Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 18.2

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r79343 = 2.0;
        double r79344 = t;
        double r79345 = 3.0;
        double r79346 = pow(r79344, r79345);
        double r79347 = l;
        double r79348 = r79347 * r79347;
        double r79349 = r79346 / r79348;
        double r79350 = k;
        double r79351 = sin(r79350);
        double r79352 = r79349 * r79351;
        double r79353 = tan(r79350);
        double r79354 = r79352 * r79353;
        double r79355 = 1.0;
        double r79356 = r79350 / r79344;
        double r79357 = pow(r79356, r79343);
        double r79358 = r79355 + r79357;
        double r79359 = r79358 - r79355;
        double r79360 = r79354 * r79359;
        double r79361 = r79343 / r79360;
        return r79361;
}

↓

double f(double t, double l, double k) {
        double r79362 = 2.0;
        double r79363 = 1.0;
        double r79364 = k;
        double r79365 = 2.0;
        double r79366 = r79362 / r79365;
        double r79367 = pow(r79364, r79366);
        double r79368 = r79363 / r79367;
        double r79369 = 1.0;
        double r79370 = pow(r79368, r79369);
        double r79371 = cos(r79364);
        double r79372 = sin(r79364);
        double r79373 = r79371 / r79372;
        double r79374 = t;
        double r79375 = pow(r79374, r79369);
        double r79376 = r79367 * r79375;
        double r79377 = r79363 / r79376;
        double r79378 = pow(r79377, r79369);
        double r79379 = l;
        double r79380 = pow(r79379, r79365);
        double r79381 = r79380 / r79372;
        double r79382 = r79378 * r79381;
        double r79383 = r79373 * r79382;
        double r79384 = r79370 * r79383;
        double r79385 = r79362 * r79384;
        return r79385;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 48.1

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

2. Simplified 40.5

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

3. Taylor expanded around inf 22.1

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

5. Applied sqr-pow 22.1

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

6. Applied associate-*l* 20.1

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

8. Applied add-sqr-sqrt 20.1

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

9. Applied times-frac 19.9

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

10. Applied unpow-prod-down 19.9

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

11. Applied associate-*l* 18.5

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

12. Simplified 18.5

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

14. Applied unpow2 18.5

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

15. Applied times-frac 18.2

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

16. Applied associate-*l* 18.2

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

17. Simplified 18.2

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

18. Final simplification 18.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))