Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 8.9

Time: 26.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r88442 = 2.0;
        double r88443 = t;
        double r88444 = 3.0;
        double r88445 = pow(r88443, r88444);
        double r88446 = l;
        double r88447 = r88446 * r88446;
        double r88448 = r88445 / r88447;
        double r88449 = k;
        double r88450 = sin(r88449);
        double r88451 = r88448 * r88450;
        double r88452 = tan(r88449);
        double r88453 = r88451 * r88452;
        double r88454 = 1.0;
        double r88455 = r88449 / r88443;
        double r88456 = pow(r88455, r88442);
        double r88457 = r88454 + r88456;
        double r88458 = r88457 - r88454;
        double r88459 = r88453 * r88458;
        double r88460 = r88442 / r88459;
        return r88460;
}

↓

double f(double t, double l, double k) {
        double r88461 = k;
        double r88462 = -4.5772359455565565e+139;
        bool r88463 = r88461 <= r88462;
        double r88464 = -2.705728249136002e-140;
        bool r88465 = r88461 <= r88464;
        double r88466 = 7.600193379940175e-155;
        bool r88467 = r88461 <= r88466;
        double r88468 = 1.1512157854309419e+132;
        bool r88469 = r88461 <= r88468;
        double r88470 = !r88469;
        bool r88471 = r88467 || r88470;
        double r88472 = !r88471;
        bool r88473 = r88465 || r88472;
        double r88474 = !r88473;
        bool r88475 = r88463 || r88474;
        double r88476 = 2.0;
        double r88477 = 1.0;
        double r88478 = 2.0;
        double r88479 = r88476 / r88478;
        double r88480 = pow(r88461, r88479);
        double r88481 = t;
        double r88482 = 1.0;
        double r88483 = pow(r88481, r88482);
        double r88484 = r88480 * r88483;
        double r88485 = r88480 * r88484;
        double r88486 = r88477 / r88485;
        double r88487 = pow(r88486, r88482);
        double r88488 = cos(r88461);
        double r88489 = sin(r88461);
        double r88490 = r88488 / r88489;
        double r88491 = l;
        double r88492 = r88490 * r88491;
        double r88493 = r88487 * r88492;
        double r88494 = r88489 / r88491;
        double r88495 = r88493 / r88494;
        double r88496 = r88476 * r88495;
        double r88497 = pow(r88461, r88476);
        double r88498 = r88477 / r88497;
        double r88499 = pow(r88498, r88482);
        double r88500 = r88477 / r88483;
        double r88501 = pow(r88500, r88482);
        double r88502 = r88501 * r88492;
        double r88503 = r88499 * r88502;
        double r88504 = r88503 / r88494;
        double r88505 = r88476 * r88504;
        double r88506 = r88475 ? r88496 : r88505;
        return r88506;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -4.5772359455565565e+139 or -2.705728249136002e-140 < k < 7.600193379940175e-155 or 1.1512157854309419e+132 < k

   1. Initial program 42.0

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 26.4

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

   5. Applied add-sqr-sqrt 45.4

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

   6. Applied unpow-prod-down 45.4

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

   7. Applied times-frac 45.4

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

   8. Simplified 45.4

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

   9. Simplified 26.2

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

   11. Applied associate-*r/ 25.8

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

   12. Applied associate-*r/ 23.9

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

   14. Applied sqr-pow 23.9

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

   15. Applied associate-*l* 14.6

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

   if -4.5772359455565565e+139 < k < -2.705728249136002e-140 or 7.600193379940175e-155 < k < 1.1512157854309419e+132

   1. Initial program 54.8

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

   2. Simplified 43.8

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

   3. Taylor expanded around inf 18.5

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

   5. Applied add-sqr-sqrt 41.0

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

   6. Applied unpow-prod-down 41.0

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

   7. Applied times-frac 41.0

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

   8. Simplified 41.0

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

   9. Simplified 16.9

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

   11. Applied associate-*r/ 15.5

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

   12. Applied associate-*r/ 8.3

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

   14. Applied *-un-lft-identity 8.3

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

   15. Applied times-frac 8.0

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

   16. Applied unpow-prod-down 8.0

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

   17. Applied associate-*l* 3.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))