Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 6.9

Time: 38.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.15499481503728959 \cdot 10^{207} \lor \neg \left(t \le 1.9403210429628964 \cdot 10^{90}\right):\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r138827 = 2.0;
        double r138828 = t;
        double r138829 = 3.0;
        double r138830 = pow(r138828, r138829);
        double r138831 = l;
        double r138832 = r138831 * r138831;
        double r138833 = r138830 / r138832;
        double r138834 = k;
        double r138835 = sin(r138834);
        double r138836 = r138833 * r138835;
        double r138837 = tan(r138834);
        double r138838 = r138836 * r138837;
        double r138839 = 1.0;
        double r138840 = r138834 / r138828;
        double r138841 = pow(r138840, r138827);
        double r138842 = r138839 + r138841;
        double r138843 = r138842 - r138839;
        double r138844 = r138838 * r138843;
        double r138845 = r138827 / r138844;
        return r138845;
}

↓

double f(double t, double l, double k) {
        double r138846 = t;
        double r138847 = -2.1549948150372896e+207;
        bool r138848 = r138846 <= r138847;
        double r138849 = 1.9403210429628964e+90;
        bool r138850 = r138846 <= r138849;
        double r138851 = !r138850;
        bool r138852 = r138848 || r138851;
        double r138853 = 2.0;
        double r138854 = 1.0;
        double r138855 = cbrt(r138854);
        double r138856 = r138855 * r138855;
        double r138857 = k;
        double r138858 = 2.0;
        double r138859 = r138853 / r138858;
        double r138860 = pow(r138857, r138859);
        double r138861 = r138856 / r138860;
        double r138862 = 1.0;
        double r138863 = pow(r138861, r138862);
        double r138864 = pow(r138846, r138862);
        double r138865 = r138855 / r138864;
        double r138866 = pow(r138865, r138862);
        double r138867 = cos(r138857);
        double r138868 = l;
        double r138869 = r138867 * r138868;
        double r138870 = sin(r138857);
        double r138871 = pow(r138870, r138858);
        double r138872 = r138869 / r138871;
        double r138873 = r138866 * r138872;
        double r138874 = r138863 * r138873;
        double r138875 = r138863 * r138874;
        double r138876 = r138853 * r138875;
        double r138877 = r138876 * r138868;
        double r138878 = r138860 * r138864;
        double r138879 = r138854 / r138878;
        double r138880 = pow(r138879, r138862);
        double r138881 = r138880 * r138869;
        double r138882 = r138863 * r138881;
        double r138883 = r138853 * r138882;
        double r138884 = r138883 * r138868;
        double r138885 = r138884 / r138871;
        double r138886 = r138852 ? r138877 : r138885;
        return r138886;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -2.1549948150372896e+207 or 1.9403210429628964e+90 < t

   1. Initial program 53.6

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

   2. Simplified 37.2

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

   3. Taylor expanded around inf 14.3

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

   5. Applied sqr-pow 14.3

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

   6. Applied associate-*l* 14.3

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

   8. Applied add-cube-cbrt 14.3

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

   9. Applied times-frac 13.9

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

   10. Applied unpow-prod-down 13.9

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

   11. Applied associate-*l* 11.7

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

   12. Simplified 11.7

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

   14. Applied add-cube-cbrt 11.7

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

   15. Applied times-frac 11.2

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

   16. Applied unpow-prod-down 11.2

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

   17. Applied associate-*l* 8.8

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

   if -2.1549948150372896e+207 < t < 1.9403210429628964e+90

   1. Initial program 45.7

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

   2. Simplified 39.2

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

   3. Taylor expanded around inf 17.3

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

   5. Applied sqr-pow 17.3

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

   6. Applied associate-*l* 12.6

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

   8. Applied add-cube-cbrt 12.6

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

   9. Applied times-frac 12.1

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

   10. Applied unpow-prod-down 12.1

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

   11. Applied associate-*l* 7.0

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

   12. Simplified 7.0

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

   14. Applied associate-*r/ 7.0

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

   15. Applied associate-*r/ 7.0

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

   16. Applied associate-*r/ 7.0

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

   17. Applied associate-*l/ 6.1

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.15499481503728959 \cdot 10^{207} \lor \neg \left(t \le 1.9403210429628964 \cdot 10^{90}\right):\\ \;\;\;\;\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot \ell}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)\right)\right) \cdot \ell}{{\left(\sin k\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))