Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 5.4

Time: 50.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -2.4692632667526504 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(2, \left(\frac{\sin k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}, \frac{\frac{k}{\ell} \cdot \sin k}{\frac{\cos k}{\frac{k}{\ell}}}\right) \cdot t\right) \cdot \sin k}\\ \mathbf{elif}\;k \le 2.837915385852385 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot t}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}} \cdot \frac{\frac{\ell}{t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\sin k \cdot \mathsf{fma}\left(2, \left(\frac{\sin k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}, \frac{\frac{k}{\ell} \cdot \sin k}{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}}\right)}\right)\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r2900432 = 2.0;
        double r2900433 = t;
        double r2900434 = 3.0;
        double r2900435 = pow(r2900433, r2900434);
        double r2900436 = l;
        double r2900437 = r2900436 * r2900436;
        double r2900438 = r2900435 / r2900437;
        double r2900439 = k;
        double r2900440 = sin(r2900439);
        double r2900441 = r2900438 * r2900440;
        double r2900442 = tan(r2900439);
        double r2900443 = r2900441 * r2900442;
        double r2900444 = 1.0;
        double r2900445 = r2900439 / r2900433;
        double r2900446 = pow(r2900445, r2900432);
        double r2900447 = r2900444 + r2900446;
        double r2900448 = r2900447 + r2900444;
        double r2900449 = r2900443 * r2900448;
        double r2900450 = r2900432 / r2900449;
        return r2900450;
}

↓

double f(double t, double l, double k) {
        double r2900451 = k;
        double r2900452 = -2.4692632667526504e-138;
        bool r2900453 = r2900451 <= r2900452;
        double r2900454 = 2.0;
        double r2900455 = sin(r2900451);
        double r2900456 = cos(r2900451);
        double r2900457 = r2900455 / r2900456;
        double r2900458 = t;
        double r2900459 = l;
        double r2900460 = r2900458 / r2900459;
        double r2900461 = r2900457 * r2900460;
        double r2900462 = r2900461 * r2900460;
        double r2900463 = r2900451 / r2900459;
        double r2900464 = r2900463 * r2900455;
        double r2900465 = r2900456 / r2900463;
        double r2900466 = r2900464 / r2900465;
        double r2900467 = fma(r2900454, r2900462, r2900466);
        double r2900468 = r2900467 * r2900458;
        double r2900469 = r2900468 * r2900455;
        double r2900470 = r2900454 / r2900469;
        double r2900471 = 2.837915385852385e-111;
        bool r2900472 = r2900451 <= r2900471;
        double r2900473 = r2900455 * r2900458;
        double r2900474 = r2900454 / r2900473;
        double r2900475 = r2900451 / r2900458;
        double r2900476 = fma(r2900475, r2900475, r2900454);
        double r2900477 = r2900459 / r2900458;
        double r2900478 = r2900476 / r2900477;
        double r2900479 = r2900474 / r2900478;
        double r2900480 = tan(r2900451);
        double r2900481 = r2900477 / r2900480;
        double r2900482 = r2900479 * r2900481;
        double r2900483 = cbrt(r2900456);
        double r2900484 = cbrt(r2900451);
        double r2900485 = cbrt(r2900459);
        double r2900486 = r2900484 / r2900485;
        double r2900487 = r2900483 / r2900486;
        double r2900488 = r2900487 * r2900487;
        double r2900489 = r2900487 * r2900488;
        double r2900490 = r2900464 / r2900489;
        double r2900491 = fma(r2900454, r2900462, r2900490);
        double r2900492 = r2900455 * r2900491;
        double r2900493 = r2900458 * r2900492;
        double r2900494 = r2900454 / r2900493;
        double r2900495 = r2900472 ? r2900482 : r2900494;
        double r2900496 = r2900453 ? r2900470 : r2900495;
        return r2900496;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -2.4692632667526504e-138

   1. Initial program 31.7

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

   2. Simplified 19.8

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

   4. Applied associate-*l/ 19.3

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

   5. Applied associate-/r/ 19.3

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

   6. Applied associate-/l* 17.4

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

   7. Taylor expanded around -inf 23.5

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

   8. Simplified 6.3

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

   10. Applied div-inv 6.3

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

   11. Applied associate-/l* 6.5

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

   12. Simplified 5.5

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

   14. Applied associate-*r* 5.6

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

   if -2.4692632667526504e-138 < k < 2.837915385852385e-111

   1. Initial program 37.5

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

   2. Simplified 28.6

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

   4. Applied associate-*l/ 21.4

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

   5. Applied associate-/r/ 21.2

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

   6. Applied associate-/l* 20.4

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

   8. Applied times-frac 6.1

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

   9. Applied *-un-lft-identity 6.1

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

   10. Applied times-frac 3.5

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

   11. Simplified 3.5

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

   if 2.837915385852385e-111 < k

   1. Initial program 30.4

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

   2. Simplified 18.7

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

   4. Applied associate-*l/ 18.4

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

   5. Applied associate-/r/ 18.3

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

   6. Applied associate-/l* 16.6

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

   7. Taylor expanded around -inf 21.9

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

   8. Simplified 6.2

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

   10. Applied div-inv 6.2

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

   11. Applied associate-/l* 6.3

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

   12. Simplified 5.7

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

   14. Applied add-cube-cbrt 5.9

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

   15. Applied add-cube-cbrt 5.9

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

   16. Applied times-frac 5.9

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

   17. Applied add-cube-cbrt 6.0

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

   18. Applied times-frac 6.0

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

   19. Simplified 6.0

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.4692632667526504 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(2, \left(\frac{\sin k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}, \frac{\frac{k}{\ell} \cdot \sin k}{\frac{\cos k}{\frac{k}{\ell}}}\right) \cdot t\right) \cdot \sin k}\\ \mathbf{elif}\;k \le 2.837915385852385 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot t}}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t}}} \cdot \frac{\frac{\ell}{t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\sin k \cdot \mathsf{fma}\left(2, \left(\frac{\sin k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}, \frac{\frac{k}{\ell} \cdot \sin k}{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{k}}{\sqrt[3]{\ell}}}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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))))