Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 8.5

Time: 38.7s

Precision: binary64

Cost: 85960

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ t_2 := {\sin k}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_4 := t \cdot t_3\\ t_5 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_6 := t_3 \cdot \frac{t}{t_5}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{\ell} \cdot \frac{\frac{2}{t_4}}{\frac{1}{\sqrt[3]{\ell}}}}{{\left(\frac{1}{\frac{t_5}{t_4}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.3 \cdot 10^{-155}:\\ \;\;\;\;{\left(\frac{t_1 \cdot t_1}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{2}{t_6}}{{t_6}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_2 \cdot t}\right)\\ \end{array} \]

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 (* (cbrt l) (cbrt (/ 1.0 k))))
        (t_2 (pow (sin k) 2.0))
        (t_3 (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t) 2.0))))))
        (t_4 (* t t_3))
        (t_5 (pow (cbrt l) 2.0))
        (t_6 (* t_3 (/ t t_5))))
   (if (<= k -5.8e+48)
     (/ 2.0 (* t_2 (* (/ k l) (/ (/ k (/ l t)) (cos k)))))
     (if (<= k -2.2e-155)
       (/
        (* (cbrt l) (/ (/ 2.0 t_4) (/ 1.0 (cbrt l))))
        (pow (/ 1.0 (/ t_5 t_4)) 2.0))
       (if (<= k 4.3e-155)
         (pow (/ (* t_1 t_1) t) 3.0)
         (if (<= k 1.15e+43)
           (/ (/ 2.0 t_6) (pow t_6 2.0))
           (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t_2 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 t_1 = cbrt(l) * cbrt((1.0 / k));
	double t_2 = pow(sin(k), 2.0);
	double t_3 = cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t), 2.0)))));
	double t_4 = t * t_3;
	double t_5 = pow(cbrt(l), 2.0);
	double t_6 = t_3 * (t / t_5);
	double tmp;
	if (k <= -5.8e+48) {
		tmp = 2.0 / (t_2 * ((k / l) * ((k / (l / t)) / cos(k))));
	} else if (k <= -2.2e-155) {
		tmp = (cbrt(l) * ((2.0 / t_4) / (1.0 / cbrt(l)))) / pow((1.0 / (t_5 / t_4)), 2.0);
	} else if (k <= 4.3e-155) {
		tmp = pow(((t_1 * t_1) / t), 3.0);
	} else if (k <= 1.15e+43) {
		tmp = (2.0 / t_6) / pow(t_6, 2.0);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t_2 * t)));
	}
	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.cbrt(l) * Math.cbrt((1.0 / k));
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0)))));
	double t_4 = t * t_3;
	double t_5 = Math.pow(Math.cbrt(l), 2.0);
	double t_6 = t_3 * (t / t_5);
	double tmp;
	if (k <= -5.8e+48) {
		tmp = 2.0 / (t_2 * ((k / l) * ((k / (l / t)) / Math.cos(k))));
	} else if (k <= -2.2e-155) {
		tmp = (Math.cbrt(l) * ((2.0 / t_4) / (1.0 / Math.cbrt(l)))) / Math.pow((1.0 / (t_5 / t_4)), 2.0);
	} else if (k <= 4.3e-155) {
		tmp = Math.pow(((t_1 * t_1) / t), 3.0);
	} else if (k <= 1.15e+43) {
		tmp = (2.0 / t_6) / Math.pow(t_6, 2.0);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t_2 * t)));
	}
	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(cbrt(l) * cbrt(Float64(1.0 / k)))
	t_2 = sin(k) ^ 2.0
	t_3 = cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))))
	t_4 = Float64(t * t_3)
	t_5 = cbrt(l) ^ 2.0
	t_6 = Float64(t_3 * Float64(t / t_5))
	tmp = 0.0
	if (k <= -5.8e+48)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k / l) * Float64(Float64(k / Float64(l / t)) / cos(k)))));
	elseif (k <= -2.2e-155)
		tmp = Float64(Float64(cbrt(l) * Float64(Float64(2.0 / t_4) / Float64(1.0 / cbrt(l)))) / (Float64(1.0 / Float64(t_5 / t_4)) ^ 2.0));
	elseif (k <= 4.3e-155)
		tmp = Float64(Float64(t_1 * t_1) / t) ^ 3.0;
	elseif (k <= 1.15e+43)
		tmp = Float64(Float64(2.0 / t_6) / (t_6 ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_2 * t))));
	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[(N[Power[l, 1/3], $MachinePrecision] * N[Power[N[(1.0 / k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(t * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 * N[(t / t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -5.8e+48], N[(2.0 / N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -2.2e-155], N[(N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(2.0 / t$95$4), $MachinePrecision] / N[(1.0 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 / N[(t$95$5 / t$95$4), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.3e-155], N[Power[N[(N[(t$95$1 * t$95$1), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k, 1.15e+43], N[(N[(2.0 / t$95$6), $MachinePrecision] / N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -5.7999999999999998e48

