Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 16.0

Time: 29.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -4.767926568016393126457677186561838999712 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot \frac{1}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 7.032624916942270176139110895833513818529 \cdot 10^{-135}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{1}}\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r118330 = 2.0;
        double r118331 = t;
        double r118332 = 3.0;
        double r118333 = pow(r118331, r118332);
        double r118334 = l;
        double r118335 = r118334 * r118334;
        double r118336 = r118333 / r118335;
        double r118337 = k;
        double r118338 = sin(r118337);
        double r118339 = r118336 * r118338;
        double r118340 = tan(r118337);
        double r118341 = r118339 * r118340;
        double r118342 = 1.0;
        double r118343 = r118337 / r118331;
        double r118344 = pow(r118343, r118330);
        double r118345 = r118342 + r118344;
        double r118346 = r118345 - r118342;
        double r118347 = r118341 * r118346;
        double r118348 = r118330 / r118347;
        return r118348;
}

↓

double f(double t, double l, double k) {
        double r118349 = l;
        double r118350 = -4.767926568016393e-144;
        bool r118351 = r118349 <= r118350;
        double r118352 = 2.0;
        double r118353 = 1.0;
        double r118354 = k;
        double r118355 = 2.0;
        double r118356 = r118352 / r118355;
        double r118357 = pow(r118354, r118356);
        double r118358 = r118353 / r118357;
        double r118359 = 1.0;
        double r118360 = pow(r118358, r118359);
        double r118361 = t;
        double r118362 = pow(r118361, r118359);
        double r118363 = r118357 * r118362;
        double r118364 = r118353 / r118363;
        double r118365 = pow(r118364, r118359);
        double r118366 = cos(r118354);
        double r118367 = pow(r118349, r118355);
        double r118368 = r118366 * r118367;
        double r118369 = r118365 * r118368;
        double r118370 = sin(r118354);
        double r118371 = pow(r118370, r118355);
        double r118372 = r118353 / r118371;
        double r118373 = r118369 * r118372;
        double r118374 = r118360 * r118373;
        double r118375 = r118352 * r118374;
        double r118376 = 7.03262491694227e-135;
        bool r118377 = r118349 <= r118376;
        double r118378 = r118357 * r118363;
        double r118379 = r118353 / r118378;
        double r118380 = pow(r118379, r118359);
        double r118381 = cbrt(r118370);
        double r118382 = 4.0;
        double r118383 = pow(r118381, r118382);
        double r118384 = r118383 / r118349;
        double r118385 = r118384 / r118349;
        double r118386 = r118366 / r118385;
        double r118387 = pow(r118381, r118355);
        double r118388 = r118386 / r118387;
        double r118389 = r118380 * r118388;
        double r118390 = r118352 * r118389;
        double r118391 = r118366 / r118370;
        double r118392 = r118365 * r118391;
        double r118393 = pow(r118370, r118353);
        double r118394 = r118367 / r118393;
        double r118395 = r118392 * r118394;
        double r118396 = r118360 * r118395;
        double r118397 = r118352 * r118396;
        double r118398 = r118377 ? r118390 : r118397;
        double r118399 = r118351 ? r118375 : r118398;
        return r118399;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -4.767926568016393e-144

   1. Initial program 49.5

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

   2. Simplified 42.3

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

   3. Taylor expanded around inf 23.8

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

   5. Applied sqr-pow 23.8

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

   6. Applied associate-*l* 20.0

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

   8. Applied *-un-lft-identity 20.0

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

   9. Applied times-frac 19.7

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

   10. Applied unpow-prod-down 19.7

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

   11. Applied associate-*l* 17.2

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

   13. Applied div-inv 17.3

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

   14. Applied associate-*r* 17.5

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

   if -4.767926568016393e-144 < l < 7.03262491694227e-135

   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 36.2

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

   3. Taylor expanded around inf 17.7

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

   5. Applied sqr-pow 17.7

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

   6. Applied associate-*l* 17.7

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

   8. Applied add-cube-cbrt 17.7

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

   9. Applied unpow-prod-down 17.7

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

   10. Applied associate-/r* 17.4

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

   11. Simplified 12.0

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

   if 7.03262491694227e-135 < l

   1. Initial program 50.8

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

   2. Simplified 43.5

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

   3. Taylor expanded around inf 26.0

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

   5. Applied sqr-pow 26.0

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

   6. Applied associate-*l* 22.2

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

   8. Applied *-un-lft-identity 22.2

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

   9. Applied times-frac 21.9

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

   10. Applied unpow-prod-down 21.9

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

   11. Applied associate-*l* 19.5

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

   13. Applied sqr-pow 19.5

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

   14. Applied times-frac 19.3

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

   15. Applied associate-*r* 19.1

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

   16. Simplified 19.1

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

4. Final simplification 16.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.767926568016393126457677186561838999712 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot \frac{1}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 7.032624916942270176139110895833513818529 \cdot 10^{-135}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{1}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))