Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 3.5

Time: 4.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 1.658246358429121552659470209720305855916 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \left(\left(\frac{\sqrt[3]{\ell}}{k} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}\right) \cdot \left(\frac{\sqrt[3]{\ell} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r108993 = 2.0;
        double r108994 = t;
        double r108995 = 3.0;
        double r108996 = pow(r108994, r108995);
        double r108997 = l;
        double r108998 = r108997 * r108997;
        double r108999 = r108996 / r108998;
        double r109000 = k;
        double r109001 = sin(r109000);
        double r109002 = r108999 * r109001;
        double r109003 = tan(r109000);
        double r109004 = r109002 * r109003;
        double r109005 = 1.0;
        double r109006 = r109000 / r108994;
        double r109007 = pow(r109006, r108993);
        double r109008 = r109005 + r109007;
        double r109009 = r109008 - r109005;
        double r109010 = r109004 * r109009;
        double r109011 = r108993 / r109010;
        return r109011;
}

↓

double f(double t, double l, double k) {
        double r109012 = k;
        double r109013 = 1.6582463584291216e+138;
        bool r109014 = r109012 <= r109013;
        double r109015 = l;
        double r109016 = sin(r109012);
        double r109017 = r109015 / r109016;
        double r109018 = tan(r109012);
        double r109019 = r109017 / r109018;
        double r109020 = 2.0;
        double r109021 = cbrt(r109015);
        double r109022 = r109021 / r109012;
        double r109023 = 1.0;
        double r109024 = t;
        double r109025 = 1.0;
        double r109026 = pow(r109024, r109025);
        double r109027 = r109023 / r109026;
        double r109028 = pow(r109027, r109025);
        double r109029 = r109022 * r109028;
        double r109030 = r109021 * r109021;
        double r109031 = r109030 / r109012;
        double r109032 = r109029 * r109031;
        double r109033 = r109020 * r109032;
        double r109034 = r109019 * r109033;
        double r109035 = r109020 * r109031;
        double r109036 = r109021 * r109028;
        double r109037 = r109036 / r109012;
        double r109038 = r109037 * r109019;
        double r109039 = r109035 * r109038;
        double r109040 = r109014 ? r109034 : r109039;
        return r109040;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < 1.6582463584291216e+138

   1. Initial program 50.6

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

   2. Simplified 38.7

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} - 0}}{\frac{{t}^{3}}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}}\]

   3. Taylor expanded around 0 9.7

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

   5. Applied sqr-pow 9.7

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

   6. Applied add-cube-cbrt 10.1

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

   7. Applied times-frac 7.6

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

   8. Applied associate-*l* 5.9

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

   9. Simplified 5.9

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

   11. Applied associate-/r* 3.2

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

   if 1.6582463584291216e+138 < k

   1. Initial program 39.5

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

   2. Simplified 32.1

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} - 0}}{\frac{{t}^{3}}{\ell}} \cdot \frac{\ell}{\sin k \cdot \tan k}}\]

   3. Taylor expanded around 0 20.5

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

   5. Applied sqr-pow 20.5

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

   6. Applied add-cube-cbrt 20.6

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

   7. Applied times-frac 11.9

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

   8. Applied associate-*l* 6.4

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

   9. Simplified 6.4

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

   11. Applied associate-/r* 6.4

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

   13. Applied pow1 6.4

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

   14. Applied pow1 6.4

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

   15. Applied pow1 6.4

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

   16. Applied pow-prod-down 6.4

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

   17. Applied pow1 6.4

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

   18. Applied pow-prod-down 6.4

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

   19. Applied pow-prod-down 6.4

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

   20. Simplified 4.5

       \[\leadsto {\color{blue}{\left(\left(2 \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}\right) \cdot \left(\frac{\sqrt[3]{\ell} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\right)\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.658246358429121552659470209720305855916 \cdot 10^{138}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \left(\left(\frac{\sqrt[3]{\ell}}{k} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{k}\right) \cdot \left(\frac{\sqrt[3]{\ell} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))