   1. Initial program 32.6

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

   2. Simplified 32.6

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

   3. Taylor expanded in t around 0 20.0

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

   4. Simplified 19.5

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{\cos k}{{\sin k}^{2}}}}} \]
      Proof
      (/.f64 2 (/.f64 (*.f64 k k) (*.f64 (/.f64 l (/.f64 t l)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 (/.f64 l (/.f64 t l)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 8 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) t)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 8 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 0 points increase in error, 8 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 4 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 8 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (pow.f64 l 2))))): 0 points increase in error, 4 points decrease in error

   5. Taylor expanded in k around inf 20.0

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

   6. Simplified 10.3

      \[\leadsto \frac{2}{\color{blue}{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}} \]
      Proof
      (/.f64 2 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 (/.f64 k l) (/.f64 (/.f64 k (/.f64 l t)) (cos.f64 k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 (/.f64 k l) (Rewrite<= associate-/r*_binary64 (/.f64 k (*.f64 (/.f64 l t) (cos.f64 k))))))): 0 points increase in error, 13 points decrease in error
      (/.f64 2 (*.f64 (pow.f64 (sin.f64 k) 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 k k) (*.f64 l (*.f64 (/.f64 l t) (cos.f64 k))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 k k)) (*.f64 l (*.f64 (/.f64 l t) (cos.f64 k)))))): 13 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2))) (*.f64 l (*.f64 (/.f64 l t) (cos.f64 k))))): 0 points increase in error, 7 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 l (/.f64 l t)) (cos.f64 k))))): 0 points increase in error, 12 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (*.f64 l (/.f64 l t)))))): 13 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 l l) t))))): 0 points increase in error, 3 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t)))): 3 points increase in error, 10 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) t)))): 13 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (*.f64 k k) (pow.f64 (sin.f64 k) 2)) t) (*.f64 (cos.f64 k) (pow.f64 l 2))))): 0 points increase in error, 13 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (pow.f64 (sin.f64 k) 2)) t) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 2 points increase in error, 0 points decrease in error

   if -5.7999999999999998e48 < k < -2.1999999999999999e-155

   1. Initial program 28.4

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

   2. Simplified 28.4

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

   3. Applied egg-rr 7.7

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

   4. Simplified 7.7

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      Proof
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1 2)) (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 8 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))))) (pow.f64 (*.f64 (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))))) (pow.f64 (*.f64 (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 8 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (pow.f64 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)) (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))))): 8 points increase in error, 0 points decrease in error

   5. Applied egg-rr 7.7

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

   6. Applied egg-rr 7.7

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

   if -2.1999999999999999e-155 < k < 4.30000000000000008e-155

   1. Initial program 37.5

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

   2. Simplified 37.5

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

   3. Taylor expanded in k around 0 62.3

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

   4. Simplified 62.0

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
      Proof
      (/.f64 2 (/.f64 (*.f64 k k) (*.f64 (/.f64 l (/.f64 t l)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 (/.f64 l (/.f64 t l)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 8 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) t)) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t) (/.f64 (cos.f64 k) (pow.f64 (sin.f64 k) 2))))): 8 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 0 points increase in error, 8 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 4 points decrease in error
      (/.f64 2 (/.f64 (pow.f64 k 2) (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 8 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (pow.f64 l 2))))): 0 points increase in error, 4 points decrease in error

