Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 11.2

Time: 3.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.90891404632138 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\sin k \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right) \cdot \frac{t}{\ell}}{\frac{1}{t}}}{\frac{\frac{\ell}{t}}{\tan k}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 7.981741095003087 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell}, \frac{t}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(\sin k \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \frac{t}{\ell}}{\frac{1}{t}}}}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r7949496 = 2.0;
        double r7949497 = t;
        double r7949498 = 3.0;
        double r7949499 = pow(r7949497, r7949498);
        double r7949500 = l;
        double r7949501 = r7949500 * r7949500;
        double r7949502 = r7949499 / r7949501;
        double r7949503 = k;
        double r7949504 = sin(r7949503);
        double r7949505 = r7949502 * r7949504;
        double r7949506 = tan(r7949503);
        double r7949507 = r7949505 * r7949506;
        double r7949508 = 1.0;
        double r7949509 = r7949503 / r7949497;
        double r7949510 = pow(r7949509, r7949496);
        double r7949511 = r7949508 + r7949510;
        double r7949512 = r7949511 + r7949508;
        double r7949513 = r7949507 * r7949512;
        double r7949514 = r7949496 / r7949513;
        return r7949514;
}

↓

double f(double t, double l, double k) {
        double r7949515 = l;
        double r7949516 = r7949515 * r7949515;
        double r7949517 = 3.90891404632138e-129;
        bool r7949518 = r7949516 <= r7949517;
        double r7949519 = 2.0;
        double r7949520 = k;
        double r7949521 = sin(r7949520);
        double r7949522 = t;
        double r7949523 = r7949520 / r7949522;
        double r7949524 = fma(r7949523, r7949523, r7949519);
        double r7949525 = r7949521 * r7949524;
        double r7949526 = r7949522 / r7949515;
        double r7949527 = r7949525 * r7949526;
        double r7949528 = 1.0;
        double r7949529 = r7949528 / r7949522;
        double r7949530 = r7949527 / r7949529;
        double r7949531 = r7949515 / r7949522;
        double r7949532 = tan(r7949520);
        double r7949533 = r7949531 / r7949532;
        double r7949534 = r7949530 / r7949533;
        double r7949535 = r7949519 / r7949534;
        double r7949536 = 7.981741095003087e-73;
        bool r7949537 = r7949516 <= r7949536;
        double r7949538 = r7949521 * r7949521;
        double r7949539 = cos(r7949520);
        double r7949540 = r7949538 / r7949539;
        double r7949541 = r7949522 * r7949522;
        double r7949542 = r7949541 * r7949522;
        double r7949543 = r7949542 / r7949516;
        double r7949544 = r7949540 * r7949543;
        double r7949545 = r7949522 / r7949516;
        double r7949546 = r7949520 * r7949520;
        double r7949547 = r7949546 * r7949538;
        double r7949548 = r7949547 / r7949539;
        double r7949549 = r7949545 * r7949548;
        double r7949550 = fma(r7949519, r7949544, r7949549);
        double r7949551 = r7949519 / r7949550;
        double r7949552 = sqrt(r7949524);
        double r7949553 = sqrt(r7949552);
        double r7949554 = r7949553 * r7949553;
        double r7949555 = r7949521 * r7949554;
        double r7949556 = r7949555 * r7949526;
        double r7949557 = r7949556 / r7949529;
        double r7949558 = r7949519 / r7949557;
        double r7949559 = r7949552 / r7949533;
        double r7949560 = r7949558 / r7949559;
        double r7949561 = r7949537 ? r7949551 : r7949560;
        double r7949562 = r7949518 ? r7949535 : r7949561;
        return r7949562;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 3.90891404632138e-129

   1. Initial program 24.1

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

   2. Simplified 11.0

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

   4. Applied clear-num 11.0

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

   5. Applied div-inv 11.0

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

   6. Applied associate-/r* 11.0

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

   7. Applied frac-times 10.8

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

   8. Applied associate-*l/ 8.5

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

   9. Applied associate-*l/ 7.9

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

   10. Simplified 7.9

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

   12. Applied associate-/r* 5.6

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

   if 3.90891404632138e-129 < (* l l) < 7.981741095003087e-73

   1. Initial program 22.6

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

   2. Simplified 20.2

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

   3. Taylor expanded around inf 13.6

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

   4. Simplified 13.8

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

   if 7.981741095003087e-73 < (* l l)

   1. Initial program 40.3

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

   2. Simplified 24.0

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

   4. Applied clear-num 24.0

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

   5. Applied div-inv 24.0

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

   6. Applied associate-/r* 24.0

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

   7. Applied frac-times 22.2

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

   8. Applied associate-*l/ 19.9

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

   9. Applied associate-*l/ 17.8

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

   10. Simplified 17.8

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

   12. Applied add-sqr-sqrt 18.0

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

   13. Applied associate-*r* 17.9

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

   14. Applied associate-*r* 17.9

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

   15. Applied times-frac 17.2

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

   16. Applied associate-/r* 16.9

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

   18. Applied add-sqr-sqrt 16.9

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

   19. Applied sqrt-prod 16.8

       \[\leadsto \frac{\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\sin k \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)}\right)}{\frac{1}{t}}}}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.90891404632138 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\sin k \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right) \cdot \frac{t}{\ell}}{\frac{1}{t}}}{\frac{\frac{\ell}{t}}{\tan k}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 7.981741095003087 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell}, \frac{t}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(\sin k \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}\right)\right) \cdot \frac{t}{\ell}}{\frac{1}{t}}}}{\frac{\sqrt{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}{\frac{\frac{\ell}{t}}{\tan k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))