Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 13.5

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -7.73053853034813306892850214854505772544 \cdot 10^{-163}:\\ \;\;\;\;\left(\left({\left(\left(\sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}} \cdot \sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}\right) \cdot \sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \le 1.026944545197601870162749096367448505013 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{{\left(\sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{1}{\ell} \cdot \frac{{t}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r110527 = 2.0;
        double r110528 = t;
        double r110529 = 3.0;
        double r110530 = pow(r110528, r110529);
        double r110531 = l;
        double r110532 = r110531 * r110531;
        double r110533 = r110530 / r110532;
        double r110534 = k;
        double r110535 = sin(r110534);
        double r110536 = r110533 * r110535;
        double r110537 = tan(r110534);
        double r110538 = r110536 * r110537;
        double r110539 = 1.0;
        double r110540 = r110534 / r110528;
        double r110541 = pow(r110540, r110527);
        double r110542 = r110539 + r110541;
        double r110543 = r110542 - r110539;
        double r110544 = r110538 * r110543;
        double r110545 = r110527 / r110544;
        return r110545;
}

↓

double f(double t, double l, double k) {
        double r110546 = l;
        double r110547 = -7.341974747858959e+153;
        bool r110548 = r110546 <= r110547;
        double r110549 = 2.0;
        double r110550 = t;
        double r110551 = cbrt(r110550);
        double r110552 = r110551 * r110551;
        double r110553 = 3.0;
        double r110554 = pow(r110552, r110553);
        double r110555 = r110554 / r110546;
        double r110556 = pow(r110551, r110553);
        double r110557 = r110556 / r110546;
        double r110558 = k;
        double r110559 = sin(r110558);
        double r110560 = r110557 * r110559;
        double r110561 = r110555 * r110560;
        double r110562 = tan(r110558);
        double r110563 = r110561 * r110562;
        double r110564 = r110549 / r110563;
        double r110565 = r110558 / r110550;
        double r110566 = pow(r110565, r110549);
        double r110567 = r110564 / r110566;
        double r110568 = -7.730538530348133e-163;
        bool r110569 = r110546 <= r110568;
        double r110570 = 1.0;
        double r110571 = 2.0;
        double r110572 = r110549 / r110571;
        double r110573 = pow(r110558, r110572);
        double r110574 = 1.0;
        double r110575 = pow(r110550, r110574);
        double r110576 = r110573 * r110575;
        double r110577 = r110570 / r110576;
        double r110578 = cbrt(r110577);
        double r110579 = r110578 * r110578;
        double r110580 = r110579 * r110578;
        double r110581 = pow(r110580, r110574);
        double r110582 = cos(r110558);
        double r110583 = pow(r110546, r110571);
        double r110584 = r110582 * r110583;
        double r110585 = pow(r110559, r110571);
        double r110586 = r110584 / r110585;
        double r110587 = r110581 * r110586;
        double r110588 = r110570 / r110573;
        double r110589 = pow(r110588, r110574);
        double r110590 = r110587 * r110589;
        double r110591 = r110590 * r110549;
        double r110592 = 1.0269445451976019e-127;
        bool r110593 = r110546 <= r110592;
        double r110594 = r110573 * r110576;
        double r110595 = r110570 / r110594;
        double r110596 = pow(r110595, r110574);
        double r110597 = r110546 / r110558;
        double r110598 = -0.16666666666666666;
        double r110599 = r110583 * r110598;
        double r110600 = fma(r110597, r110597, r110599);
        double r110601 = r110596 * r110600;
        double r110602 = r110549 * r110601;
        double r110603 = 4.1624864010972877e+148;
        bool r110604 = r110546 <= r110603;
        double r110605 = pow(r110577, r110574);
        double r110606 = r110605 * r110584;
        double r110607 = r110606 * r110589;
        double r110608 = r110607 / r110585;
        double r110609 = r110549 * r110608;
        double r110610 = r110570 / r110546;
        double r110611 = pow(r110550, r110553);
        double r110612 = r110611 / r110546;
        double r110613 = r110610 * r110612;
        double r110614 = r110613 * r110559;
        double r110615 = r110614 * r110562;
        double r110616 = r110549 / r110615;
        double r110617 = r110616 / r110566;
        double r110618 = r110604 ? r110609 : r110617;
        double r110619 = r110593 ? r110602 : r110618;
        double r110620 = r110569 ? r110591 : r110619;
        double r110621 = r110548 ? r110567 : r110620;
        return r110621;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if l < -7.341974747858959e+153

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied add-cube-cbrt 64.0

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

   5. Applied unpow-prod-down 64.0

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

   6. Applied times-frac 50.5

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

   7. Applied associate-*l* 50.5

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

   if -7.341974747858959e+153 < l < -7.730538530348133e-163

   1. Initial program 44.4

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

   2. Simplified 35.5

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

   3. Taylor expanded around inf 10.3

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

   5. Applied sqr-pow 10.3

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

   6. Applied associate-*l* 6.6

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

   8. Applied add-sqr-sqrt 6.6

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

   9. Applied times-frac 6.4

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

   10. Applied unpow-prod-down 6.4

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

   11. Applied associate-*l* 3.8

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

   12. Simplified 3.8

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

   14. Applied add-cube-cbrt 4.1

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

   if -7.730538530348133e-163 < l < 1.0269445451976019e-127

   1. Initial program 46.1

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

   2. Simplified 37.2

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

   3. Taylor expanded around inf 18.1

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

   5. Applied sqr-pow 18.1

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

   6. Applied associate-*l* 18.1

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

   7. Taylor expanded around 0 18.7

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} - \frac{1}{6} \cdot {\ell}^{2}\right)}\right)\]

   8. Simplified 9.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)}\right)\]

   if 1.0269445451976019e-127 < l < 4.1624864010972877e+148

   1. Initial program 44.5

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

   2. Simplified 35.2

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

   3. Taylor expanded around inf 11.8

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

   5. Applied sqr-pow 11.8

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

   6. Applied associate-*l* 7.0

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

   8. Applied add-sqr-sqrt 7.0

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

   9. Applied times-frac 6.8

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

   10. Applied unpow-prod-down 6.8

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

   11. Applied associate-*l* 3.5

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

   12. Simplified 3.5

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

   14. Applied associate-*r/ 4.0

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

   15. Applied associate-*r/ 4.0

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

   16. Simplified 4.0

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

   if 4.1624864010972877e+148 < l

   1. Initial program 63.6

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

   2. Simplified 63.3

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

   4. Applied *-un-lft-identity 63.3

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

   5. Applied unpow-prod-down 63.3

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

   6. Applied times-frac 52.0

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

   7. Simplified 52.0

      \[\leadsto \frac{\frac{2}{\left(\left(\color{blue}{\frac{1}{\ell}} \cdot \frac{{t}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le -7.73053853034813306892850214854505772544 \cdot 10^{-163}:\\ \;\;\;\;\left(\left({\left(\left(\sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}} \cdot \sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}\right) \cdot \sqrt[3]{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \le 1.026944545197601870162749096367448505013 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{{\left(\sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{1}{\ell} \cdot \frac{{t}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))))