Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 5.8

Time: 50.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.3958085813923515 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k \cdot t}}{\frac{\tan k}{\frac{\ell}{t}}}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 1.5748242425025286 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{\frac{2}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\ell} \cdot \frac{k}{\ell}, \left(\left(\left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right) \cdot \left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right)\right) \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot t}}{\frac{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2643304 = 2.0;
        double r2643305 = t;
        double r2643306 = 3.0;
        double r2643307 = pow(r2643305, r2643306);
        double r2643308 = l;
        double r2643309 = r2643308 * r2643308;
        double r2643310 = r2643307 / r2643309;
        double r2643311 = k;
        double r2643312 = sin(r2643311);
        double r2643313 = r2643310 * r2643312;
        double r2643314 = tan(r2643311);
        double r2643315 = r2643313 * r2643314;
        double r2643316 = 1.0;
        double r2643317 = r2643311 / r2643305;
        double r2643318 = pow(r2643317, r2643304);
        double r2643319 = r2643316 + r2643318;
        double r2643320 = r2643319 + r2643316;
        double r2643321 = r2643315 * r2643320;
        double r2643322 = r2643304 / r2643321;
        return r2643322;
}

↓

double f(double t, double l, double k) {
        double r2643323 = t;
        double r2643324 = -5.3958085813923515e+84;
        bool r2643325 = r2643323 <= r2643324;
        double r2643326 = 2.0;
        double r2643327 = k;
        double r2643328 = sin(r2643327);
        double r2643329 = r2643328 * r2643323;
        double r2643330 = r2643326 / r2643329;
        double r2643331 = tan(r2643327);
        double r2643332 = l;
        double r2643333 = r2643332 / r2643323;
        double r2643334 = r2643331 / r2643333;
        double r2643335 = r2643330 / r2643334;
        double r2643336 = r2643327 / r2643323;
        double r2643337 = fma(r2643336, r2643336, r2643326);
        double r2643338 = r2643337 / r2643333;
        double r2643339 = r2643335 / r2643338;
        double r2643340 = 1.5748242425025286e+51;
        bool r2643341 = r2643323 <= r2643340;
        double r2643342 = 1.0;
        double r2643343 = r2643342 / r2643323;
        double r2643344 = r2643326 / r2643328;
        double r2643345 = cos(r2643327);
        double r2643346 = r2643328 / r2643345;
        double r2643347 = r2643327 / r2643332;
        double r2643348 = r2643347 * r2643347;
        double r2643349 = r2643323 / r2643332;
        double r2643350 = cbrt(r2643346);
        double r2643351 = r2643349 * r2643350;
        double r2643352 = r2643351 * r2643351;
        double r2643353 = r2643352 * r2643350;
        double r2643354 = r2643353 * r2643326;
        double r2643355 = fma(r2643346, r2643348, r2643354);
        double r2643356 = r2643344 / r2643355;
        double r2643357 = r2643343 * r2643356;
        double r2643358 = r2643337 * r2643331;
        double r2643359 = r2643358 / r2643333;
        double r2643360 = r2643359 / r2643333;
        double r2643361 = r2643330 / r2643360;
        double r2643362 = r2643341 ? r2643357 : r2643361;
        double r2643363 = r2643325 ? r2643339 : r2643362;
        return r2643363;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -5.3958085813923515e+84

   1. Initial program 22.4

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

   2. Simplified 9.7

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

   4. Applied associate-*l/ 6.2

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

   5. Applied associate-/r/ 5.9

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

   6. Applied associate-/l* 5.1

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

   8. Applied times-frac 1.5

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

   9. Applied associate-/r* 1.2

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

   if -5.3958085813923515e+84 < t < 1.5748242425025286e+51

   1. Initial program 41.9

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

   2. Simplified 31.9

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

   4. Applied associate-*l/ 31.9

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

   5. Applied associate-/r/ 32.0

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

   6. Applied associate-/l* 29.1

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

   7. Taylor expanded around -inf 29.5

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

   8. Simplified 12.9

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

   10. Applied add-cube-cbrt 13.0

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

   11. Applied associate-*r* 13.0

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

   12. Simplified 11.2

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

   14. Applied *-un-lft-identity 11.2

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

   15. Applied *-un-lft-identity 11.2

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

   16. Applied times-frac 11.2

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

   17. Applied times-frac 10.2

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

   18. Simplified 10.2

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

   if 1.5748242425025286e+51 < t

   1. Initial program 22.3

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

   2. Simplified 9.5

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

   4. Applied associate-*l/ 6.0

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

   5. Applied associate-/r/ 5.8

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

   6. Applied associate-/l* 4.8

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

   8. Applied associate-/r* 2.0

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

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.3958085813923515 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k \cdot t}}{\frac{\tan k}{\frac{\ell}{t}}}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 1.5748242425025286 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{t} \cdot \frac{\frac{2}{\sin k}}{\mathsf{fma}\left(\frac{\sin k}{\cos k}, \frac{k}{\ell} \cdot \frac{k}{\ell}, \left(\left(\left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right) \cdot \left(\frac{t}{\ell} \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right)\right) \cdot \sqrt[3]{\frac{\sin k}{\cos k}}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot t}}{\frac{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k}{\frac{\ell}{t}}}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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))))