Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 4.1

Time: 1.1min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{k}{\frac{\cos k}{\frac{t_1}{\ell}}}\\ t_3 := \sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\\ \mathbf{if}\;\ell \leq -7.412556353546589 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{t_3}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \frac{2}{k \cdot t}\\ \mathbf{if}\;\ell \leq 2.1536861594204514 \cdot 10^{-118}:\\ \;\;\;\;\frac{t_4}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{t_3}}}} \cdot \frac{\ell}{\frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := \frac{1}{\sqrt{\ell}}\\ \mathbf{if}\;\ell \leq 5.057396345476647 \cdot 10^{+255}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot t_5\right) \cdot \frac{t}{\sqrt{\ell}}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{t_5} \cdot \frac{\ell}{\frac{k}{\frac{\cos k}{\frac{t_1}{\sqrt{\ell}}}}}\\ \end{array}\\ \end{array}\\ \end{array} \]

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 (pow (sin k) 2.0))
        (t_2 (/ k (/ (cos k) (/ t_1 l))))
        (t_3 (* (cbrt l) (cbrt l))))
   (if (<= l -7.412556353546589e+58)
     (/
      (/ 2.0 (* (* k (/ (* (cbrt t) (cbrt t)) t_3)) (/ (cbrt t) (cbrt l))))
      t_2)
     (let* ((t_4 (/ 2.0 (* k t))))
       (if (<= l 2.1536861594204514e-118)
         (*
          (/ t_4 (/ (* (cbrt k) (cbrt k)) (/ 1.0 (/ (sin k) t_3))))
          (/ l (/ (cbrt k) (/ (cos k) (/ (sin k) (cbrt l))))))
         (let* ((t_5 (/ 1.0 (sqrt l))))
           (if (<= l 5.057396345476647e+255)
             (/ (/ 2.0 (* (* k t_5) (/ t (sqrt l)))) t_2)
             (* (/ t_4 t_5) (/ l (/ k (/ (cos k) (/ t_1 (sqrt l)))))))))))))

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 = pow(sin(k), 2.0);
	double t_2 = k / (cos(k) / (t_1 / l));
	double t_3 = cbrt(l) * cbrt(l);
	double tmp;
	if (l <= -7.412556353546589e+58) {
		tmp = (2.0 / ((k * ((cbrt(t) * cbrt(t)) / t_3)) * (cbrt(t) / cbrt(l)))) / t_2;
	} else {
		double t_4 = 2.0 / (k * t);
		double tmp_1;
		if (l <= 2.1536861594204514e-118) {
			tmp_1 = (t_4 / ((cbrt(k) * cbrt(k)) / (1.0 / (sin(k) / t_3)))) * (l / (cbrt(k) / (cos(k) / (sin(k) / cbrt(l)))));
		} else {
			double t_5 = 1.0 / sqrt(l);
			double tmp_2;
			if (l <= 5.057396345476647e+255) {
				tmp_2 = (2.0 / ((k * t_5) * (t / sqrt(l)))) / t_2;
			} else {
				tmp_2 = (t_4 / t_5) * (l / (k / (cos(k) / (t_1 / sqrt(l)))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -7.41255635354658943e58

   1. Initial program 55.9

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

   2. Simplified 52.9

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

   3. Taylor expanded in t around 0 46.0

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

   4. Simplified 46.0

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

   6. Applied associate-/l*_binary64 46.8

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

   7. Simplified 46.8

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}}}} \]
   8. Using strategy rm

   9. Applied times-frac_binary64 37.5

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

   10. Applied *-un-lft-identity_binary64 37.5

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}}}} \]

   11. Applied times-frac_binary64 37.5

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{\frac{t}{\ell}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   12. Applied times-frac_binary64 11.8

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{1}{\frac{t}{\ell}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   13. Applied associate-/r*_binary64 11.3

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{1}{\frac{t}{\ell}}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   14. Simplified 10.4

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

   16. Applied add-cube-cbrt_binary64 10.9

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

   17. Applied add-cube-cbrt_binary64 11.0

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

   18. Applied times-frac_binary64 11.0

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

   19. Applied associate-*r*_binary64 2.8

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

   if -7.41255635354658943e58 < l < 2.15368615942045137e-118

   1. Initial program 45.5

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

   2. Simplified 36.0

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

   3. Taylor expanded in t around 0 15.6

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

   4. Simplified 15.6

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

   6. Applied associate-/l*_binary64 14.5

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

   7. Simplified 14.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}}}} \]
   8. Using strategy rm

   9. Applied times-frac_binary64 9.2

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

   10. Applied *-un-lft-identity_binary64 9.2

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}}}} \]

   11. Applied times-frac_binary64 9.0

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{\frac{t}{\ell}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   12. Applied times-frac_binary64 7.1

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{1}{\frac{t}{\ell}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   13. Applied associate-/r*_binary64 6.8

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{1}{\frac{t}{\ell}}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   14. Simplified 6.7

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

   16. Applied add-cube-cbrt_binary64 7.0

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

   17. Applied unpow2_binary64 7.0

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

   18. Applied times-frac_binary64 6.3

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\frac{\cos k}{\color{blue}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sin k}{\sqrt[3]{\ell}}}}}} \]

   19. Applied *-un-lft-identity_binary64 6.3

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sin k}{\sqrt[3]{\ell}}}}} \]

