Toniolo and Linder, Equation (10+)

Average Error: 31.6 → 7.8

Time: 43.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -7.646207435524129 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}}{\cos k}, \frac{\sin k \cdot k}{\cos k} \cdot \frac{\sin k \cdot k}{\ell}\right)}{\ell}}}{t}\\ \mathbf{elif}\;k \le 454432954895.17645:\\ \;\;\;\;\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \sqrt{2}\right)\right) \cdot \frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \sqrt{2}}{\mathsf{fma}\left(2, \frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, \frac{\left(\sin k \cdot k\right) \cdot \frac{\sin k \cdot k}{\cos k}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}}{\cos k}, \frac{\sin k \cdot k}{\cos k} \cdot \frac{\sin k \cdot k}{\ell}\right)}{\ell}}}{t}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2969794 = 2.0;
        double r2969795 = t;
        double r2969796 = 3.0;
        double r2969797 = pow(r2969795, r2969796);
        double r2969798 = l;
        double r2969799 = r2969798 * r2969798;
        double r2969800 = r2969797 / r2969799;
        double r2969801 = k;
        double r2969802 = sin(r2969801);
        double r2969803 = r2969800 * r2969802;
        double r2969804 = tan(r2969801);
        double r2969805 = r2969803 * r2969804;
        double r2969806 = 1.0;
        double r2969807 = r2969801 / r2969795;
        double r2969808 = pow(r2969807, r2969794);
        double r2969809 = r2969806 + r2969808;
        double r2969810 = r2969809 + r2969806;
        double r2969811 = r2969805 * r2969810;
        double r2969812 = r2969794 / r2969811;
        return r2969812;
}

↓

double f(double t, double l, double k) {
        double r2969813 = k;
        double r2969814 = -7.646207435524129e+170;
        bool r2969815 = r2969813 <= r2969814;
        double r2969816 = 2.0;
        double r2969817 = sin(r2969813);
        double r2969818 = t;
        double r2969819 = r2969817 * r2969818;
        double r2969820 = r2969819 * r2969819;
        double r2969821 = l;
        double r2969822 = r2969820 / r2969821;
        double r2969823 = cos(r2969813);
        double r2969824 = r2969822 / r2969823;
        double r2969825 = r2969817 * r2969813;
        double r2969826 = r2969825 / r2969823;
        double r2969827 = r2969825 / r2969821;
        double r2969828 = r2969826 * r2969827;
        double r2969829 = fma(r2969816, r2969824, r2969828);
        double r2969830 = r2969829 / r2969821;
        double r2969831 = r2969816 / r2969830;
        double r2969832 = r2969831 / r2969818;
        double r2969833 = 454432954895.17645;
        bool r2969834 = r2969813 <= r2969833;
        double r2969835 = cbrt(r2969821);
        double r2969836 = cbrt(r2969818);
        double r2969837 = r2969835 / r2969836;
        double r2969838 = sqrt(r2969816);
        double r2969839 = r2969837 * r2969838;
        double r2969840 = r2969837 * r2969839;
        double r2969841 = r2969819 / r2969821;
        double r2969842 = r2969819 / r2969823;
        double r2969843 = r2969841 * r2969842;
        double r2969844 = r2969825 * r2969826;
        double r2969845 = r2969844 / r2969821;
        double r2969846 = fma(r2969816, r2969843, r2969845);
        double r2969847 = r2969839 / r2969846;
        double r2969848 = r2969840 * r2969847;
        double r2969849 = r2969834 ? r2969848 : r2969832;
        double r2969850 = r2969815 ? r2969832 : r2969849;
        return r2969850;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -7.646207435524129e+170 or 454432954895.17645 < k

   1. Initial program 32.1

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

   2. Simplified 20.5

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

   4. Applied associate-*l/ 20.5

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

   5. Applied associate-*l/ 20.5

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

   6. Applied associate-*l/ 18.5

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

   7. Taylor expanded around inf 17.1

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

   8. Simplified 17.1

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

   10. Applied times-frac 13.4

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

   12. Applied associate-/r/ 11.3

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

   13. Applied associate-/r* 11.3

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

   if -7.646207435524129e+170 < k < 454432954895.17645

   1. Initial program 31.2

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

   2. Simplified 14.3

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

   4. Applied associate-*l/ 12.7

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

   5. Applied associate-*l/ 10.2

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

   6. Applied associate-*l/ 9.2

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

   7. Taylor expanded around inf 24.5

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

   8. Simplified 10.5

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

   10. Applied times-frac 9.1

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

   12. Applied add-cube-cbrt 9.4

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

   13. Applied add-cube-cbrt 9.5

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

   14. Applied times-frac 9.5

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

   15. Applied *-un-lft-identity 9.5

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

   16. Applied times-frac 8.8

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

   17. Applied add-sqr-sqrt 8.8

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

   18. Applied times-frac 8.5

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

   19. Simplified 8.6

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

   20. Simplified 4.8

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -7.646207435524129 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}}{\cos k}, \frac{\sin k \cdot k}{\cos k} \cdot \frac{\sin k \cdot k}{\ell}\right)}{\ell}}}{t}\\ \mathbf{elif}\;k \le 454432954895.17645:\\ \;\;\;\;\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \sqrt{2}\right)\right) \cdot \frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \sqrt{2}}{\mathsf{fma}\left(2, \frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, \frac{\left(\sin k \cdot k\right) \cdot \frac{\sin k \cdot k}{\cos k}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}}{\cos k}, \frac{\sin k \cdot k}{\cos k} \cdot \frac{\sin k \cdot k}{\ell}\right)}{\ell}}}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))