Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 8.9

Time: 25.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r87492 = 2.0;
        double r87493 = t;
        double r87494 = 3.0;
        double r87495 = pow(r87493, r87494);
        double r87496 = l;
        double r87497 = r87496 * r87496;
        double r87498 = r87495 / r87497;
        double r87499 = k;
        double r87500 = sin(r87499);
        double r87501 = r87498 * r87500;
        double r87502 = tan(r87499);
        double r87503 = r87501 * r87502;
        double r87504 = 1.0;
        double r87505 = r87499 / r87493;
        double r87506 = pow(r87505, r87492);
        double r87507 = r87504 + r87506;
        double r87508 = r87507 - r87504;
        double r87509 = r87503 * r87508;
        double r87510 = r87492 / r87509;
        return r87510;
}

↓

double f(double t, double l, double k) {
        double r87511 = k;
        double r87512 = -4.5772359455565565e+139;
        bool r87513 = r87511 <= r87512;
        double r87514 = -2.705728249136002e-140;
        bool r87515 = r87511 <= r87514;
        double r87516 = 7.600193379940175e-155;
        bool r87517 = r87511 <= r87516;
        double r87518 = 1.1512157854309419e+132;
        bool r87519 = r87511 <= r87518;
        double r87520 = !r87519;
        bool r87521 = r87517 || r87520;
        double r87522 = !r87521;
        bool r87523 = r87515 || r87522;
        double r87524 = !r87523;
        bool r87525 = r87513 || r87524;
        double r87526 = 2.0;
        double r87527 = 1.0;
        double r87528 = 2.0;
        double r87529 = r87526 / r87528;
        double r87530 = pow(r87511, r87529);
        double r87531 = t;
        double r87532 = 1.0;
        double r87533 = pow(r87531, r87532);
        double r87534 = r87530 * r87533;
        double r87535 = r87530 * r87534;
        double r87536 = r87527 / r87535;
        double r87537 = pow(r87536, r87532);
        double r87538 = cos(r87511);
        double r87539 = sin(r87511);
        double r87540 = fabs(r87539);
        double r87541 = r87538 / r87540;
        double r87542 = l;
        double r87543 = r87541 * r87542;
        double r87544 = r87537 * r87543;
        double r87545 = r87540 / r87542;
        double r87546 = r87544 / r87545;
        double r87547 = r87526 * r87546;
        double r87548 = cbrt(r87527);
        double r87549 = r87548 * r87548;
        double r87550 = pow(r87511, r87526);
        double r87551 = r87549 / r87550;
        double r87552 = pow(r87551, r87532);
        double r87553 = r87548 / r87533;
        double r87554 = pow(r87553, r87532);
        double r87555 = r87554 * r87543;
        double r87556 = r87552 * r87555;
        double r87557 = r87556 / r87545;
        double r87558 = r87526 * r87557;
        double r87559 = r87525 ? r87547 : r87558;
        return r87559;
}

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 26.4

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

   6. Applied times-frac 26.4

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

   7. Simplified 26.4

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

   8. Simplified 26.2

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

   10. Applied associate-*r/ 25.8

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

   11. Applied associate-*r/ 23.9

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

   13. Applied sqr-pow 23.9

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

   14. Applied associate-*l* 14.6

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

   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 18.5

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

   6. Applied times-frac 18.5

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

   7. Simplified 18.5

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

   8. Simplified 16.9

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

   10. Applied associate-*r/ 15.5

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

   11. Applied associate-*r/ 8.3

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

   13. Applied add-cube-cbrt 8.3

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

   14. Applied times-frac 8.0

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

   15. Applied unpow-prod-down 8.0

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

   16. Applied associate-*l* 3.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

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))))