Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 6.4

Time: 45.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.1925115852584323 \cdot 10^{-209}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;t \le 2.7233600055589613 \cdot 10^{-164}:\\ \;\;\;\;\frac{2 \cdot \ell}{\frac{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}{\frac{1}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2803688 = 2.0;
        double r2803689 = t;
        double r2803690 = 3.0;
        double r2803691 = pow(r2803689, r2803690);
        double r2803692 = l;
        double r2803693 = r2803692 * r2803692;
        double r2803694 = r2803691 / r2803693;
        double r2803695 = k;
        double r2803696 = sin(r2803695);
        double r2803697 = r2803694 * r2803696;
        double r2803698 = tan(r2803695);
        double r2803699 = r2803697 * r2803698;
        double r2803700 = 1.0;
        double r2803701 = r2803695 / r2803689;
        double r2803702 = pow(r2803701, r2803688);
        double r2803703 = r2803700 + r2803702;
        double r2803704 = r2803703 + r2803700;
        double r2803705 = r2803699 * r2803704;
        double r2803706 = r2803688 / r2803705;
        return r2803706;
}

↓

double f(double t, double l, double k) {
        double r2803707 = t;
        double r2803708 = -2.1925115852584323e-209;
        bool r2803709 = r2803707 <= r2803708;
        double r2803710 = l;
        double r2803711 = r2803710 / r2803707;
        double r2803712 = 2.0;
        double r2803713 = k;
        double r2803714 = sin(r2803713);
        double r2803715 = r2803714 * r2803707;
        double r2803716 = r2803715 / r2803710;
        double r2803717 = cos(r2803713);
        double r2803718 = r2803715 / r2803717;
        double r2803719 = r2803716 * r2803718;
        double r2803720 = r2803713 * r2803714;
        double r2803721 = r2803720 / r2803710;
        double r2803722 = r2803720 / r2803717;
        double r2803723 = r2803721 * r2803722;
        double r2803724 = fma(r2803719, r2803712, r2803723);
        double r2803725 = r2803712 / r2803724;
        double r2803726 = r2803711 * r2803725;
        double r2803727 = 2.7233600055589613e-164;
        bool r2803728 = r2803707 <= r2803727;
        double r2803729 = r2803712 * r2803710;
        double r2803730 = 1.0;
        double r2803731 = r2803730 / r2803707;
        double r2803732 = r2803724 / r2803731;
        double r2803733 = r2803729 / r2803732;
        double r2803734 = r2803711 / r2803724;
        double r2803735 = r2803730 / r2803734;
        double r2803736 = r2803712 / r2803735;
        double r2803737 = r2803728 ? r2803733 : r2803736;
        double r2803738 = r2803709 ? r2803726 : r2803737;
        return r2803738;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.1925115852584323e-209

   1. Initial program 29.1

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

   2. Simplified 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/ 12.8

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

   5. Applied associate-*l/ 11.6

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

   6. Applied associate-*l/ 10.4

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

   7. Taylor expanded around inf 20.6

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

   8. Simplified 12.7

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

   10. Applied *-un-lft-identity 12.7

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

   11. Applied *-un-lft-identity 12.7

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

   12. Applied times-frac 12.7

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

   13. Applied associate-/r* 12.7

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

   14. Simplified 5.7

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

   16. Applied associate-/r/ 5.6

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

   if -2.1925115852584323e-209 < t < 2.7233600055589613e-164

   1. Initial program 62.6

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

   2. Simplified 50.8

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

   4. Applied associate-*l/ 50.8

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

   5. Applied associate-*l/ 51.1

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

   6. Applied associate-*l/ 48.9

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

   7. Taylor expanded around inf 24.1

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

   8. Simplified 24.1

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

   10. Applied *-un-lft-identity 24.1

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

   11. Applied *-un-lft-identity 24.1

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

   12. Applied times-frac 24.1

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

   13. Applied associate-/r* 24.1

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

   14. Simplified 21.4

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

   16. Applied div-inv 21.5

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

   17. Applied *-un-lft-identity 21.5

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

   18. Applied times-frac 16.1

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

   19. Applied associate-/r* 16.1

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

   20. Simplified 16.1

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

   if 2.7233600055589613e-164 < t

   1. Initial program 26.0

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

   2. Simplified 11.8

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

   4. Applied associate-*l/ 10.6

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

   5. Applied associate-*l/ 9.0

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

   6. Applied associate-*l/ 8.0

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

   7. Taylor expanded around inf 20.6

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

   8. Simplified 12.2

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

   10. Applied *-un-lft-identity 12.2

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

   11. Applied *-un-lft-identity 12.2

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

   12. Applied times-frac 12.2

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

   13. Applied associate-/r* 12.2

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

   14. Simplified 4.4

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

   16. Applied clear-num 4.4

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.1925115852584323 \cdot 10^{-209}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;t \le 2.7233600055589613 \cdot 10^{-164}:\\ \;\;\;\;\frac{2 \cdot \ell}{\frac{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}{\frac{1}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\mathsf{fma}\left(\frac{\sin k \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\cos k}, 2, \frac{k \cdot \sin k}{\ell} \cdot \frac{k \cdot \sin k}{\cos k}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))))