Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 12.6

Time: 54.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 2.227208593675140378978460642398774691024 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left({\left(\sin k\right)}^{2} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r123083 = 2.0;
        double r123084 = t;
        double r123085 = 3.0;
        double r123086 = pow(r123084, r123085);
        double r123087 = l;
        double r123088 = r123087 * r123087;
        double r123089 = r123086 / r123088;
        double r123090 = k;
        double r123091 = sin(r123090);
        double r123092 = r123089 * r123091;
        double r123093 = tan(r123090);
        double r123094 = r123092 * r123093;
        double r123095 = 1.0;
        double r123096 = r123090 / r123084;
        double r123097 = pow(r123096, r123083);
        double r123098 = r123095 + r123097;
        double r123099 = r123098 + r123095;
        double r123100 = r123094 * r123099;
        double r123101 = r123083 / r123100;
        return r123101;
}

↓

double f(double t, double l, double k) {
        double r123102 = k;
        double r123103 = 2.2272085936751404e-135;
        bool r123104 = r123102 <= r123103;
        double r123105 = 2.0;
        double r123106 = sqrt(r123105);
        double r123107 = t;
        double r123108 = cbrt(r123107);
        double r123109 = 3.0;
        double r123110 = pow(r123108, r123109);
        double r123111 = l;
        double r123112 = r123110 / r123111;
        double r123113 = sin(r123102);
        double r123114 = r123112 * r123113;
        double r123115 = r123110 * r123114;
        double r123116 = r123106 / r123115;
        double r123117 = tan(r123102);
        double r123118 = r123106 / r123117;
        double r123119 = r123111 / r123110;
        double r123120 = 2.0;
        double r123121 = 1.0;
        double r123122 = r123102 / r123107;
        double r123123 = pow(r123122, r123105);
        double r123124 = fma(r123120, r123121, r123123);
        double r123125 = r123119 / r123124;
        double r123126 = r123118 * r123125;
        double r123127 = r123116 * r123126;
        double r123128 = sqrt(r123124);
        double r123129 = r123128 * r123110;
        double r123130 = pow(r123113, r123120);
        double r123131 = r123130 * r123112;
        double r123132 = r123129 * r123131;
        double r123133 = r123105 / r123132;
        double r123134 = cos(r123102);
        double r123135 = r123119 * r123134;
        double r123136 = r123135 / r123128;
        double r123137 = r123133 * r123136;
        double r123138 = r123104 ? r123127 : r123137;
        return r123138;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < 2.2272085936751404e-135

   1. Initial program 34.0

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

   2. Simplified 34.0

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

   4. Applied add-cube-cbrt 34.2

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

   5. Applied unpow-prod-down 34.2

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

   6. Applied times-frac 27.2

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

   7. Applied associate-*l* 24.3

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

   9. Applied unpow-prod-down 24.3

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

   10. Applied associate-/l* 18.3

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

   12. Applied *-un-lft-identity 18.3

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

   13. Applied associate-*l/ 16.9

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

   14. Applied associate-*l/ 15.4

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

   15. Applied associate-/r/ 15.5

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

   16. Applied times-frac 14.2

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

   17. Simplified 14.2

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

   19. Applied add-sqr-sqrt 14.2

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

   20. Applied times-frac 14.3

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

   21. Applied associate-*l* 12.3

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

   if 2.2272085936751404e-135 < k

   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 31.0

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

   4. Applied add-cube-cbrt 31.1

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

   5. Applied unpow-prod-down 31.1

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

   6. Applied times-frac 24.0

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

   7. Applied associate-*l* 23.5

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

   9. Applied unpow-prod-down 23.5

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

   10. Applied associate-/l* 18.0

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

   12. Applied add-sqr-sqrt 18.0

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

   13. Applied tan-quot 18.0

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

   14. Applied associate-*l/ 17.9

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

   15. Applied frac-times 17.2

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

   16. Applied associate-/r/ 17.1

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

   17. Applied times-frac 15.1

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

   18. Simplified 13.1

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.227208593675140378978460642398774691024 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left({\left(\sin k\right)}^{2} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \cos k}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))