Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 9.5

Time: 6.6min

Precision: binary64

Cost: 46792

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.158281865103833 \cdot 10^{-96} \lor \neg \left(t \leq 4.743251944033027 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{\cos k} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\ell \cdot \ell}}\\ \end{array}\]

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 (or (<= t -1.158281865103833e-96) (not (<= t 4.743251944033027e-121)))
   (*
    (/ 1.0 (* t (* (/ t l) (sin k))))
    (* (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))) (/ (/ l t) (tan k))))
   (/
    2.0
    (/
     (+
      (* 2.0 (/ (* (pow t 3.0) (pow (sin k) 2.0)) (cos k)))
      (/ (* (* k k) (* t (pow (sin k) 2.0))) (cos k)))
     (* l 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 tmp;
	if ((t <= -1.158281865103833e-96) || !(t <= 4.743251944033027e-121)) {
		tmp = (1.0 / (t * ((t / l) * sin(k)))) * ((2.0 / (2.0 + pow((k / t), 2.0))) * ((l / t) / tan(k)));
	} else {
		tmp = 2.0 / (((2.0 * ((pow(t, 3.0) * pow(sin(k), 2.0)) / cos(k))) + (((k * k) * (t * pow(sin(k), 2.0))) / cos(k))) / (l * l));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Alternatives

Alternative 1
Error	9.5
Cost	21064

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5155131008313367 \cdot 10^{-97} \lor \neg \left(t \leq 1.019490745297331 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 2
Error	10.4
Cost	21250

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6667884751615243 \cdot 10^{-88}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}\\ \mathbf{elif}\;t \leq 6.0002321974644175 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}\\ \end{array}\]

Alternative 3
Error	10.3
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9578879993731398 \cdot 10^{-88} \lor \neg \left(t \leq 4.180345886249698 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 4
Error	11.2
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7872461029704355 \cdot 10^{-88} \lor \neg \left(t \leq 1.5942796493750423 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 5
Error	12.6
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -1.810752780242958 \cdot 10^{-96} \lor \neg \left(t \leq 1.880166588690414 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 6
Error	12.6
Cost	21571

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4246415369043404 \cdot 10^{+184}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{t \cdot \tan k}\\ \mathbf{elif}\;t \leq -2.7623975969650846 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.2624909142672755 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{t \cdot \tan k}\\ \end{array}\]

Alternative 7
Error	11.8
Cost	20936

\[\begin{array}{l} \mathbf{if}\;t \leq -7.670056855252757 \cdot 10^{-88} \lor \neg \left(t \leq 9.985821249938532 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\ell}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{t \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 8
Error	15.9
Cost	21508

\[\begin{array}{l} \mathbf{if}\;k \leq -0.5129642332068288:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{elif}\;k \leq -3.087448234319778 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{elif}\;k \leq 8.38670249307762 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq 4.2121847874261014 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array}\]

Alternative 9
Error	15.9
Cost	21508

\[\begin{array}{l} \mathbf{if}\;k \leq -4.022794147924689:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq -2.769766079922634 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{elif}\;k \leq 7.210503642306518 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq 1.2465856195046439 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Alternative 10
Error	15.9
Cost	21508

\[\begin{array}{l} \mathbf{if}\;k \leq -170.12582257842217:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq -3.087448234319778 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{elif}\;k \leq 3.2399814008711825 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq 1.2465856195046439 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Alternative 11
Error	19.3
Cost	14536

\[\begin{array}{l} \mathbf{if}\;k \leq -3.3025876039701445 \cdot 10^{-94} \lor \neg \left(k \leq 8.074394723383298 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}\\ \end{array}\]

Alternative 12
Error	19.6
Cost	14208

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

Alternative 13
Error	19.5
Cost	1730

\[\begin{array}{l} \mathbf{if}\;k \leq -0.09886546233103015:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 0.11895529779941548:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\frac{\ell}{t}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 14
Error	23.1
Cost	2372

\[\begin{array}{l} \mathbf{if}\;t \leq -3.785048231465137 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{k \cdot \left(t \cdot k\right)} \cdot \frac{\frac{\ell}{t}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -1.9578879993731398 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.598151039956418 \cdot 10^{-71}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 3.036099748585488 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(t \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(t \cdot k\right)}{\frac{\frac{\ell}{t}}{\frac{t}{\ell}}}}\\ \end{array}\]

Alternative 15
Error	26.4
Cost	1923

\[\begin{array}{l} \mathbf{if}\;k \leq -1.0719328307788216 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{k \cdot \left(t \cdot k\right)} \cdot \frac{\frac{\ell}{t}}{\frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 1.6950306735437461 \cdot 10^{-164}:\\ \;\;\;\;\frac{1}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 0.12752623020513543:\\ \;\;\;\;\frac{1}{k \cdot \left(t \cdot k\right)} \cdot \frac{\frac{\ell}{t}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 16
Error	25.8
Cost	1288

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.114548850981367 \cdot 10^{-139} \lor \neg \left(\ell \leq 7.878230629541253 \cdot 10^{-122}\right):\\ \;\;\;\;\frac{1}{k \cdot \left(t \cdot k\right)} \cdot \frac{\frac{\ell}{t}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 17
Error	28.2
Cost	1474

\[\begin{array}{l} \mathbf{if}\;t \leq -0.0003104848319893586:\\ \;\;\;\;\frac{1}{k \cdot \left(t \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{t \cdot t}\right)\\ \mathbf{elif}\;t \leq 3.29263803328796 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t \cdot t}{\ell}}}{k \cdot \left(t \cdot k\right)}\\ \end{array}\]

Alternative 18
Error	28.2
Cost	1160

\[\begin{array}{l} \mathbf{if}\;t \leq -246.96991901702776 \lor \neg \left(t \leq 3.917555686498155 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\frac{\ell}{\frac{t \cdot t}{\ell}}}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 19
Error	27.5
Cost	1474

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00023345068908161738:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 0.04709131752104685:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(t \cdot k\right)}}{\frac{t \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 20
Error	30.6
Cost	64

\[0\]

Alternative 21
Error	61.9
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if t < -1.1582818651038331e-96 or 4.7432519440330267e-121 < t

   1. Initial program 23.6

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

   2. Simplified 23.6

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

   4. Applied cube-mult_binary64_790 23.6

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

   5. Applied times-frac_binary64_766 16.9

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

   6. Applied associate-*l*_binary64_701 14.6

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

   8. Applied *-un-lft-identity_binary64_760 14.6

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

   9. Applied times-frac_binary64_766 9.8

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

   10. Applied associate-*l*_binary64_701 8.7

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

   12. Applied *-un-lft-identity_binary64_760 8.7

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

   13. Applied times-frac_binary64_766 8.7

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

   14. Simplified 6.7

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

   16. Applied *-un-lft-identity_binary64_760 6.7

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

   17. Applied times-frac_binary64_766 5.6

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

   18. Applied associate-*l*_binary64_701 4.5

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

   19. Simplified 4.5

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

   20. Simplified 4.5

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

   if -1.1582818651038331e-96 < t < 4.7432519440330267e-121

   1. Initial program 62.9

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

   2. Simplified 62.9

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

   4. Applied add-log-exp_binary64_799 63.5

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

   5. Simplified 63.5

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

   6. Taylor expanded around inf 27.0

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

   7. Simplified 27.0

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

   8. Simplified 27.0

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.158281865103833 \cdot 10^{-96} \lor \neg \left(t \leq 4.743251944033027 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{1}{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)} \cdot \left(\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{\cos k} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\ell \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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))))