Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 4.7

Time: 38.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.2768326759006626 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{\mathsf{fma}\left(2, \frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), \left(\frac{k}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{k}{\ell}\right)}{\frac{2}{\sin k}}}\\ \mathbf{elif}\;k \le 6.996202509808683 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{t}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sin k}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t \cdot \mathsf{fma}\left(2, \frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), \left(\frac{k}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{k}{\ell}\right)\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2343963 = 2.0;
        double r2343964 = t;
        double r2343965 = 3.0;
        double r2343966 = pow(r2343964, r2343965);
        double r2343967 = l;
        double r2343968 = r2343967 * r2343967;
        double r2343969 = r2343966 / r2343968;
        double r2343970 = k;
        double r2343971 = sin(r2343970);
        double r2343972 = r2343969 * r2343971;
        double r2343973 = tan(r2343970);
        double r2343974 = r2343972 * r2343973;
        double r2343975 = 1.0;
        double r2343976 = r2343970 / r2343964;
        double r2343977 = pow(r2343976, r2343963);
        double r2343978 = r2343975 + r2343977;
        double r2343979 = r2343978 + r2343975;
        double r2343980 = r2343974 * r2343979;
        double r2343981 = r2343963 / r2343980;
        return r2343981;
}

↓

double f(double t, double l, double k) {
        double r2343982 = k;
        double r2343983 = -1.2768326759006626e-98;
        bool r2343984 = r2343982 <= r2343983;
        double r2343985 = 1.0;
        double r2343986 = t;
        double r2343987 = r2343985 / r2343986;
        double r2343988 = 2.0;
        double r2343989 = sin(r2343982);
        double r2343990 = cos(r2343982);
        double r2343991 = r2343989 / r2343990;
        double r2343992 = l;
        double r2343993 = r2343986 / r2343992;
        double r2343994 = r2343993 * r2343993;
        double r2343995 = r2343991 * r2343994;
        double r2343996 = r2343982 / r2343992;
        double r2343997 = r2343996 * r2343991;
        double r2343998 = r2343997 * r2343996;
        double r2343999 = fma(r2343988, r2343995, r2343998);
        double r2344000 = r2343988 / r2343989;
        double r2344001 = r2343999 / r2344000;
        double r2344002 = r2343987 / r2344001;
        double r2344003 = 6.996202509808683e-64;
        bool r2344004 = r2343982 <= r2344003;
        double r2344005 = sqrt(r2343988);
        double r2344006 = r2344005 / r2343986;
        double r2344007 = tan(r2343982);
        double r2344008 = r2343992 / r2343986;
        double r2344009 = r2344007 / r2344008;
        double r2344010 = r2344006 / r2344009;
        double r2344011 = r2344005 / r2343989;
        double r2344012 = r2343982 / r2343986;
        double r2344013 = fma(r2344012, r2344012, r2343988);
        double r2344014 = r2344013 / r2344008;
        double r2344015 = r2344011 / r2344014;
        double r2344016 = r2344010 * r2344015;
        double r2344017 = r2343986 * r2343999;
        double r2344018 = r2343989 * r2344017;
        double r2344019 = r2343988 / r2344018;
        double r2344020 = r2344004 ? r2344016 : r2344019;
        double r2344021 = r2343984 ? r2344002 : r2344020;
        return r2344021;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.2768326759006626e-98

   1. Initial program 29.6

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

   2. Simplified 18.1

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

   4. Applied associate-*l/ 17.8

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

   5. Applied associate-/r/ 17.8

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

   6. Applied associate-/l* 16.2

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

   7. Taylor expanded around inf 20.4

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

   8. Simplified 5.4

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

   10. Applied associate-*r* 5.2

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

   12. Applied *-un-lft-identity 5.2

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

   13. Applied times-frac 5.2

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

   14. Applied associate-/l* 4.9

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

   if -1.2768326759006626e-98 < k < 6.996202509808683e-64

   1. Initial program 35.0

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

   2. Simplified 25.2

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

   4. Applied associate-*l/ 18.2

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

   5. Applied associate-/r/ 18.2

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

   6. Applied associate-/l* 17.4

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

   8. Applied times-frac 6.5

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

   9. Applied add-sqr-sqrt 6.8

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

   10. Applied times-frac 6.7

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

   11. Applied times-frac 3.1

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

   if 6.996202509808683e-64 < k

   1. Initial program 30.8

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

   2. Simplified 19.1

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

   4. Applied associate-*l/ 19.0

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

   5. Applied associate-/r/ 18.8

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

   6. Applied associate-/l* 17.1

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

   7. Taylor expanded around inf 21.8

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

   8. Simplified 5.2

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

   10. Applied associate-*r* 5.2

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

   12. Applied div-inv 5.2

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

   13. Applied associate-/l* 5.4

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

   14. Simplified 5.4

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.2768326759006626 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{t}}{\frac{\mathsf{fma}\left(2, \frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), \left(\frac{k}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{k}{\ell}\right)}{\frac{2}{\sin k}}}\\ \mathbf{elif}\;k \le 6.996202509808683 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{t}}{\frac{\tan k}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sin k}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t \cdot \mathsf{fma}\left(2, \frac{\sin k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right), \left(\frac{k}{\ell} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{k}{\ell}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))))