Toniolo and Linder, Equation (10+)

Average Error: 31.8 → 12.8

Time: 17.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \cos k\\ t_2 := \frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{t_1}}\\ \mathbf{if}\;k \leq -1.7335409660968685 \cdot 10^{-70}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \sin k \cdot \frac{t}{\ell}\\ t_4 := t \cdot t_3\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq 3.8487644440104676 \cdot 10^{-296}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(t_4 \cdot \tan k\right) \cdot t_5}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_6 := \frac{t}{\frac{\ell}{t}}\\ \mathbf{if}\;k \leq 1.7455086026036075 \cdot 10^{-246}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(\tan k \cdot \left(t_6 \cdot \frac{k \cdot t}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 4.375811370949229 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{t_5 \cdot \left(\sin k \cdot t_4\right)} \cdot \frac{2}{\frac{1}{t_1}}\\ \mathbf{elif}\;k \leq 4.4055434024518627 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t_5 \cdot \left(\sin k \cdot \left(t_3 \cdot t_6\right)\right)}\\ \end{array}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l t) (cos k)))
        (t_2
         (/
          2.0
          (/ (/ (* (pow (sin k) 2.0) (fma k k (* 2.0 (* t t)))) l) t_1))))
   (if (<= k -1.7335409660968685e-70)
     t_2
     (let* ((t_3 (* (sin k) (/ t l)))
            (t_4 (* t t_3))
            (t_5 (+ 2.0 (pow (/ k t) 2.0))))
       (if (<= k 3.8487644440104676e-296)
         (* (/ l t) (/ 2.0 (* (* t_4 (tan k)) t_5)))
         (let* ((t_6 (/ t (/ l t))))
           (if (<= k 1.7455086026036075e-246)
             (/ 2.0 (* t_5 (* (tan k) (* t_6 (/ (* k t) l)))))
             (if (<= k 4.375811370949229e+19)
               (* (/ 1.0 (* t_5 (* (sin k) t_4))) (/ 2.0 (/ 1.0 t_1)))
               (if (<= k 4.4055434024518627e+151)
                 t_2
                 (* (cos k) (/ 2.0 (* t_5 (* (sin k) (* t_3 t_6))))))))))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = (l / t) * cos(k);
	double t_2 = 2.0 / (((pow(sin(k), 2.0) * fma(k, k, (2.0 * (t * t)))) / l) / t_1);
	double tmp;
	if (k <= -1.7335409660968685e-70) {
		tmp = t_2;
	} else {
		double t_3 = sin(k) * (t / l);
		double t_4 = t * t_3;
		double t_5 = 2.0 + pow((k / t), 2.0);
		double tmp_1;
		if (k <= 3.8487644440104676e-296) {
			tmp_1 = (l / t) * (2.0 / ((t_4 * tan(k)) * t_5));
		} else {
			double t_6 = t / (l / t);
			double tmp_2;
			if (k <= 1.7455086026036075e-246) {
				tmp_2 = 2.0 / (t_5 * (tan(k) * (t_6 * ((k * t) / l))));
			} else if (k <= 4.375811370949229e+19) {
				tmp_2 = (1.0 / (t_5 * (sin(k) * t_4))) * (2.0 / (1.0 / t_1));
			} else if (k <= 4.4055434024518627e+151) {
				tmp_2 = t_2;
			} else {
				tmp_2 = cos(k) * (2.0 / (t_5 * (sin(k) * (t_3 * t_6))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if k < -1.7335409660968685e-70 or 43758113709492290000 < k < 4.4055434024518627e151

   1. Initial program 30.6

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

   2. Simplified 30.6

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

   3. Applied unpow3_binary64 30.6

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

   4. Applied times-frac_binary64 23.5

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

   5. Applied associate-*l*_binary64 23.4

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

   6. Applied associate-/l*_binary64 17.9

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

   7. Applied tan-quot_binary64 17.9

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

   8. Applied associate-*l/_binary64 17.9

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

   9. Applied frac-times_binary64 17.8

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

   10. Applied associate-*l/_binary64 15.7

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

   11. Taylor expanded in l around inf 11.6

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

   12. Simplified 11.6

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

   if -1.7335409660968685e-70 < k < 3.84876444401046764e-296

   1. Initial program 34.1

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

   2. Simplified 34.1

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

   3. Applied unpow3_binary64 34.1

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

   4. Applied times-frac_binary64 29.3

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

   5. Applied associate-*l*_binary64 22.3

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

   6. Applied associate-/l*_binary64 15.4

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

   7. Applied associate-*l/_binary64 12.1

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

   8. Applied associate-*l/_binary64 8.4

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

   9. Applied associate-*l/_binary64 8.4

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

   10. Applied associate-/r/_binary64 8.3

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

   if 3.84876444401046764e-296 < k < 1.74550860260360747e-246

   1. Initial program 40.7

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

   2. Simplified 40.7

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

   3. Applied unpow3_binary64 40.7

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

   4. Applied times-frac_binary64 39.9

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

   5. Applied associate-*l*_binary64 26.8

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

   6. Applied associate-/l*_binary64 23.6

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

   7. Taylor expanded in k around 0 25.4

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

   if 1.74550860260360747e-246 < k < 43758113709492290000

   1. Initial program 30.0

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

   2. Simplified 30.0

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

   3. Applied unpow3_binary64 30.0

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

   4. Applied times-frac_binary64 22.8

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

   5. Applied associate-*l*_binary64 19.6

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

   6. Applied associate-/l*_binary64 13.4

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

   7. Applied tan-quot_binary64 13.4

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

   8. Applied associate-*l/_binary64 12.0

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

   9. Applied frac-times_binary64 9.0

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

   10. Applied associate-*l/_binary64 8.4

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

   11. Applied div-inv_binary64 8.4

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

   12. Applied *-un-lft-identity_binary64 8.4

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

   13. Applied times-frac_binary64 8.2

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

   if 4.4055434024518627e151 < k

   1. Initial program 34.4

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

   2. Simplified 34.4

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

   3. Applied unpow3_binary64 34.4

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

   4. Applied times-frac_binary64 27.4

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

   5. Applied associate-*l*_binary64 27.4

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

   6. Applied associate-/l*_binary64 22.5

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

   7. Applied tan-quot_binary64 22.5

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

   8. Applied associate-*r/_binary64 22.5

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

   9. Applied associate-*l/_binary64 22.5

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

   10. Applied associate-/r/_binary64 22.5

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.7335409660968685 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 3.8487644440104676 \cdot 10^{-296}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.7455086026036075 \cdot 10^{-246}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k \cdot t}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 4.375811370949229 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)} \cdot \frac{2}{\frac{1}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 4.4055434024518627 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))