Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 5.5

Time: 38.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.155631462029534724175353991865915612942 \cdot 10^{-190}:\\ \;\;\;\;\frac{2}{\frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} + 2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot t\right)}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \le 3.297004377141025967006934297574193329242 \cdot 10^{95}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right)}{\cos k} + \frac{t}{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} + 2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot t\right)}{\cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3172451 = 2.0;
        double r3172452 = t;
        double r3172453 = 3.0;
        double r3172454 = pow(r3172452, r3172453);
        double r3172455 = l;
        double r3172456 = r3172455 * r3172455;
        double r3172457 = r3172454 / r3172456;
        double r3172458 = k;
        double r3172459 = sin(r3172458);
        double r3172460 = r3172457 * r3172459;
        double r3172461 = tan(r3172458);
        double r3172462 = r3172460 * r3172461;
        double r3172463 = 1.0;
        double r3172464 = r3172458 / r3172452;
        double r3172465 = pow(r3172464, r3172451);
        double r3172466 = r3172463 + r3172465;
        double r3172467 = r3172466 + r3172463;
        double r3172468 = r3172462 * r3172467;
        double r3172469 = r3172451 / r3172468;
        return r3172469;
}

↓

double f(double t, double l, double k) {
        double r3172470 = l;
        double r3172471 = r3172470 * r3172470;
        double r3172472 = 9.155631462029535e-190;
        bool r3172473 = r3172471 <= r3172472;
        double r3172474 = 2.0;
        double r3172475 = t;
        double r3172476 = k;
        double r3172477 = r3172470 / r3172476;
        double r3172478 = r3172477 * r3172477;
        double r3172479 = cos(r3172476);
        double r3172480 = sin(r3172476);
        double r3172481 = r3172480 * r3172480;
        double r3172482 = r3172479 / r3172481;
        double r3172483 = r3172478 * r3172482;
        double r3172484 = r3172475 / r3172483;
        double r3172485 = r3172480 / r3172470;
        double r3172486 = r3172485 * r3172475;
        double r3172487 = r3172486 * r3172475;
        double r3172488 = r3172486 * r3172487;
        double r3172489 = r3172488 / r3172479;
        double r3172490 = r3172474 * r3172489;
        double r3172491 = r3172484 + r3172490;
        double r3172492 = r3172474 / r3172491;
        double r3172493 = 3.297004377141026e+95;
        bool r3172494 = r3172471 <= r3172493;
        double r3172495 = r3172486 * r3172486;
        double r3172496 = r3172475 * r3172495;
        double r3172497 = r3172496 / r3172479;
        double r3172498 = r3172474 * r3172497;
        double r3172499 = r3172477 * r3172470;
        double r3172500 = r3172499 * r3172482;
        double r3172501 = r3172475 / r3172500;
        double r3172502 = r3172501 * r3172476;
        double r3172503 = r3172498 + r3172502;
        double r3172504 = r3172474 / r3172503;
        double r3172505 = r3172494 ? r3172504 : r3172492;
        double r3172506 = r3172473 ? r3172492 : r3172505;
        return r3172506;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (* l l) < 9.155631462029535e-190 or 3.297004377141026e+95 < (* l l)

   1. Initial program 35.7

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

   3. Applied add-cube-cbrt 35.7

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

   4. Applied unpow-prod-down 35.7

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

   5. Applied times-frac 26.9

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

   6. Taylor expanded around inf 36.8

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

   7. Simplified 21.3

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

   9. Applied associate-*r/ 21.3

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

   10. Simplified 7.3

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

   12. Applied associate-*l* 6.8

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

   if 9.155631462029535e-190 < (* l l) < 3.297004377141026e+95

   1. Initial program 24.5

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

   3. Applied add-cube-cbrt 24.8

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

   4. Applied unpow-prod-down 24.8

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

   5. Applied times-frac 23.3

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

   6. Taylor expanded around inf 16.5

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

   7. Simplified 15.2

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

   9. Applied associate-*r/ 15.2

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

   10. Simplified 6.6

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

   12. Applied associate-*r/ 6.6

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

   13. Applied associate-*r/ 6.6

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

   14. Applied associate-/r/ 1.8

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.155631462029534724175353991865915612942 \cdot 10^{-190}:\\ \;\;\;\;\frac{2}{\frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} + 2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot t\right)}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \le 3.297004377141025967006934297574193329242 \cdot 10^{95}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right)}{\cos k} + \frac{t}{\left(\frac{\ell}{k} \cdot \ell\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}} + 2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot t\right)}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))