Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 12.1

Time: 1.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2362365607704.0986328125:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{1}{\cos k}\right) \cdot \frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}, \frac{{\left(\frac{1}{{-1}^{2}}\right)}^{1}}{\cos k} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3857019 = 2.0;
        double r3857020 = t;
        double r3857021 = 3.0;
        double r3857022 = pow(r3857020, r3857021);
        double r3857023 = l;
        double r3857024 = r3857023 * r3857023;
        double r3857025 = r3857022 / r3857024;
        double r3857026 = k;
        double r3857027 = sin(r3857026);
        double r3857028 = r3857025 * r3857027;
        double r3857029 = tan(r3857026);
        double r3857030 = r3857028 * r3857029;
        double r3857031 = 1.0;
        double r3857032 = r3857026 / r3857020;
        double r3857033 = pow(r3857032, r3857019);
        double r3857034 = r3857031 + r3857033;
        double r3857035 = r3857034 + r3857031;
        double r3857036 = r3857030 * r3857035;
        double r3857037 = r3857019 / r3857036;
        return r3857037;
}

↓

double f(double t, double l, double k) {
        double r3857038 = t;
        double r3857039 = 2362365607704.0986;
        bool r3857040 = r3857038 <= r3857039;
        double r3857041 = 2.0;
        double r3857042 = 1.0;
        double r3857043 = -1.0;
        double r3857044 = pow(r3857043, r3857041);
        double r3857045 = r3857042 / r3857044;
        double r3857046 = 1.0;
        double r3857047 = pow(r3857045, r3857046);
        double r3857048 = k;
        double r3857049 = cos(r3857048);
        double r3857050 = r3857042 / r3857049;
        double r3857051 = r3857047 * r3857050;
        double r3857052 = sin(r3857048);
        double r3857053 = r3857052 * r3857038;
        double r3857054 = r3857053 * r3857053;
        double r3857055 = l;
        double r3857056 = r3857054 / r3857055;
        double r3857057 = r3857051 * r3857056;
        double r3857058 = r3857047 / r3857049;
        double r3857059 = r3857052 * r3857048;
        double r3857060 = r3857059 * r3857059;
        double r3857061 = r3857060 / r3857055;
        double r3857062 = r3857058 * r3857061;
        double r3857063 = fma(r3857041, r3857057, r3857062);
        double r3857064 = cbrt(r3857038);
        double r3857065 = 3.0;
        double r3857066 = pow(r3857064, r3857065);
        double r3857067 = r3857055 / r3857066;
        double r3857068 = r3857063 / r3857067;
        double r3857069 = r3857041 / r3857068;
        double r3857070 = r3857066 / r3857055;
        double r3857071 = r3857070 * r3857052;
        double r3857072 = r3857071 * r3857066;
        double r3857073 = cbrt(r3857055);
        double r3857074 = cbrt(r3857066);
        double r3857075 = r3857073 / r3857074;
        double r3857076 = r3857072 / r3857075;
        double r3857077 = r3857048 / r3857038;
        double r3857078 = pow(r3857077, r3857041);
        double r3857079 = r3857046 + r3857078;
        double r3857080 = r3857079 + r3857046;
        double r3857081 = tan(r3857048);
        double r3857082 = r3857080 * r3857081;
        double r3857083 = r3857082 / r3857075;
        double r3857084 = r3857076 * r3857083;
        double r3857085 = r3857084 / r3857075;
        double r3857086 = r3857041 / r3857085;
        double r3857087 = r3857040 ? r3857069 : r3857086;
        return r3857087;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 2362365607704.0986

   1. Initial program 36.6

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

   3. Applied add-cube-cbrt 36.7

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

   4. Applied unpow-prod-down 36.7

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

   5. Applied times-frac 29.6

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

   6. Applied associate-*l* 28.0

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

   8. Applied unpow-prod-down 28.0

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

   9. Applied associate-/l* 22.5

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

   11. Applied associate-*l/ 21.9

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

   12. Applied associate-*l/ 21.5

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

   13. Applied associate-*l/ 19.5

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

   14. Taylor expanded around -inf 21.3

       \[\leadsto \frac{2}{\frac{\color{blue}{{\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{k}^{2} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} + 2 \cdot \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left({t}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right)}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\]

   15. Simplified 15.7

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

   if 2362365607704.0986 < t

   1. Initial program 22.5

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

   3. Applied add-cube-cbrt 22.6

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

   4. Applied unpow-prod-down 22.6

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

   5. Applied times-frac 16.3

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

   6. Applied associate-*l* 14.4

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

   8. Applied unpow-prod-down 14.4

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

   9. Applied associate-/l* 8.2

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

   11. Applied associate-*l/ 6.8

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

   12. Applied associate-*l/ 3.8

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

   13. Applied associate-*l/ 3.5

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

   15. Applied add-cube-cbrt 3.5

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

   16. Applied add-cube-cbrt 3.5

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

   17. Applied times-frac 3.5

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

   18. Applied associate-/r* 3.5

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

   19. Simplified 3.5

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2362365607704.0986328125:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \left({\left(\frac{1}{{-1}^{2}}\right)}^{1} \cdot \frac{1}{\cos k}\right) \cdot \frac{\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)}{\ell}, \frac{{\left(\frac{1}{{-1}^{2}}\right)}^{1}}{\cos k} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))