Toniolo and Linder, Equation (10+)

Average Error: 31.9 → 13.0

Time: 16.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{t}{\ell}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.0645803590118758 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{\left(t_1 \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \cdot \frac{2}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{t}{\ell} \cdot \sin k\\ \mathbf{if}\;t \leq -1276127.0504908424:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t_3\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq -3.3849052858131366 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{t_2}\\ \mathbf{elif}\;t \leq 3.6258818320847586 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(t_1 \cdot t_3\right)}}{t_2}\\ \end{array}\\ \end{array} \]

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 (* t (/ t l))) (t_2 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -1.0645803590118758e+126)
     (* (/ 1.0 (* (* t_1 (/ (* t (sin k)) l)) (tan k))) (/ 2.0 t_2))
     (let* ((t_3 (* (/ t l) (sin k))))
       (if (<= t -1276127.0504908424)
         (/ 2.0 (/ (* t_2 (* (sin k) (* (* t t) t_3))) (* l (cos k))))
         (if (<= t -3.3849052858131366e-93)
           (/ (/ 2.0 (* (tan k) (/ (/ (pow t 3.0) l) (/ l (sin k))))) t_2)
           (if (<= t 3.6258818320847586e-101)
             (/
              2.0
              (*
               (/ (pow (sin k) 2.0) (* l l))
               (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
             (/ (/ 2.0 (* (tan k) (* t_1 t_3))) t_2))))))))

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 = t * (t / l);
	double t_2 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -1.0645803590118758e+126) {
		tmp = (1.0 / ((t_1 * ((t * sin(k)) / l)) * tan(k))) * (2.0 / t_2);
	} else {
		double t_3 = (t / l) * sin(k);
		double tmp_1;
		if (t <= -1276127.0504908424) {
			tmp_1 = 2.0 / ((t_2 * (sin(k) * ((t * t) * t_3))) / (l * cos(k)));
		} else if (t <= -3.3849052858131366e-93) {
			tmp_1 = (2.0 / (tan(k) * ((pow(t, 3.0) / l) / (l / sin(k))))) / t_2;
		} else if (t <= 3.6258818320847586e-101) {
			tmp_1 = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
		} else {
			tmp_1 = (2.0 / (tan(k) * (t_1 * t_3))) / t_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 t < -1.0645803590118758e126

   1. Initial program 22.1

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

   2. Simplified 22.1

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

   4. Applied unpow3_binary64 22.1

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

   5. Applied times-frac_binary64 16.4

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

   6. Applied associate-*l*_binary64 16.1

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

   7. Simplified 16.1

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

   9. Applied *-un-lft-identity_binary64 16.1

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

   10. Applied times-frac_binary64 6.4

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

   11. Simplified 6.4

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

   13. Applied associate-*r/_binary64 6.4

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

   14. Simplified 6.4

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

   16. Applied *-un-lft-identity_binary64 6.4

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

   17. Applied times-frac_binary64 6.3

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

   if -1.0645803590118758e126 < t < -1276127.05049084243

   1. Initial program 20.9

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

   2. Simplified 20.9

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

   4. Applied unpow3_binary64 20.9

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

   5. Applied times-frac_binary64 12.9

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

   6. Applied associate-*l*_binary64 8.9

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

   7. Simplified 8.9

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

   9. Applied tan-quot_binary64 8.9

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

   10. Applied associate-*l/_binary64 8.9

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

   11. Applied frac-times_binary64 6.4

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

   12. Applied associate-*l/_binary64 6.4

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

   if -1276127.05049084243 < t < -3.384905285813137e-93

   1. Initial program 26.5

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

   2. Simplified 26.5

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

   4. Applied unpow3_binary64 26.6

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

   5. Applied times-frac_binary64 22.1

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

   6. Applied associate-*l*_binary64 18.1

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

   7. Simplified 18.1

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

   9. Applied *-un-lft-identity_binary64 18.1

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

   10. Applied times-frac_binary64 18.1

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

   11. Simplified 18.1

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

   13. Applied associate-/r*_binary64 18.1

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

   14. Simplified 18.0

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

   if -3.384905285813137e-93 < t < 3.6258818320847586e-101

   1. Initial program 61.8

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

   2. Simplified 61.8

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

   4. Applied unpow3_binary64 61.8

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

   5. Applied times-frac_binary64 53.0

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

   6. Applied associate-*l*_binary64 52.9

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

   7. Simplified 52.9

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

   8. Taylor expanded around 0 41.1

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

   9. Simplified 26.2

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

   if 3.6258818320847586e-101 < t

   1. Initial program 22.6

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

   2. Simplified 22.6

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

   4. Applied unpow3_binary64 22.6

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

   5. Applied times-frac_binary64 16.3

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

   6. Applied associate-*l*_binary64 14.2

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

   7. Simplified 14.2

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

   9. Applied *-un-lft-identity_binary64 14.2

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

   10. Applied times-frac_binary64 9.3

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

   11. Simplified 9.3

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

   13. Applied associate-/r*_binary64 9.3

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0645803590118758 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -1276127.0504908424:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq -3.3849052858131366 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.6258818320847586 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021196 
(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))))