Toniolo and Linder, Equation (10+)

Average Error: 33.2 → 11.8

Time: 25.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\frac{\ell}{t}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k}\\ t_3 := \frac{2}{t_1}\\ \mathbf{if}\;t \leq -6.387471119151526 \cdot 10^{+136}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{elif}\;t \leq -3.1111375128457297 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 1.7556874143821332 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 24609664.9974396:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.916775720671396 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\tan k \cdot \left(t \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)} \cdot \left(\ell \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{t_1}\\ \end{array} \]

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 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ (/ l t) (* (* t (/ (* t (sin k)) l)) (tan k))))
        (t_3 (/ 2.0 t_1)))
   (if (<= t -6.387471119151526e+136)
     (* t_2 t_3)
     (if (<= t -3.1111375128457297e-148)
       (/ 2.0 (* (* (/ t l) (* (sin k) (/ (* t t) l))) (* (tan k) t_1)))
       (if (<= t 1.7556874143821332e-146)
         (/
          2.0
          (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (cos k) (pow l 2.0))))
         (if (<= t 24609664.9974396)
           (/ 2.0 (* t_1 (* (tan k) (* (/ t l) (* t (* t (/ (sin k) l)))))))
           (if (<= t 2.916775720671396e+152)
             (* (/ 1.0 (* (tan k) (* t (* t (* (sin k) (/ t l)))))) (* l t_3))
             (* 2.0 (/ t_2 t_1)))))))))

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 = 2.0 + pow((k / t), 2.0);
	double t_2 = (l / t) / ((t * ((t * sin(k)) / l)) * tan(k));
	double t_3 = 2.0 / t_1;
	double tmp;
	if (t <= -6.387471119151526e+136) {
		tmp = t_2 * t_3;
	} else if (t <= -3.1111375128457297e-148) {
		tmp = 2.0 / (((t / l) * (sin(k) * ((t * t) / l))) * (tan(k) * t_1));
	} else if (t <= 1.7556874143821332e-146) {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)));
	} else if (t <= 24609664.9974396) {
		tmp = 2.0 / (t_1 * (tan(k) * ((t / l) * (t * (t * (sin(k) / l))))));
	} else if (t <= 2.916775720671396e+152) {
		tmp = (1.0 / (tan(k) * (t * (t * (sin(k) * (t / l)))))) * (l * t_3);
	} else {
		tmp = 2.0 * (t_2 / t_1);
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 6 regimes

2. if t < -6.3874711191515265e136

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   3. Applied cube-mult_binary64 23.0

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

   4. Applied times-frac_binary64 17.0

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

   5. Applied associate-*l*_binary64 16.7

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

   6. Applied *-un-lft-identity_binary64 16.7

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

   7. Applied times-frac_binary64 6.0

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

   8. Applied associate-*l*_binary64 3.9

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

   9. Applied *-un-lft-identity_binary64 3.9

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

   10. Applied times-frac_binary64 3.9

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

   11. Simplified 0.4

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

   if -6.3874711191515265e136 < t < -3.11113751284572967e-148

   1. Initial program 28.8

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

   2. Simplified 28.8

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

   3. Applied cube-mult_binary64 28.8

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

   4. Applied times-frac_binary64 18.5

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

   5. Applied associate-*l*_binary64 15.3

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

   6. Applied associate-*l*_binary64 15.2

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

   if -3.11113751284572967e-148 < t < 1.7556874143821332e-146

   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

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

   3. Taylor expanded in t around 0 29.1

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

   if 1.7556874143821332e-146 < t < 24609664.997439601

   1. Initial program 36.3

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

   2. Simplified 36.3

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

   3. Applied cube-mult_binary64 36.4

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

   4. Applied times-frac_binary64 25.7

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

   5. Applied associate-*l*_binary64 22.9

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

   6. Applied *-un-lft-identity_binary64 22.9

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

   7. Applied times-frac_binary64 22.9

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

   8. Applied associate-*l*_binary64 22.9

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

   9. Applied div-inv_binary64 22.9

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

   10. Applied associate-*l*_binary64 22.9

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

   11. Simplified 22.9

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

   if 24609664.997439601 < t < 2.91677572067139585e152

   1. Initial program 25.1

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

   2. Simplified 25.1

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

   3. Applied cube-mult_binary64 25.1

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

   4. Applied times-frac_binary64 13.2

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

   5. Applied associate-*l*_binary64 9.1

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

   6. Applied *-un-lft-identity_binary64 9.1

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

   7. Applied times-frac_binary64 9.1

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

   8. Applied associate-*l*_binary64 8.9

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

   9. Applied *-un-lft-identity_binary64 8.9

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

   10. Applied times-frac_binary64 8.8

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

   11. Applied associate-*l/_binary64 8.8

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

   12. Applied frac-times_binary64 9.2

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

   13. Applied associate-*l/_binary64 6.3

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

   14. Applied associate-/r/_binary64 6.1

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

   15. Applied associate-*l*_binary64 5.3

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

   16. Simplified 5.3

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

   if 2.91677572067139585e152 < 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. Applied cube-mult_binary64 22.6

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

   4. Applied times-frac_binary64 18.1

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

   5. Applied associate-*l*_binary64 18.1

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

   6. Applied *-un-lft-identity_binary64 18.1

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

   7. Applied times-frac_binary64 6.7

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

   8. Applied associate-*l*_binary64 4.2

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

   9. Applied div-inv_binary64 4.2

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

   10. Simplified 0.4

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.387471119151526 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -3.1111375128457297 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.7556874143821332 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 24609664.9974396:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.916775720671396 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\tan k \cdot \left(t \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)} \cdot \left(\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021225 
(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))))