Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 20.8

Time: 27.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.31826692498424758 \cdot 10^{38} \lor \neg \left(t \le 1.25152897564453319 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{1} \cdot \left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r210 = 2.0;
        double r211 = t;
        double r212 = 3.0;
        double r213 = pow(r211, r212);
        double r214 = l;
        double r215 = r214 * r214;
        double r216 = r213 / r215;
        double r217 = k;
        double r218 = sin(r217);
        double r219 = r216 * r218;
        double r220 = tan(r217);
        double r221 = r219 * r220;
        double r222 = 1.0;
        double r223 = r217 / r211;
        double r224 = pow(r223, r210);
        double r225 = r222 + r224;
        double r226 = r225 + r222;
        double r227 = r221 * r226;
        double r228 = r210 / r227;
        return r228;
}

↓

double f(double t, double l, double k) {
        double r229 = t;
        double r230 = -1.3182669249842476e+38;
        bool r231 = r229 <= r230;
        double r232 = 1.2515289756445332e-36;
        bool r233 = r229 <= r232;
        double r234 = !r233;
        bool r235 = r231 || r234;
        double r236 = 2.0;
        double r237 = cbrt(r236);
        double r238 = r237 * r237;
        double r239 = sqrt(r238);
        double r240 = cbrt(r229);
        double r241 = r240 * r240;
        double r242 = 3.0;
        double r243 = 2.0;
        double r244 = r242 / r243;
        double r245 = pow(r241, r244);
        double r246 = r239 / r245;
        double r247 = sqrt(r237);
        double r248 = r247 / r245;
        double r249 = sqrt(r236);
        double r250 = pow(r240, r242);
        double r251 = k;
        double r252 = sin(r251);
        double r253 = r250 * r252;
        double r254 = r249 / r253;
        double r255 = l;
        double r256 = r254 * r255;
        double r257 = r248 * r256;
        double r258 = r246 * r257;
        double r259 = tan(r251);
        double r260 = r258 / r259;
        double r261 = 1.0;
        double r262 = r251 / r229;
        double r263 = pow(r262, r236);
        double r264 = fma(r243, r261, r263);
        double r265 = r255 / r264;
        double r266 = r260 * r265;
        double r267 = pow(r241, r242);
        double r268 = r249 / r267;
        double r269 = 1.0;
        double r270 = r268 / r269;
        double r271 = r256 / r259;
        double r272 = r271 * r265;
        double r273 = r270 * r272;
        double r274 = r235 ? r266 : r273;
        return r274;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -1.3182669249842476e+38 or 1.2515289756445332e-36 < t

   1. Initial program 22.6

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

   2. Simplified 22.1

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

   4. Applied *-un-lft-identity 22.1

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

   5. Applied times-frac 20.9

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

   6. Applied associate-*r* 17.3

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

   7. Simplified 16.4

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

   9. Applied add-cube-cbrt 16.7

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

   10. Applied unpow-prod-down 16.7

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

   11. Applied associate-*l* 14.7

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

   13. Applied add-sqr-sqrt 14.7

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

   14. Applied times-frac 14.5

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

   15. Applied associate-*l* 12.1

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

   17. Applied sqr-pow 12.1

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

   18. Applied add-cube-cbrt 12.1

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

   19. Applied sqrt-prod 12.1

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

   20. Applied times-frac 11.8

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

   21. Applied associate-*l* 8.9

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

   if -1.3182669249842476e+38 < t < 1.2515289756445332e-36

   1. Initial program 48.3

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

   2. Simplified 49.6

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

   4. Applied *-un-lft-identity 49.6

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

   5. Applied times-frac 49.0

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

   6. Applied associate-*r* 48.5

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

   7. Simplified 46.9

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

   9. Applied add-cube-cbrt 47.1

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

   10. Applied unpow-prod-down 47.1

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

   11. Applied associate-*l* 47.1

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

   13. Applied add-sqr-sqrt 47.1

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

   14. Applied times-frac 47.1

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

   15. Applied associate-*l* 43.0

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

   17. Applied *-un-lft-identity 43.0

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

   18. Applied times-frac 43.1

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

   19. Applied associate-*l* 40.3

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

4. Final simplification 20.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.31826692498424758 \cdot 10^{38} \lor \neg \left(t \le 1.25152897564453319 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{1} \cdot \left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))