Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 11.3

Time: 13.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -2.1807370113174654 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \frac{1}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq -3.568572040991893 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 5.138813199555828 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 5.388805501647768 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 7.433691841824543 \cdot 10^{+192}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \frac{1}{\cos k \cdot \frac{\ell}{t}}}\\ \end{array}\]

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
 (if (<= k -2.1807370113174654e+141)
   (/
    2.0
    (*
     (* (+ 2.0 (pow (/ k t) 2.0)) (* (sin k) (* t (* (sin k) (/ t l)))))
     (/ 1.0 (* (cos k) (/ l t)))))
   (if (<= k -3.568572040991893e-152)
     (/
      2.0
      (/
       (+
        (* 2.0 (* t (* (/ t l) (pow (sin k) 2.0))))
        (/ (* (pow (sin k) 2.0) (* k k)) l))
       (* (cos k) (/ l t))))
     (if (<= k 5.138813199555828e-131)
       (/
        2.0
        (*
         (+ 2.0 (pow (/ k t) 2.0))
         (* (* t (* (/ t l) (* (sin k) (/ t l)))) (tan k))))
       (if (<= k 5.388805501647768e+143)
         (/
          2.0
          (/
           (+
            (* 2.0 (* t (* (/ t l) (pow (sin k) 2.0))))
            (/ (* (pow (sin k) 2.0) (* k k)) l))
           (* (cos k) (/ l t))))
         (if (<= k 7.433691841824543e+192)
           (/
            2.0
            (*
             (/ (pow (sin k) 2.0) (cos k))
             (+ (/ (* k (* k t)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l l))))))
           (/
            2.0
            (*
             (*
              (+ 2.0 (pow (/ k t) 2.0))
              (* (sin k) (* t (* (sin k) (/ t l)))))
             (/ 1.0 (* (cos k) (/ l t)))))))))))

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 tmp;
	if (k <= -2.1807370113174654e+141) {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (sin(k) * (t * (sin(k) * (t / l))))) * (1.0 / (cos(k) * (l / t))));
	} else if (k <= -3.568572040991893e-152) {
		tmp = 2.0 / (((2.0 * (t * ((t / l) * pow(sin(k), 2.0)))) + ((pow(sin(k), 2.0) * (k * k)) / l)) / (cos(k) * (l / t)));
	} else if (k <= 5.138813199555828e-131) {
		tmp = 2.0 / ((2.0 + pow((k / t), 2.0)) * ((t * ((t / l) * (sin(k) * (t / l)))) * tan(k)));
	} else if (k <= 5.388805501647768e+143) {
		tmp = 2.0 / (((2.0 * (t * ((t / l) * pow(sin(k), 2.0)))) + ((pow(sin(k), 2.0) * (k * k)) / l)) / (cos(k) * (l / t)));
	} else if (k <= 7.433691841824543e+192) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((k * (k * t)) / (l * l)) + (2.0 * (pow(t, 3.0) / (l * l)))));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (sin(k) * (t * (sin(k) * (t / l))))) * (1.0 / (cos(k) * (l / t))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if k < -2.1807370113174654e141 or 7.43369184182454322e192 < k

   1. Initial program 33.7

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

   2. Simplified 33.7

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

   4. Applied unpow3_binary64 33.7

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

   5. Applied times-frac_binary64 27.0

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

   6. Applied associate-*l*_binary64 27.0

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

   8. Applied associate-/l*_binary64 21.7

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

   10. Applied tan-quot_binary64 21.7

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

   11. Applied associate-*l/_binary64 21.7

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

   12. Applied frac-times_binary64 21.7

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

   13. Applied associate-*l/_binary64 20.0

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

   14. Simplified 20.0

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

   16. Applied div-inv_binary64 20.0

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

   17. Simplified 20.0

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

   if -2.1807370113174654e141 < k < -3.5685720409918931e-152 or 5.13881319955582764e-131 < k < 5.38880550164776784e143

   1. Initial program 30.3

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

   2. Simplified 30.3

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

   4. Applied unpow3_binary64 30.3

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

   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 21.3

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

   8. Applied associate-/l*_binary64 14.6

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

   10. Applied tan-quot_binary64 14.6

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

   11. Applied associate-*l/_binary64 14.4

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

   12. Applied frac-times_binary64 13.5

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

   13. Applied associate-*l/_binary64 11.7

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

   14. Simplified 11.7

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

   15. Taylor expanded around inf 9.9

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

   16. Simplified 3.9

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

   if -3.5685720409918931e-152 < k < 5.13881319955582764e-131

   1. Initial program 38.3

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

   2. Simplified 38.3

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

   4. Applied unpow3_binary64 38.3

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

   5. Applied times-frac_binary64 33.8

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

   6. Applied associate-*l*_binary64 24.1

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

   8. Applied associate-/l*_binary64 18.1

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

   10. Applied div-inv_binary64 18.2

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

   11. Applied associate-*l*_binary64 13.5

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

   12. Simplified 13.5

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

   if 5.38880550164776784e143 < k < 7.43369184182454322e192

   1. Initial program 31.2

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

   2. Simplified 31.2

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

   4. Applied unpow3_binary64 31.2

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

   5. Applied times-frac_binary64 24.1

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

   6. Applied associate-*l*_binary64 24.1

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

   8. Applied associate-/l*_binary64 18.5

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

   10. Applied tan-quot_binary64 18.5

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

   11. Applied associate-*l/_binary64 18.5

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

   12. Applied frac-times_binary64 18.5

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

   13. Applied associate-*l/_binary64 16.5

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

   14. Simplified 16.5

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

   15. Taylor expanded around inf 30.0

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

   16. Simplified 22.4

       \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.1807370113174654 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \frac{1}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq -3.568572040991893 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 5.138813199555828 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 5.388805501647768 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 7.433691841824543 \cdot 10^{+192}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \frac{1}{\cos k \cdot \frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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))))