Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 9.4

Time: 12.2min

Precision: binary64

Cost: 40962

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -7.590115960093544 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}}}\\ \mathbf{elif}\;t \leq 4.9188446717816575 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}}{\ell}}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \frac{\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}{\sqrt[3]{\frac{\ell}{t}}}}\\ \end{array}\]

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 (<= t -7.590115960093544e-49)
   (/
    2.0
    (/
     1.0
     (/
      (/ l t)
      (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (* (/ t l) (sin k))))))))
   (if (<= t 4.9188446717816575e-52)
     (/
      2.0
      (/
       (/
        (+
         (/ (* (pow (sin k) 2.0) (* k k)) (cos k))
         (* 2.0 (/ (* (pow (sin k) 2.0) (* t t)) (cos k))))
        l)
       (/ l t)))
     (/
      2.0
      (*
       (/ (+ 2.0 (pow (/ k t) 2.0)) (* (cbrt (/ l t)) (cbrt (/ l t))))
       (/ (* (tan k) (* t (* (/ t l) (sin k)))) (cbrt (/ 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 (t <= -7.590115960093544e-49) {
		tmp = 2.0 / (1.0 / ((l / t) / ((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * ((t / l) * sin(k)))))));
	} else if (t <= 4.9188446717816575e-52) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * (k * k)) / cos(k)) + (2.0 * ((pow(sin(k), 2.0) * (t * t)) / cos(k)))) / l) / (l / t));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) / (cbrt(l / t) * cbrt(l / t))) * ((tan(k) * (t * ((t / l) * sin(k)))) / cbrt(l / t)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Alternatives

Alternative 1
Error	9.4
Cost	40456

\[\begin{array}{l} \mathbf{if}\;t \leq -2.290681488355177 \cdot 10^{-49} \lor \neg \left(t \leq 1.2217593768261278 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}}{\ell}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 2
Error	9.4
Cost	21064

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3146038142536228 \cdot 10^{-69} \lor \neg \left(t \leq 6.010803212440071 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 3
Error	9.4
Cost	21250

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6282454218159167 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 2.7346199221904777 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 4
Error	9.4
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -2.277556820959288 \cdot 10^{-66} \lor \neg \left(t \leq 3.159270465779861 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 5
Error	11.1
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2842414582231446 \cdot 10^{-63} \lor \neg \left(t \leq 1.3104733160998375 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 6
Error	12.4
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -9.73272946386767 \cdot 10^{-50} \lor \neg \left(t \leq 2.916613012300213 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{t \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 7
Error	15.3
Cost	20552

\[\begin{array}{l} \mathbf{if}\;t \leq -0.45837407220548676 \lor \neg \left(t \leq 2.127976288491358 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 8
Error	16.9
Cost	20552

\[\begin{array}{l} \mathbf{if}\;k \leq -4.880209785807149 \cdot 10^{+59} \lor \neg \left(k \leq 26.46224254970775\right):\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Alternative 9
Error	18.8
Cost	14208

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

Alternative 10
Error	23.7
Cost	14530

\[\begin{array}{l} \mathbf{if}\;t \leq -3.496607597689733 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.786932960485934 \cdot 10^{-57}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{{t}^{1.5}}} \cdot \frac{k}{\frac{\ell}{{t}^{1.5}}}\right)}\\ \end{array}\]

Alternative 11
Error	24.0
Cost	1602

\[\begin{array}{l} \mathbf{if}\;t \leq -1.354451279343827 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.554858491562186 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array}\]

Alternative 12
Error	23.6
Cost	1288

\[\begin{array}{l} \mathbf{if}\;t \leq -1.168471794670234 \cdot 10^{-69} \lor \neg \left(t \leq 1.0700984684013482 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{1}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 13
Error	25.0
Cost	1602

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6903813489031153 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 3.2805991925196846 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \frac{k \cdot \frac{t}{\ell}}{\ell}}\\ \end{array}\]

Alternative 14
Error	25.0
Cost	1288

\[\begin{array}{l} \mathbf{if}\;t \leq -5.554707144313222 \cdot 10^{-68} \lor \neg \left(t \leq 3.2805991925196846 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{1}{\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 15
Error	30.6
Cost	64

\[0\]

Alternative 16
Error	61.9
Cost	64

\[-1\]

Derivation

1. Split input into 3 regimes

2. if t < -7.59011596009354375e-49

   1. Initial program 22.4

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

   3. Applied unpow3_binary64_826 22.4

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

   4. Applied times-frac_binary64_766 15.8

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

   5. Applied associate-*l*_binary64_701 13.3

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

   7. Applied associate-/l*_binary64_705 7.6

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

   9. Applied associate-*l/_binary64_703 6.5

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

   10. Applied associate-*l/_binary64_703 4.7

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

   11. Applied associate-*l/_binary64_703 4.2

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

   12. Simplified 4.2

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

   14. Applied clear-num_binary64_759 4.2

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

   15. Simplified 4.2

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

   if -7.59011596009354375e-49 < t < 4.91884467178165751e-52

   1. Initial program 55.0

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

   3. Applied unpow3_binary64_826 55.0

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

   4. Applied times-frac_binary64_766 45.6

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

   5. Applied associate-*l*_binary64_701 45.0

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

   7. Applied associate-/l*_binary64_705 38.3

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

   9. Applied associate-*l/_binary64_703 38.3

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

   10. Applied associate-*l/_binary64_703 39.7

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

   11. Applied associate-*l/_binary64_703 36.2

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

   12. Simplified 36.2

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

   13. Taylor expanded around inf 22.2

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

   14. Simplified 22.2

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

   15. Simplified 22.2

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

   if 4.91884467178165751e-52 < t

   1. Initial program 23.2

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

   3. Applied unpow3_binary64_826 23.2

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

   4. Applied times-frac_binary64_766 16.8

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

   5. Applied associate-*l*_binary64_701 14.5

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

   7. Applied associate-/l*_binary64_705 8.5

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

   9. Applied associate-*l/_binary64_703 7.4

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

   10. Applied associate-*l/_binary64_703 4.6

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

   11. Applied associate-*l/_binary64_703 4.1

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

   12. Simplified 4.1

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

   14. Applied add-cube-cbrt_binary64_795 4.5

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

   15. Applied times-frac_binary64_766 4.3

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

   16. Simplified 4.3

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.590115960093544 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)}}}\\ \mathbf{elif}\;t \leq 4.9188446717816575 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot t\right)}{\cos k}}{\ell}}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \frac{\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}{\sqrt[3]{\frac{\ell}{t}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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))))