Toniolo and Linder, Equation (10-)

Average Error: 48.7 → 11.5

Time: 1.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.429270327298531141168205995229653525329 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r193512 = 2.0;
        double r193513 = t;
        double r193514 = 3.0;
        double r193515 = pow(r193513, r193514);
        double r193516 = l;
        double r193517 = r193516 * r193516;
        double r193518 = r193515 / r193517;
        double r193519 = k;
        double r193520 = sin(r193519);
        double r193521 = r193518 * r193520;
        double r193522 = tan(r193519);
        double r193523 = r193521 * r193522;
        double r193524 = 1.0;
        double r193525 = r193519 / r193513;
        double r193526 = pow(r193525, r193512);
        double r193527 = r193524 + r193526;
        double r193528 = r193527 - r193524;
        double r193529 = r193523 * r193528;
        double r193530 = r193512 / r193529;
        return r193530;
}

↓

double f(double t, double l, double k) {
        double r193531 = l;
        double r193532 = r193531 * r193531;
        double r193533 = 0.0;
        bool r193534 = r193532 <= r193533;
        double r193535 = 2.0;
        double r193536 = 1.0;
        double r193537 = k;
        double r193538 = 2.0;
        double r193539 = r193535 / r193538;
        double r193540 = pow(r193537, r193539);
        double r193541 = t;
        double r193542 = 1.0;
        double r193543 = pow(r193541, r193542);
        double r193544 = r193540 * r193543;
        double r193545 = r193536 / r193544;
        double r193546 = pow(r193545, r193542);
        double r193547 = r193536 / r193540;
        double r193548 = pow(r193547, r193542);
        double r193549 = sin(r193537);
        double r193550 = pow(r193549, r193538);
        double r193551 = cos(r193537);
        double r193552 = r193531 * r193551;
        double r193553 = r193550 / r193552;
        double r193554 = r193553 / r193531;
        double r193555 = r193548 / r193554;
        double r193556 = r193546 * r193555;
        double r193557 = r193535 * r193556;
        double r193558 = 2.429270327298531e-217;
        bool r193559 = r193532 <= r193558;
        double r193560 = pow(r193537, r193535);
        double r193561 = r193560 * r193543;
        double r193562 = r193536 / r193561;
        double r193563 = pow(r193562, r193542);
        double r193564 = r193551 / r193549;
        double r193565 = r193549 / r193531;
        double r193566 = r193531 / r193565;
        double r193567 = r193564 * r193566;
        double r193568 = r193563 * r193567;
        double r193569 = r193535 * r193568;
        double r193570 = r193548 * r193552;
        double r193571 = r193531 * r193570;
        double r193572 = r193571 / r193550;
        double r193573 = r193547 / r193543;
        double r193574 = pow(r193573, r193542);
        double r193575 = r193572 * r193574;
        double r193576 = r193535 * r193575;
        double r193577 = r193559 ? r193569 : r193576;
        double r193578 = r193534 ? r193557 : r193577;
        return r193578;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 0.0

   1. Initial program 46.5

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 20.5

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

   5. Applied sqr-pow 20.5

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

   6. Applied associate-*r* 20.5

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

   7. Simplified 20.5

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

   9. Applied *-un-lft-identity 20.5

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

   10. Applied times-frac 20.5

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

   11. Applied unpow-prod-down 20.5

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

   12. Applied associate-*l* 20.4

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

   13. Simplified 20.4

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

   15. Applied associate-/l* 20.4

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

   16. Simplified 13.3

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

   if 0.0 < (* l l) < 2.429270327298531e-217

   1. Initial program 45.5

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

   2. Simplified 34.6

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

   3. Taylor expanded around inf 5.3

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

   5. Applied add-sqr-sqrt 35.1

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

   6. Applied unpow-prod-down 35.1

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

   7. Applied times-frac 35.1

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

   8. Simplified 35.1

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

   9. Simplified 4.4

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

   if 2.429270327298531e-217 < (* l l)

   1. Initial program 50.4

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

   2. Simplified 43.6

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

   3. Taylor expanded around inf 26.2

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

   5. Applied sqr-pow 26.2

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

   6. Applied associate-*r* 22.2

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

   7. Simplified 22.2

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

   9. Applied *-un-lft-identity 22.2

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

   10. Applied times-frac 22.0

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

   11. Applied unpow-prod-down 22.0

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

   12. Applied associate-*l* 19.7

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

   13. Simplified 19.6

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

   15. Applied *-un-lft-identity 19.6

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

   16. Applied associate-*l* 19.6

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

   17. Simplified 11.8

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

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.429270327298531141168205995229653525329 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))))