   20. Applied times-frac_binary64 6.3

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\color{blue}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}}} \]

   21. Applied add-cube-cbrt_binary64 6.4

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{\color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \sqrt[3]{k}}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}} \]

   22. Applied times-frac_binary64 6.4

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\color{blue}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}}} \]

   23. Applied associate-*r/_binary64 5.2

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}} \]

   24. Applied associate-/r/_binary64 5.4

       \[\leadsto \frac{\color{blue}{\frac{2}{k \cdot t} \cdot \ell}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}} \]

   25. Applied times-frac_binary64 3.6

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot t}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \frac{\ell}{\frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}}} \]

   if 2.15368615942045137e-118 < l < 5.0573963454766469e255

   1. Initial program 48.9

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

   2. Simplified 41.3

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

   3. Taylor expanded in t around 0 22.7

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

   4. Simplified 22.7

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

   6. Applied associate-/l*_binary64 22.0

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

   7. Simplified 22.1

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}}}} \]
   8. Using strategy rm

   9. Applied times-frac_binary64 18.4

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

   10. Applied *-un-lft-identity_binary64 18.4

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}}}} \]

   11. Applied times-frac_binary64 18.4

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{\frac{t}{\ell}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   12. Applied times-frac_binary64 7.5

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{1}{\frac{t}{\ell}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   13. Applied associate-/r*_binary64 7.0

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{1}{\frac{t}{\ell}}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   14. Simplified 6.5

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

   16. Applied add-sqr-sqrt_binary64 6.5

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

   17. Applied *-un-lft-identity_binary64 6.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{\color{blue}{1 \cdot t}}{\sqrt{\ell} \cdot \sqrt{\ell}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}} \]

   18. Applied times-frac_binary64 6.5

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

   19. Applied associate-*r*_binary64 3.6

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

   if 5.0573963454766469e255 < l

   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

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

   3. Taylor expanded in t around 0 64.0

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

   4. Simplified 64.0

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

   6. Applied associate-/l*_binary64 64.0

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

   7. Simplified 64.0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \ell}}}}} \]
   8. Using strategy rm

   9. Applied times-frac_binary64 57.4

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

   10. Applied *-un-lft-identity_binary64 57.4

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}}}} \]

   11. Applied times-frac_binary64 57.4

       \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{\frac{t}{\ell}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   12. Applied times-frac_binary64 18.5

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{1}{\frac{t}{\ell}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   13. Applied associate-/r*_binary64 18.1

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{1}{\frac{t}{\ell}}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}} \]

   14. Simplified 17.4

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

   16. Applied add-sqr-sqrt_binary64 17.5

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

   17. Applied *-un-lft-identity_binary64 17.5

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

   18. Applied unpow-prod-down_binary64 17.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\frac{\cos k}{\frac{\color{blue}{{1}^{2} \cdot {\sin k}^{2}}}{\sqrt{\ell} \cdot \sqrt{\ell}}}}} \]

   19. Applied times-frac_binary64 17.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\frac{\cos k}{\color{blue}{\frac{{1}^{2}}{\sqrt{\ell}} \cdot \frac{{\sin k}^{2}}{\sqrt{\ell}}}}}} \]

   20. Applied *-un-lft-identity_binary64 17.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\frac{\color{blue}{1 \cdot \cos k}}{\frac{{1}^{2}}{\sqrt{\ell}} \cdot \frac{{\sin k}^{2}}{\sqrt{\ell}}}}} \]

   21. Applied times-frac_binary64 17.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{k}{\color{blue}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}}} \]

   22. Applied *-un-lft-identity_binary64 17.5

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\frac{\color{blue}{1 \cdot k}}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}} \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}} \]

   23. Applied times-frac_binary64 16.8

       \[\leadsto \frac{\frac{2}{k \cdot \frac{t}{\ell}}}{\color{blue}{\frac{1}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}}} \]

   24. Applied associate-*r/_binary64 24.6

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k \cdot t}{\ell}}}}{\frac{1}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}} \]

   25. Applied associate-/r/_binary64 24.6

       \[\leadsto \frac{\color{blue}{\frac{2}{k \cdot t} \cdot \ell}}{\frac{1}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}}} \cdot \frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}} \]

   26. Applied times-frac_binary64 31.3

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot t}}{\frac{1}{\frac{1}{\frac{{1}^{2}}{\sqrt{\ell}}}}} \cdot \frac{\ell}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}}} \]

   27. Simplified 31.3

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot t}}{\frac{1}{\sqrt{\ell}}}} \cdot \frac{\ell}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.412556353546589 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}\\ \mathbf{elif}\;\ell \leq 2.1536861594204514 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{2}{k \cdot t}}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\frac{1}{\frac{\sin k}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \frac{\ell}{\frac{\sqrt[3]{k}}{\frac{\cos k}{\frac{\sin k}{\sqrt[3]{\ell}}}}}\\ \mathbf{elif}\;\ell \leq 5.057396345476647 \cdot 10^{+255}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot \frac{1}{\sqrt{\ell}}\right) \cdot \frac{t}{\sqrt{\ell}}}}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot t}}{\frac{1}{\sqrt{\ell}}} \cdot \frac{\ell}{\frac{k}{\frac{\cos k}{\frac{{\sin k}^{2}}{\sqrt{\ell}}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))))