Toniolo and Linder, Equation (10+)

Average Error: 33.0 → 10.0

Time: 32.8s

Precision: binary64

Cost: 33940

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\\ t_3 := 1 + \left(1 + t_1\right)\\ \mathbf{if}\;t \leq -1.876647700209357 \cdot 10^{+154}:\\ \;\;\;\;\frac{{t_2}^{2}}{k} \cdot \frac{t_2}{k}\\ \mathbf{elif}\;t \leq -9.210115154097725 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t_3}\\ \mathbf{elif}\;t \leq -3.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;t \leq 10000000:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\frac{\ell}{k}}{t}}{k}}{\sin k \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

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
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2 (/ (cbrt l) (/ t (cbrt l))))
        (t_3 (+ 1.0 (+ 1.0 t_1))))
   (if (<= t -1.876647700209357e+154)
     (* (/ (pow t_2 2.0) k) (/ t_2 k))
     (if (<= t -9.210115154097725e+87)
       (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k)) (tan k)) t_3))
       (if (<= t -3.25e+34)
         (* (/ l k) (/ (/ l k) (pow t 3.0)))
         (if (<= t -1e-70)
           (/
            (/ l (sin k))
            (* (tan k) (* (+ 2.0 t_1) (* (/ (pow t 3.0) l) 0.5))))
           (if (<= t 10000000.0)
             (/ (/ (* 2.0 (/ (/ l k) t)) k) (* (sin k) (/ (tan k) l)))
             (/
              2.0
              (*
               t_3
               (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))))))))))

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((k / t), 2.0);
	double t_2 = cbrt(l) / (t / cbrt(l));
	double t_3 = 1.0 + (1.0 + t_1);
	double tmp;
	if (t <= -1.876647700209357e+154) {
		tmp = (pow(t_2, 2.0) / k) * (t_2 / k);
	} else if (t <= -9.210115154097725e+87) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k)) * tan(k)) * t_3);
	} else if (t <= -3.25e+34) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (t <= -1e-70) {
		tmp = (l / sin(k)) / (tan(k) * ((2.0 + t_1) * ((pow(t, 3.0) / l) * 0.5)));
	} else if (t <= 10000000.0) {
		tmp = ((2.0 * ((l / k) / t)) / k) / (sin(k) * (tan(k) / l));
	} else {
		tmp = 2.0 / (t_3 * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = Math.cbrt(l) / (t / Math.cbrt(l));
	double t_3 = 1.0 + (1.0 + t_1);
	double tmp;
	if (t <= -1.876647700209357e+154) {
		tmp = (Math.pow(t_2, 2.0) / k) * (t_2 / k);
	} else if (t <= -9.210115154097725e+87) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k)) * Math.tan(k)) * t_3);
	} else if (t <= -3.25e+34) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (t <= -1e-70) {
		tmp = (l / Math.sin(k)) / (Math.tan(k) * ((2.0 + t_1) * ((Math.pow(t, 3.0) / l) * 0.5)));
	} else if (t <= 10000000.0) {
		tmp = ((2.0 * ((l / k) / t)) / k) / (Math.sin(k) * (Math.tan(k) / l));
	} else {
		tmp = 2.0 / (t_3 * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(cbrt(l) / Float64(t / cbrt(l)))
	t_3 = Float64(1.0 + Float64(1.0 + t_1))
	tmp = 0.0
	if (t <= -1.876647700209357e+154)
		tmp = Float64(Float64((t_2 ^ 2.0) / k) * Float64(t_2 / k));
	elseif (t <= -9.210115154097725e+87)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k)) * tan(k)) * t_3));
	elseif (t <= -3.25e+34)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (t <= -1e-70)
		tmp = Float64(Float64(l / sin(k)) / Float64(tan(k) * Float64(Float64(2.0 + t_1) * Float64(Float64((t ^ 3.0) / l) * 0.5))));
	elseif (t <= 10000000.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(l / k) / t)) / k) / Float64(sin(k) * Float64(tan(k) / l)));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[l, 1/3], $MachinePrecision] / N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.876647700209357e+154], N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t$95$2 / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -9.210115154097725e+87], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.25e+34], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-70], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 10000000.0], N[(N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.876647700209357e154

   1. Initial program 21.5

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

   2. Simplified 21.2

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) l) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k))))): 16 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 34 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 1 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 37 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 1 points decrease in error

   3. Taylor expanded in k around 0 28.0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Simplified 23.7

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k \cdot k}} \]
      Proof
      (/.f64 (/.f64 l (/.f64 (pow.f64 t 3) l)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (pow.f64 t 3))) (*.f64 k k)): 32 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 t 3)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 t 3)) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 6 points increase in error, 6 points decrease in error

   5. Applied egg-rr 9.1

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

   if -1.876647700209357e154 < t < -9.21011515409772518e87

   1. Initial program 29.3

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

   2. Applied egg-rr 10.5

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

   if -9.21011515409772518e87 < t < -3.25000000000000008e34

   1. Initial program 19.0

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

   2. Simplified 13.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) l) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k))))): 16 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 34 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 1 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 37 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 1 points decrease in error

