Toniolo and Linder, Equation (10+)

Average Error: 31.6 → 8.6

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0865304981214163 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \frac{\sin k}{\sqrt{\sqrt{2}}}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \le 0.02437188434168144:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{k \cdot k}{\sqrt{2} \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos k} + 2 \cdot \frac{t \cdot t}{\frac{\sqrt{2} \cdot \left(\cos k \cdot \ell\right)}{\sin k \cdot \sin k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k}{\frac{\ell}{t}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r4081457 = 2.0;
        double r4081458 = t;
        double r4081459 = 3.0;
        double r4081460 = pow(r4081458, r4081459);
        double r4081461 = l;
        double r4081462 = r4081461 * r4081461;
        double r4081463 = r4081460 / r4081462;
        double r4081464 = k;
        double r4081465 = sin(r4081464);
        double r4081466 = r4081463 * r4081465;
        double r4081467 = tan(r4081464);
        double r4081468 = r4081466 * r4081467;
        double r4081469 = 1.0;
        double r4081470 = r4081464 / r4081458;
        double r4081471 = pow(r4081470, r4081457);
        double r4081472 = r4081469 + r4081471;
        double r4081473 = r4081472 + r4081469;
        double r4081474 = r4081468 * r4081473;
        double r4081475 = r4081457 / r4081474;
        return r4081475;
}

↓

double f(double t, double l, double k) {
        double r4081476 = t;
        double r4081477 = -1.0865304981214163e-31;
        bool r4081478 = r4081476 <= r4081477;
        double r4081479 = 2.0;
        double r4081480 = sqrt(r4081479);
        double r4081481 = k;
        double r4081482 = tan(r4081481);
        double r4081483 = l;
        double r4081484 = r4081483 / r4081476;
        double r4081485 = r4081482 / r4081484;
        double r4081486 = r4081476 * r4081485;
        double r4081487 = sqrt(r4081480);
        double r4081488 = r4081481 / r4081476;
        double r4081489 = r4081488 * r4081488;
        double r4081490 = r4081479 + r4081489;
        double r4081491 = r4081487 / r4081490;
        double r4081492 = r4081486 / r4081491;
        double r4081493 = sin(r4081481);
        double r4081494 = r4081493 / r4081487;
        double r4081495 = r4081492 * r4081494;
        double r4081496 = r4081480 / r4081495;
        double r4081497 = r4081496 * r4081484;
        double r4081498 = 0.02437188434168144;
        bool r4081499 = r4081476 <= r4081498;
        double r4081500 = r4081481 * r4081481;
        double r4081501 = r4081480 * r4081483;
        double r4081502 = r4081500 / r4081501;
        double r4081503 = r4081493 * r4081493;
        double r4081504 = cos(r4081481);
        double r4081505 = r4081503 / r4081504;
        double r4081506 = r4081502 * r4081505;
        double r4081507 = r4081476 * r4081476;
        double r4081508 = r4081504 * r4081483;
        double r4081509 = r4081480 * r4081508;
        double r4081510 = r4081509 / r4081503;
        double r4081511 = r4081507 / r4081510;
        double r4081512 = r4081479 * r4081511;
        double r4081513 = r4081506 + r4081512;
        double r4081514 = r4081480 / r4081513;
        double r4081515 = r4081514 * r4081484;
        double r4081516 = r4081479 / r4081490;
        double r4081517 = r4081486 * r4081493;
        double r4081518 = r4081517 / r4081484;
        double r4081519 = r4081516 / r4081518;
        double r4081520 = r4081499 ? r4081515 : r4081519;
        double r4081521 = r4081478 ? r4081497 : r4081520;
        return r4081521;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.0865304981214163e-31

   1. Initial program 21.5

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

   2. Simplified 7.8

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

   4. Applied associate-*l/ 6.6

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

   5. Applied associate-*r/ 4.0

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

   6. Applied associate-/r/ 3.4

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

   8. Applied *-un-lft-identity 3.4

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

   9. Applied add-sqr-sqrt 3.5

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

   10. Applied times-frac 3.5

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

   11. Applied associate-/l* 3.6

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

   13. Applied *-un-lft-identity 3.6

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

   14. Applied add-sqr-sqrt 3.5

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

   15. Applied times-frac 3.5

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

   16. Applied times-frac 3.5

       \[\leadsto \frac{\frac{\sqrt{2}}{1}}{\color{blue}{\frac{\sin k}{\frac{\sqrt{\sqrt{2}}}{1}} \cdot \frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \frac{\ell}{t}\]

   17. Simplified 3.5

       \[\leadsto \frac{\frac{\sqrt{2}}{1}}{\color{blue}{\frac{\sin k}{\sqrt{\sqrt{2}}}} \cdot \frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\ell}{t}\]

   if -1.0865304981214163e-31 < t < 0.02437188434168144

   1. Initial program 49.4

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

   2. Simplified 35.1

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

   4. Applied associate-*l/ 35.1

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

   5. Applied associate-*r/ 36.2

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

   6. Applied associate-/r/ 33.2

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

   8. Applied *-un-lft-identity 33.2

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

   9. Applied add-sqr-sqrt 33.2

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

   10. Applied times-frac 33.2

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

   11. Applied associate-/l* 33.2

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

   13. Applied *-un-lft-identity 33.2

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

   14. Applied add-sqr-sqrt 33.2

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

   15. Applied times-frac 33.2

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

   16. Applied times-frac 32.4

       \[\leadsto \frac{\frac{\sqrt{2}}{1}}{\color{blue}{\frac{\sin k}{\frac{\sqrt{\sqrt{2}}}{1}} \cdot \frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}}} \cdot \frac{\ell}{t}\]

   17. Simplified 32.4

       \[\leadsto \frac{\frac{\sqrt{2}}{1}}{\color{blue}{\frac{\sin k}{\sqrt{\sqrt{2}}}} \cdot \frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\ell}{t}\]

   18. Taylor expanded around -inf 21.5

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

   19. Simplified 18.4

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

   if 0.02437188434168144 < t

   1. Initial program 22.4

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

   2. Simplified 7.4

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

   4. Applied associate-*l/ 5.9

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

   5. Applied associate-*r/ 3.1

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0865304981214163 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{t \cdot \frac{\tan k}{\frac{\ell}{t}}}{\frac{\sqrt{\sqrt{2}}}{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \frac{\sin k}{\sqrt{\sqrt{2}}}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \le 0.02437188434168144:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{k \cdot k}{\sqrt{2} \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos k} + 2 \cdot \frac{t \cdot t}{\frac{\sqrt{2} \cdot \left(\cos k \cdot \ell\right)}{\sin k \cdot \sin k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\left(t \cdot \frac{\tan k}{\frac{\ell}{t}}\right) \cdot \sin k}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))))