Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 12.0

Time: 15.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.0421023210157143 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{elif}\;t \leq 6.4743981145055726 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \mathbf{elif}\;t \leq 2.4659146346821276 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}{\tan k}\right)\\ \mathbf{elif}\;t \leq 6.839250237191215 \cdot 10^{-165}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \end{array}\]

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 (<= t -1.0421023210157143e-243)
   (*
    (/ 1.0 (* t (* (/ t l) (sin k))))
    (* (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))) (/ (/ l t) (tan k))))
   (if (<= t 6.4743981145055726e-223)
     (/
      2.0
      (+
       (/ (* (* k k) (* t (pow (sin k) 2.0))) (* (* l l) (cos k)))
       (* 2.0 (/ (pow t 3.0) (/ (* l l) (/ (pow (sin k) 2.0) (cos k)))))))
     (if (<= t 2.4659146346821276e-207)
       (*
        (/
         (/ (* (cbrt l) (cbrt l)) (* (cbrt t) (cbrt t)))
         (* t (* (/ t l) (sin k))))
        (*
         (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))
         (/ (/ (cbrt l) (cbrt t)) (tan k))))
       (if (<= t 6.839250237191215e-165)
         (/
          2.0
          (*
           (/ (pow (sin k) 2.0) (cos k))
           (+ (/ (* t (* k k)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l l))))))
         (*
          (/ 1.0 (* t (* (/ t l) (sin k))))
          (* (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))) (/ (/ l t) (tan 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 (t <= -1.0421023210157143e-243) {
		tmp = (1.0 / (t * ((t / l) * sin(k)))) * ((2.0 / (2.0 + pow((k / t), 2.0))) * ((l / t) / tan(k)));
	} else if (t <= 6.4743981145055726e-223) {
		tmp = 2.0 / ((((k * k) * (t * pow(sin(k), 2.0))) / ((l * l) * cos(k))) + (2.0 * (pow(t, 3.0) / ((l * l) / (pow(sin(k), 2.0) / cos(k))))));
	} else if (t <= 2.4659146346821276e-207) {
		tmp = (((cbrt(l) * cbrt(l)) / (cbrt(t) * cbrt(t))) / (t * ((t / l) * sin(k)))) * ((2.0 / (2.0 + pow((k / t), 2.0))) * ((cbrt(l) / cbrt(t)) / tan(k)));
	} else if (t <= 6.839250237191215e-165) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((t * (k * k)) / (l * l)) + (2.0 * (pow(t, 3.0) / (l * l)))));
	} else {
		tmp = (1.0 / (t * ((t / l) * sin(k)))) * ((2.0 / (2.0 + pow((k / t), 2.0))) * ((l / t) / tan(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 t < -1.04210232101571429e-243 or 6.83925023719121535e-165 < t

   1. Initial program 28.7

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

   2. Simplified 28.7

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

   4. Applied cube-mult_binary64_449 28.7

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

   5. Applied times-frac_binary64_425 20.3

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

   6. Applied associate-*l*_binary64_360 18.6

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

   8. Applied *-un-lft-identity_binary64_419 18.6

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

   9. Applied times-frac_binary64_425 13.4

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

   10. Applied associate-*l*_binary64_360 12.4

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

   12. Applied *-un-lft-identity_binary64_419 12.4

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

   13. Applied times-frac_binary64_425 12.5

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

   14. Simplified 11.0

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

   16. Applied *-un-lft-identity_binary64_419 11.0

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

   17. Applied times-frac_binary64_425 9.9

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

   18. Applied associate-*l*_binary64_360 8.3

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

   19. Simplified 8.3

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

   if -1.04210232101571429e-243 < t < 6.47439811450557262e-223

   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. Using strategy rm

   4. Applied cube-mult_binary64_449 64.0

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

   5. Applied times-frac_binary64_425 64.0

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

   6. Applied associate-*l*_binary64_360 64.0

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

   8. Applied *-un-lft-identity_binary64_419 64.0

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

   9. Applied times-frac_binary64_425 57.4

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

   10. Applied associate-*l*_binary64_360 57.4

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

   11. Taylor expanded around inf 44.6

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

   12. Simplified 44.6

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

   if 6.47439811450557262e-223 < t < 2.4659146346821276e-207

   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. Using strategy rm

   4. Applied cube-mult_binary64_449 64.0

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

   5. Applied times-frac_binary64_425 64.0

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

   6. Applied associate-*l*_binary64_360 64.0

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

   8. Applied *-un-lft-identity_binary64_419 64.0

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

   9. Applied times-frac_binary64_425 51.2

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

   10. Applied associate-*l*_binary64_360 51.2

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

   12. Applied *-un-lft-identity_binary64_419 51.2

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

   13. Applied times-frac_binary64_425 52.6

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

   14. Simplified 53.6

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

   16. Applied add-cube-cbrt_binary64_454 53.6

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

   17. Applied add-cube-cbrt_binary64_454 53.6

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

   18. Applied times-frac_binary64_425 53.6

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

   19. Applied times-frac_binary64_425 52.7

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

   20. Applied associate-*l*_binary64_360 49.4

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

   21. Simplified 49.4

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

   if 2.4659146346821276e-207 < t < 6.83925023719121535e-165

   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 around inf 37.3

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

   4. Simplified 35.3

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0421023210157143 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{elif}\;t \leq 6.4743981145055726 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \mathbf{elif}\;t \leq 2.4659146346821276 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}{\tan k}\right)\\ \mathbf{elif}\;t \leq 6.839250237191215 \cdot 10^{-165}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020352 
(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))))