   3. Taylor expanded in k around 0 28.4

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Simplified 28.6

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k \cdot k}} \]
      Proof
      (/.f64 (/.f64 l (/.f64 (pow.f64 t 3) l)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (pow.f64 t 3))) (*.f64 k k)): 32 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 t 3)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 t 3)) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 6 points increase in error, 6 points decrease in error

   5. Applied egg-rr 28.7

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

   6. Taylor expanded in t around 0 28.4

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   7. Simplified 18.8

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]
      Proof
      (*.f64 (/.f64 l k) (/.f64 (/.f64 l k) (pow.f64 t 3))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 k (pow.f64 t 3))))): 13 points increase in error, 5 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k (*.f64 k (pow.f64 t 3))))): 56 points increase in error, 4 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k (*.f64 k (pow.f64 t 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 l 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) (pow.f64 t 3)))): 30 points increase in error, 3 points decrease in error
      (/.f64 (pow.f64 l 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (pow.f64 t 3))): 0 points increase in error, 0 points decrease in error

   if -3.25000000000000008e34 < t < -9.99999999999999996e-71

   1. Initial program 24.2

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

   2. Simplified 13.8

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) l) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k))))): 16 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 34 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 1 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 37 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 1 points decrease in error

   3. Applied egg-rr 11.5

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k}}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right) \cdot \tan k}} \]

   if -9.99999999999999996e-71 < t < 1e7

   1. Initial program 53.1

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

   2. Simplified 49.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 23 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) l) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k))))): 16 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 34 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 1 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 37 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 1 points decrease in error

   3. Taylor expanded in t around 0 20.6

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

   4. Simplified 6.7

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 l k) (*.f64 t k))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 l k) k) t))): 45 points increase in error, 37 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 k k))) t)): 38 points increase in error, 19 points decrease in error
      (*.f64 2 (/.f64 (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 k 2))) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 24 points increase in error, 32 points decrease in error

   5. Applied egg-rr 6.7

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

   6. Applied egg-rr 5.2

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

   if 1e7 < t

   1. Initial program 23.4

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

   2. Applied egg-rr 13.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.876647700209357 \cdot 10^{+154}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\right)}^{2}}{k} \cdot \frac{\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}}{k}\\ \mathbf{elif}\;t \leq -9.210115154097725 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -3.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;t \leq 10000000:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\frac{\ell}{k}}{t}}{k}}{\sin k \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	33156

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{t}{\sqrt[3]{\ell}}\\ t_3 := \frac{\sqrt[3]{\ell}}{t_2}\\ \mathbf{if}\;t \leq -1.876647700209357 \cdot 10^{+154}:\\ \;\;\;\;\frac{{t_3}^{2}}{k} \cdot \frac{t_3}{k}\\ \mathbf{elif}\;t \leq -9.210115154097725 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq -3.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\frac{\ell}{k}}{t}}{k}}{\sin k \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_2}^{-2}}{k} \cdot \frac{\ell}{\frac{k}{\frac{\sqrt[3]{\ell}}{t}}}\\ \end{array} \]

Alternative 2
Error	11.9
Cost	20360

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{k}\\ t_2 := \sin k \cdot \frac{\tan k}{\ell}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-48}:\\ \;\;\;\;\frac{t_1}{t_2 \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 10000:\\ \;\;\;\;\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{-2}}{k} \cdot \frac{\ell}{\frac{k}{\frac{\sqrt[3]{\ell}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t \cdot k}}{t_2}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	20236

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{k}\\ t_2 := \sin k \cdot \frac{\tan k}{\ell}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-48}:\\ \;\;\;\;\frac{t_1}{t_2 \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 10^{-142}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;k \leq 10000:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\right)}^{3}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t \cdot k}}{t_2}\\ \end{array} \]

Alternative 4
Error	16.3
Cost	14156

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{k \cdot k} \cdot \left(2 \cdot \frac{\ell}{\sin k}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 5
Error	12.0
Cost	14156

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\frac{\ell}{k}}{t}}{k}}{\sin k \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 6
Error	20.5
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{t \cdot k} \cdot \frac{1}{k}\right)\right) \cdot \mathsf{fma}\left(\ell, -0.16666666666666666, \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 7
Error	22.3
Cost	8012

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\left(2 \cdot \left(\frac{\ell}{t \cdot k} \cdot \frac{1}{k}\right)\right) \cdot \mathsf{fma}\left(\ell, -0.16666666666666666, \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.3
Cost	7884

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\ell, -0.16666666666666666, \frac{\ell}{k \cdot k}\right) \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	22.3
Cost	7436

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{if}\;t \leq -2.2444348010889995 \cdot 10^{+106}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.0
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	23.9
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	28.7
Cost	1224

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t \cdot \frac{t \cdot t}{\ell}}}{k \cdot k}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 82000000:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	34.1
Cost	960

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

Reproduce

herbie shell --seed 2022313 
(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))))