 22.5

   \[\leadsto 2 \cdot \left({\left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

6. Applied times-frac 22.5

   \[\leadsto 2 \cdot \left({\color{blue}{\left(\frac{\sqrt{1}}{{t}^{1}} \cdot \frac{\sqrt{1}}{{k}^{2}}\right)}}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

7. Applied unpow-prod-down 22.5

   \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot {\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1}\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

8. Applied associate-*l* 23.1

   \[\leadsto 2 \cdot \color{blue}{\left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)}\]

9. Simplified 20.1

   \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \color{blue}{\frac{{\left(\frac{1}{{k}^{2}}\right)}^{1}}{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\cos k}}}\right)\]
10. Using strategy rm

11. Applied *-un-lft-identity 20.1

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1}}{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\color{blue}{1 \cdot \cos k}}}\right)\]

12. Applied times-frac 20.1

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1}}{\color{blue}{\frac{\frac{\sin k}{\ell}}{1} \cdot \frac{\frac{\sin k}{\ell}}{\cos k}}}\right)\]

13. Applied sqr-pow 20.1

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{1} \cdot \frac{\frac{\sin k}{\ell}}{\cos k}}\right)\]

14. Applied add-sqr-sqrt 20.1

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{{k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{1} \cdot \frac{\frac{\sin k}{\ell}}{\cos k}}\right)\]

15. Applied times-frac 20.0

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\color{blue}{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}} \cdot \frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}}^{1}}{\frac{\frac{\sin k}{\ell}}{1} \cdot \frac{\frac{\sin k}{\ell}}{\cos k}}\right)\]

16. Applied unpow-prod-down 20.0

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\color{blue}{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot {\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}{\frac{\frac{\sin k}{\ell}}{1} \cdot \frac{\frac{\sin k}{\ell}}{\cos k}}\right)\]

17. Applied times-frac 7.4

    \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \color{blue}{\left(\frac{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{1}} \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{\cos k}}\right)}\right)\]

18. Applied associate-*r* 0.5

    \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{1}}\right) \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{\cos k}}\right)}\]
19. Using strategy rm

20. Applied clear-num 0.5

    \[\leadsto 2 \cdot \left(\left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell}}{1}}\right) \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{\sin k}{\ell}}{\cos k}}{{\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}}\right)\]

21. Final simplification 0.5

    \[\leadsto 2 \cdot \left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}}\right) \cdot \frac{1}{\frac{\frac{\frac{\sin k}{\ell}}{\cos k}}{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}\right)\]

Reproduce

herbie shell --seed 2019174 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))