Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 15.0

Time: 30.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.88131 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 7.7895926479734956 \cdot 10^{303}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1}}\right)}^{\left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r91327 = 2.0;
        double r91328 = t;
        double r91329 = 3.0;
        double r91330 = pow(r91328, r91329);
        double r91331 = l;
        double r91332 = r91331 * r91331;
        double r91333 = r91330 / r91332;
        double r91334 = k;
        double r91335 = sin(r91334);
        double r91336 = r91333 * r91335;
        double r91337 = tan(r91334);
        double r91338 = r91336 * r91337;
        double r91339 = 1.0;
        double r91340 = r91334 / r91328;
        double r91341 = pow(r91340, r91327);
        double r91342 = r91339 + r91341;
        double r91343 = r91342 - r91339;
        double r91344 = r91338 * r91343;
        double r91345 = r91327 / r91344;
        return r91345;
}

↓

double f(double t, double l, double k) {
        double r91346 = l;
        double r91347 = r91346 * r91346;
        double r91348 = 9.8813129168249e-324;
        bool r91349 = r91347 <= r91348;
        double r91350 = 2.0;
        double r91351 = 1.0;
        double r91352 = k;
        double r91353 = 2.0;
        double r91354 = r91350 / r91353;
        double r91355 = pow(r91352, r91354);
        double r91356 = t;
        double r91357 = 1.0;
        double r91358 = pow(r91356, r91357);
        double r91359 = r91355 * r91358;
        double r91360 = r91355 * r91359;
        double r91361 = r91351 / r91360;
        double r91362 = pow(r91361, r91357);
        double r91363 = cos(r91352);
        double r91364 = sin(r91352);
        double r91365 = cbrt(r91364);
        double r91366 = 4.0;
        double r91367 = pow(r91365, r91366);
        double r91368 = r91367 / r91346;
        double r91369 = r91368 / r91346;
        double r91370 = r91363 / r91369;
        double r91371 = pow(r91365, r91353);
        double r91372 = r91370 / r91371;
        double r91373 = r91362 * r91372;
        double r91374 = r91350 * r91373;
        double r91375 = 7.789592647973496e+303;
        bool r91376 = r91347 <= r91375;
        double r91377 = sqrt(r91351);
        double r91378 = r91377 / r91355;
        double r91379 = pow(r91378, r91357);
        double r91380 = r91351 / r91359;
        double r91381 = pow(r91380, r91357);
        double r91382 = r91363 / r91364;
        double r91383 = pow(r91346, r91353);
        double r91384 = r91383 / r91364;
        double r91385 = r91382 * r91384;
        double r91386 = r91381 * r91385;
        double r91387 = r91379 * r91386;
        double r91388 = r91350 * r91387;
        double r91389 = pow(r91352, r91350);
        double r91390 = r91358 * r91389;
        double r91391 = r91351 / r91390;
        double r91392 = pow(r91391, r91357);
        double r91393 = exp(r91392);
        double r91394 = r91363 * r91383;
        double r91395 = pow(r91364, r91353);
        double r91396 = r91394 / r91395;
        double r91397 = pow(r91393, r91396);
        double r91398 = log(r91397);
        double r91399 = r91350 * r91398;
        double r91400 = r91376 ? r91388 : r91399;
        double r91401 = r91349 ? r91374 : r91400;
        return r91401;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 9.8813129168249e-324

   1. Initial program 46.0

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

   2. Simplified 37.2

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

   3. Taylor expanded around inf 19.7

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 19.7

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

   6. Applied associate-*l* 19.7

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

   8. Applied add-cube-cbrt 19.7

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

   9. Applied unpow-prod-down 19.7

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

   10. Applied associate-/r* 19.6

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

   11. Simplified 13.5

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   if 9.8813129168249e-324 < (* l l) < 7.789592647973496e+303

   1. Initial program 44.4

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

   2. Simplified 35.3

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

   3. Taylor expanded around inf 11.6

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 11.6

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

   6. Applied associate-*l* 7.3

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

   8. Applied add-sqr-sqrt 7.3

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

   9. Applied times-frac 6.9

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

   10. Applied unpow-prod-down 6.9

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

   11. Applied associate-*l* 4.0

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

   12. Simplified 4.0

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

   14. Applied add-sqr-sqrt 35.0

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

   15. Applied unpow-prod-down 35.0

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

   16. Applied times-frac 34.8

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

   17. Simplified 34.7

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

   18. Simplified 3.6

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

   if 7.789592647973496e+303 < (* l l)

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   3. Taylor expanded around inf 63.7

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 63.7

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

   6. Applied associate-*l* 63.7

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

   8. Applied add-log-exp 63.8

      \[\leadsto 2 \cdot \color{blue}{\log \left(e^{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}\right)}\]

   9. Simplified 58.2

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

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.88131 \cdot 10^{-324}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 7.7895926479734956 \cdot 10^{303}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left({\left(e^{{\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1}}\right)}^{\left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))