Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 7.1

Time: 43.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -3.663093566121438 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \le 9.580968836515877 \cdot 10^{+85}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)} \cdot \ell\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3521822 = 2.0;
        double r3521823 = t;
        double r3521824 = 3.0;
        double r3521825 = pow(r3521823, r3521824);
        double r3521826 = l;
        double r3521827 = r3521826 * r3521826;
        double r3521828 = r3521825 / r3521827;
        double r3521829 = k;
        double r3521830 = sin(r3521829);
        double r3521831 = r3521828 * r3521830;
        double r3521832 = tan(r3521829);
        double r3521833 = r3521831 * r3521832;
        double r3521834 = 1.0;
        double r3521835 = r3521829 / r3521823;
        double r3521836 = pow(r3521835, r3521822);
        double r3521837 = r3521834 + r3521836;
        double r3521838 = r3521837 + r3521834;
        double r3521839 = r3521833 * r3521838;
        double r3521840 = r3521822 / r3521839;
        return r3521840;
}

↓

double f(double t, double l, double k) {
        double r3521841 = k;
        double r3521842 = -3.663093566121438e+79;
        bool r3521843 = r3521841 <= r3521842;
        double r3521844 = 2.0;
        double r3521845 = 1.0;
        double r3521846 = l;
        double r3521847 = r3521845 / r3521846;
        double r3521848 = t;
        double r3521849 = sin(r3521841);
        double r3521850 = r3521848 * r3521849;
        double r3521851 = cos(r3521841);
        double r3521852 = r3521850 / r3521851;
        double r3521853 = r3521850 / r3521846;
        double r3521854 = r3521852 * r3521853;
        double r3521855 = r3521841 * r3521849;
        double r3521856 = r3521855 / r3521851;
        double r3521857 = r3521855 / r3521846;
        double r3521858 = r3521856 * r3521857;
        double r3521859 = fma(r3521844, r3521854, r3521858);
        double r3521860 = r3521848 * r3521859;
        double r3521861 = r3521847 * r3521860;
        double r3521862 = r3521844 / r3521861;
        double r3521863 = 9.580968836515877e+85;
        bool r3521864 = r3521841 <= r3521863;
        double r3521865 = r3521846 / r3521848;
        double r3521866 = r3521859 / r3521865;
        double r3521867 = r3521844 / r3521866;
        double r3521868 = r3521844 / r3521860;
        double r3521869 = r3521868 * r3521846;
        double r3521870 = r3521864 ? r3521867 : r3521869;
        double r3521871 = r3521843 ? r3521862 : r3521870;
        return r3521871;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -3.663093566121438e+79

   1. Initial program 32.7

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

   2. Simplified 21.2

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

   4. Applied associate-*l/ 21.2

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

   5. Applied associate-*l/ 21.2

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

   6. Applied associate-*l/ 19.5

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

   7. Taylor expanded around inf 18.0

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

   8. Simplified 18.0

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

   10. Applied div-inv 18.1

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

   11. Applied *-un-lft-identity 18.1

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

   12. Applied times-frac 17.2

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

   13. Simplified 11.2

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

   if -3.663093566121438e+79 < k < 9.580968836515877e+85

   1. Initial program 31.0

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

   2. Simplified 13.8

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

   4. Applied associate-*l/ 12.3

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

   5. Applied associate-*l/ 10.1

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

   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 9.5

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

   10. Applied *-un-lft-identity 9.5

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

   11. Applied *-un-lft-identity 9.5

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

   12. Applied times-frac 9.5

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

   13. Applied associate-/r* 9.5

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

   14. Simplified 3.5

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

   if 9.580968836515877e+85 < k

   1. Initial program 33.2

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

   2. Simplified 21.0

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

   4. Applied associate-*l/ 21.0

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

   5. Applied associate-*l/ 21.0

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

   6. Applied associate-*l/ 19.2

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

   7. Taylor expanded around inf 18.0

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

   8. Simplified 18.0

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

   10. Applied div-inv 18.0

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

   11. Applied *-un-lft-identity 18.0

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

   12. Applied times-frac 17.3

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

   13. Applied *-un-lft-identity 17.3

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

   14. Applied times-frac 17.3

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

   15. Simplified 17.2

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

   16. Simplified 10.6

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

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -3.663093566121438 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \le 9.580968836515877 \cdot 10^{+85}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(2, \frac{t \cdot \sin k}{\cos k} \cdot \frac{t \cdot \sin k}{\ell}, \frac{k \cdot \sin k}{\cos k} \cdot \frac{k \cdot \sin k}{\ell}\right)} \cdot \ell\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))