Toniolo and Linder, Equation (10+)

Average Error: 33.0 → 8.7

Time: 51.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6870817860977717 \lor \neg \left(t \le 1.676533032936832680217902058483758764851 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \tan k\right)}{\frac{\ell}{{t}^{\left(3 \cdot \frac{1}{3}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k\right)}^{2} \cdot \left(t \cdot t\right)}{\cos k \cdot \ell}, 2, \frac{{\left(\sin k\right)}^{2}}{\frac{\cos k}{k} \cdot \frac{\ell}{k}}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r117481 = 2.0;
        double r117482 = t;
        double r117483 = 3.0;
        double r117484 = pow(r117482, r117483);
        double r117485 = l;
        double r117486 = r117485 * r117485;
        double r117487 = r117484 / r117486;
        double r117488 = k;
        double r117489 = sin(r117488);
        double r117490 = r117487 * r117489;
        double r117491 = tan(r117488);
        double r117492 = r117490 * r117491;
        double r117493 = 1.0;
        double r117494 = r117488 / r117482;
        double r117495 = pow(r117494, r117481);
        double r117496 = r117493 + r117495;
        double r117497 = r117496 + r117493;
        double r117498 = r117492 * r117497;
        double r117499 = r117481 / r117498;
        return r117499;
}

↓

double f(double t, double l, double k) {
        double r117500 = t;
        double r117501 = -6870817860977717.0;
        bool r117502 = r117500 <= r117501;
        double r117503 = 1.6765330329368327e-05;
        bool r117504 = r117500 <= r117503;
        double r117505 = !r117504;
        bool r117506 = r117502 || r117505;
        double r117507 = 2.0;
        double r117508 = cbrt(r117500);
        double r117509 = 3.0;
        double r117510 = pow(r117508, r117509);
        double r117511 = k;
        double r117512 = sin(r117511);
        double r117513 = r117510 * r117512;
        double r117514 = l;
        double r117515 = r117513 / r117514;
        double r117516 = r117515 * r117510;
        double r117517 = 1.0;
        double r117518 = r117511 / r117500;
        double r117519 = pow(r117518, r117507);
        double r117520 = r117519 + r117517;
        double r117521 = r117517 + r117520;
        double r117522 = tan(r117511);
        double r117523 = r117521 * r117522;
        double r117524 = r117516 * r117523;
        double r117525 = 0.3333333333333333;
        double r117526 = r117509 * r117525;
        double r117527 = pow(r117500, r117526);
        double r117528 = r117514 / r117527;
        double r117529 = r117524 / r117528;
        double r117530 = r117507 / r117529;
        double r117531 = 2.0;
        double r117532 = pow(r117512, r117531);
        double r117533 = r117500 * r117500;
        double r117534 = r117532 * r117533;
        double r117535 = cos(r117511);
        double r117536 = r117535 * r117514;
        double r117537 = r117534 / r117536;
        double r117538 = r117535 / r117511;
        double r117539 = r117514 / r117511;
        double r117540 = r117538 * r117539;
        double r117541 = r117532 / r117540;
        double r117542 = fma(r117537, r117507, r117541);
        double r117543 = r117514 / r117510;
        double r117544 = r117542 / r117543;
        double r117545 = r117507 / r117544;
        double r117546 = r117506 ? r117530 : r117545;
        return r117546;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -6870817860977717.0 or 1.6765330329368327e-05 < t

   1. Initial program 23.2

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

   3. Applied add-cube-cbrt 23.3

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

   4. Applied unpow-prod-down 23.3

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

   5. Applied times-frac 16.3

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

   6. Applied associate-*l* 14.5

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

   8. Applied unpow-prod-down 14.5

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

   9. Applied associate-/l* 8.0

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

   11. Applied associate-*l/ 6.7

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

   12. Applied associate-*l/ 3.9

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

   13. Applied associate-*l/ 3.7

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

   14. Simplified 3.6

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

   16. Applied pow1/3 33.5

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

   17. Applied pow-pow 3.5

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

   if -6870817860977717.0 < t < 1.6765330329368327e-05

   1. Initial program 48.5

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

   3. Applied add-cube-cbrt 48.6

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

   4. Applied unpow-prod-down 48.6

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

   5. Applied times-frac 41.0

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

   6. Applied associate-*l* 39.6

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

   8. Applied unpow-prod-down 39.6

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

   9. Applied associate-/l* 34.5

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

   11. Applied associate-*l/ 34.5

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

   12. Applied associate-*l/ 36.1

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

   13. Applied associate-*l/ 32.6

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

   14. Simplified 31.9

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

   15. Taylor expanded around inf 20.8

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

   16. Simplified 16.8

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6870817860977717 \lor \neg \left(t \le 1.676533032936832680217902058483758764851 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \tan k\right)}{\frac{\ell}{{t}^{\left(3 \cdot \frac{1}{3}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{{\left(\sin k\right)}^{2} \cdot \left(t \cdot t\right)}{\cos k \cdot \ell}, 2, \frac{{\left(\sin k\right)}^{2}}{\frac{\cos k}{k} \cdot \frac{\ell}{k}}\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))