Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 11.0

Time: 13.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -1.5767169438438566 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 6.597454098759182 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.1117120927669026 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 3.8305734146775315 \cdot 10^{+227}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\cos k}\right)}\\ \end{array}\]

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 -1.5767169438438566e-152)
   (/
    2.0
    (/
     (+
      (* 2.0 (* t (* (pow (sin k) 2.0) (/ t l))))
      (/ (* (pow (sin k) 2.0) (* k k)) l))
     (* (/ l t) (cos k))))
   (if (<= k 6.597454098759182e-81)
     (/
      2.0
      (*
       (* (* t (* (/ t l) (* (sin k) (/ t l)))) (tan k))
       (+ 2.0 (pow (/ k t) 2.0))))
     (if (<= k 1.1117120927669026e+194)
       (/
        2.0
        (/
         (+
          (* 2.0 (* t (* (pow (sin k) 2.0) (/ t l))))
          (/ (* (pow (sin k) 2.0) (* k k)) l))
         (* (/ l t) (cos k))))
       (if (<= k 3.8305734146775315e+227)
         (/
          2.0
          (*
           (/ (pow (sin k) 2.0) (cos k))
           (+ (/ (* k (* k t)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l l))))))
         (/
          2.0
          (*
           t
           (*
            (* (/ t l) (* (pow (sin k) 2.0) (/ t l)))
            (/ (+ 2.0 (pow (/ k t) 2.0)) (cos k))))))))))

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 <= -1.5767169438438566e-152) {
		tmp = 2.0 / (((2.0 * (t * (pow(sin(k), 2.0) * (t / l)))) + ((pow(sin(k), 2.0) * (k * k)) / l)) / ((l / t) * cos(k)));
	} else if (k <= 6.597454098759182e-81) {
		tmp = 2.0 / (((t * ((t / l) * (sin(k) * (t / l)))) * tan(k)) * (2.0 + pow((k / t), 2.0)));
	} else if (k <= 1.1117120927669026e+194) {
		tmp = 2.0 / (((2.0 * (t * (pow(sin(k), 2.0) * (t / l)))) + ((pow(sin(k), 2.0) * (k * k)) / l)) / ((l / t) * cos(k)));
	} else if (k <= 3.8305734146775315e+227) {
		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 / (t * (((t / l) * (pow(sin(k), 2.0) * (t / l))) * ((2.0 + pow((k / t), 2.0)) / cos(k))));
	}
	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 < -1.5767169438438566e-152 or 6.5974540987591818e-81 < k < 1.11171209276690263e194

   1. Initial program 31.4

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

   2. Simplified 31.4

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

   4. Applied unpow3_binary64_144 31.4

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

   5. Applied times-frac_binary64_84 23.5

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

   6. Applied associate-*l*_binary64_19 23.0

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

   8. Applied associate-/l*_binary64_23 17.0

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

   10. Applied tan-quot_binary64_237 17.0

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

   11. Applied associate-*l/_binary64_21 17.0

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

   12. Applied frac-times_binary64_88 16.3

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

   13. Applied associate-*l/_binary64_21 14.5

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

   14. Simplified 14.5

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

   15. Taylor expanded around inf 12.4

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

   16. Simplified 9.1

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

   if -1.5767169438438566e-152 < k < 6.5974540987591818e-81

   1. Initial program 36.8

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

   2. Simplified 36.8

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

   4. Applied unpow3_binary64_144 36.8

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

   5. Applied times-frac_binary64_84 31.8

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

   6. Applied associate-*l*_binary64_19 23.6

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

   8. Applied associate-/l*_binary64_23 16.3

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

   10. Applied div-inv_binary64_75 16.3

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

   11. Applied associate-*l*_binary64_19 13.0

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

   12. Simplified 13.0

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

   if 1.11171209276690263e194 < k < 3.8305734146775315e227

   1. Initial program 30.9

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

   2. Simplified 30.9

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

   4. Applied unpow3_binary64_144 30.9

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

   5. Applied times-frac_binary64_84 24.8

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

   6. Applied associate-*l*_binary64_19 24.8

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

   8. Applied associate-/l*_binary64_23 19.1

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

   10. Applied tan-quot_binary64_237 19.2

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

   11. Applied associate-*l/_binary64_21 19.2

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

   12. Applied frac-times_binary64_88 19.2

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

   13. Applied associate-*l/_binary64_21 17.3

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

   14. Simplified 17.3

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

   15. Taylor expanded around inf 27.4

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

   16. Simplified 21.8

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

   if 3.8305734146775315e227 < k

   1. Initial program 31.3

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

   2. Simplified 31.3

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

   4. Applied unpow3_binary64_144 31.3

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

   5. Applied times-frac_binary64_84 25.8

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

   6. Applied associate-*l*_binary64_19 25.8

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

   8. Applied associate-/l*_binary64_23 21.0

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

   10. Applied tan-quot_binary64_237 21.0

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

   11. Applied associate-*l/_binary64_21 21.0

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

   12. Applied frac-times_binary64_88 20.9

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

   13. Applied associate-*l/_binary64_21 19.5

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

   14. Simplified 19.5

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

   16. Applied associate-*l/_binary64_21 19.5

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

   17. Applied associate-/r/_binary64_24 19.5

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

   18. Simplified 17.3

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.5767169438438566 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 6.597454098759182 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.1117120927669026 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 3.8305734146775315 \cdot 10^{+227}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\cos k}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(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))))