Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 7.1

Time: 41.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.7670378365958354 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k \cdot \sqrt[3]{t}}{\frac{\ell}{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;t \le 1.295357818168273 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \sin k}{\cos k} \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right), \frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k \cdot \sqrt[3]{t}}{\frac{\ell}{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1320348 = 2.0;
        double r1320349 = t;
        double r1320350 = 3.0;
        double r1320351 = pow(r1320349, r1320350);
        double r1320352 = l;
        double r1320353 = r1320352 * r1320352;
        double r1320354 = r1320351 / r1320353;
        double r1320355 = k;
        double r1320356 = sin(r1320355);
        double r1320357 = r1320354 * r1320356;
        double r1320358 = tan(r1320355);
        double r1320359 = r1320357 * r1320358;
        double r1320360 = 1.0;
        double r1320361 = r1320355 / r1320349;
        double r1320362 = pow(r1320361, r1320348);
        double r1320363 = r1320360 + r1320362;
        double r1320364 = r1320363 + r1320360;
        double r1320365 = r1320359 * r1320364;
        double r1320366 = r1320348 / r1320365;
        return r1320366;
}

↓

double f(double t, double l, double k) {
        double r1320367 = t;
        double r1320368 = -1.7670378365958354e-22;
        bool r1320369 = r1320367 <= r1320368;
        double r1320370 = 2.0;
        double r1320371 = k;
        double r1320372 = sin(r1320371);
        double r1320373 = cbrt(r1320367);
        double r1320374 = r1320372 * r1320373;
        double r1320375 = l;
        double r1320376 = r1320375 / r1320367;
        double r1320377 = r1320374 / r1320376;
        double r1320378 = r1320373 * r1320373;
        double r1320379 = r1320377 * r1320378;
        double r1320380 = r1320379 * r1320372;
        double r1320381 = r1320371 / r1320367;
        double r1320382 = fma(r1320381, r1320381, r1320370);
        double r1320383 = r1320380 * r1320382;
        double r1320384 = cos(r1320371);
        double r1320385 = r1320376 * r1320384;
        double r1320386 = r1320383 / r1320385;
        double r1320387 = r1320370 / r1320386;
        double r1320388 = 1.295357818168273e+59;
        bool r1320389 = r1320367 <= r1320388;
        double r1320390 = r1320372 * r1320372;
        double r1320391 = r1320390 / r1320384;
        double r1320392 = r1320367 * r1320367;
        double r1320393 = r1320392 / r1320375;
        double r1320394 = r1320367 / r1320375;
        double r1320395 = r1320393 * r1320394;
        double r1320396 = r1320391 * r1320395;
        double r1320397 = r1320375 / r1320371;
        double r1320398 = r1320397 * r1320397;
        double r1320399 = r1320384 / r1320390;
        double r1320400 = r1320398 * r1320399;
        double r1320401 = r1320367 / r1320400;
        double r1320402 = fma(r1320370, r1320396, r1320401);
        double r1320403 = r1320370 / r1320402;
        double r1320404 = r1320389 ? r1320403 : r1320387;
        double r1320405 = r1320369 ? r1320387 : r1320404;
        return r1320405;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -1.7670378365958354e-22 or 1.295357818168273e+59 < t

   1. Initial program 21.6

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

   2. Simplified 10.7

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

   4. Applied add-cube-cbrt 10.8

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

   5. Applied times-frac 10.3

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

   6. Applied associate-*l* 6.3

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

   8. Applied tan-quot 6.3

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

   9. Applied associate-*l/ 6.1

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

   10. Applied frac-times 4.0

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

   11. Applied associate-*l/ 3.7

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

   13. Applied *-un-lft-identity 3.7

       \[\leadsto \frac{2}{\frac{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \color{blue}{\left(1 \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   14. Applied associate-*r* 3.7

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot 1\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   15. Simplified 2.9

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

   if -1.7670378365958354e-22 < t < 1.295357818168273e+59

   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 35.1

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

   4. Applied add-cube-cbrt 35.2

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

   5. Applied times-frac 34.1

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

   6. Applied associate-*l* 32.6

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

   8. Applied tan-quot 32.6

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

   9. Applied associate-*l/ 32.6

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

   10. Applied frac-times 34.0

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

   11. Applied associate-*l/ 31.0

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

   12. Taylor expanded around inf 33.5

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

   13. Simplified 13.0

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

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.7670378365958354 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k \cdot \sqrt[3]{t}}{\frac{\ell}{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;t \le 1.295357818168273 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{\sin k \cdot \sin k}{\cos k} \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right), \frac{t}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\sin k \cdot \sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k \cdot \sqrt[3]{t}}{\frac{\ell}{t}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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))))