   5. Applied egg-rr 64.0

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

   6. Applied egg-rr 4.0

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

   7. Simplified 4.0

      \[\leadsto {\left(\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\right)}}{t}\right)}^{3} \]
      Proof
      (pow.f64 (/.f64 (*.f64 (*.f64 (cbrt.f64 l) (cbrt.f64 (/.f64 1 k))) (*.f64 (cbrt.f64 l) (cbrt.f64 (/.f64 1 k)))) t) 3): 0 points increase in error, 0 points decrease in error
      (pow.f64 (/.f64 (*.f64 (*.f64 (cbrt.f64 l) (cbrt.f64 (Rewrite<= unpow-1_binary64 (pow.f64 k -1)))) (*.f64 (cbrt.f64 l) (cbrt.f64 (/.f64 1 k)))) t) 3): 0 points increase in error, 0 points decrease in error
      (pow.f64 (/.f64 (*.f64 (*.f64 (cbrt.f64 l) (cbrt.f64 (pow.f64 k -1))) (*.f64 (cbrt.f64 l) (cbrt.f64 (Rewrite<= unpow-1_binary64 (pow.f64 k -1))))) t) 3): 0 points increase in error, 0 points decrease in error

   if 4.30000000000000008e-155 < k < 1.1500000000000001e43

   1. Initial program 27.1

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

   2. Simplified 27.1

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

   3. Applied egg-rr 6.8

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

   4. Simplified 6.8

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      Proof
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (/.f64 2 (*.f64 (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1 2)) (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 8 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))))) (pow.f64 (*.f64 (cbrt.f64 (*.f64 (sin.f64 k) (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))))) (pow.f64 (*.f64 (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 0 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1 (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))))) (pow.f64 (*.f64 (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k))))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)): 8 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (pow.f64 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2))) 2)) (/.f64 2 (*.f64 (cbrt.f64 (*.f64 (tan.f64 k) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (sin.f64 k)))) (/.f64 t (pow.f64 (cbrt.f64 l) 2)))))): 8 points increase in error, 0 points decrease in error

   if 1.1500000000000001e43 < k

   1. Initial program 33.3

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

   2. Simplified 33.3

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

   3. Taylor expanded in k around inf 20.1

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

   4. Simplified 10.3

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 (cos.f64 k) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 (cos.f64 k) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 7 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) (/.f64 (cos.f64 k) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (cos.f64 k) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 7 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (pow.f64 k 2)) (/.f64 (cos.f64 k) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 7 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 7 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 7 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{\ell} \cdot \frac{\frac{2}{t \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}{\frac{1}{\sqrt[3]{\ell}}}}{{\left(\frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.3 \cdot 10^{-155}:\\ \;\;\;\;{\left(\frac{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\right)}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.4
Cost	85904

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_4 := \frac{2 \cdot \frac{t_2}{t \cdot t_3}}{{\left(t_3 \cdot \frac{t}{t_2}\right)}^{2}}\\ t_5 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ \mathbf{if}\;k \leq -1.3 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -1.16 \cdot 10^{-154}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;{\left(\frac{t_5 \cdot t_5}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \end{array} \]

Alternative 2
Error	8.4
Cost	85904

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_4 := t_3 \cdot \frac{t}{t_2}\\ t_5 := {t_4}^{2}\\ t_6 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{2 \cdot \frac{t_2}{t \cdot t_3}}{t_5}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-156}:\\ \;\;\;\;{\left(\frac{t_6 \cdot t_6}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\frac{2}{t_4}}{t_5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \end{array} \]

Alternative 3
Error	8.5
Cost	85904

\[\begin{array}{l} t_1 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ t_2 := {\sin k}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_4 := t_3 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_5 := {t_4}^{2}\\ \mathbf{if}\;k \leq -1.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{\ell} \cdot \frac{\frac{2}{t \cdot t_3}}{\frac{1}{\sqrt[3]{\ell}}}}{t_5}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-155}:\\ \;\;\;\;{\left(\frac{t_1 \cdot t_1}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{2}{t_4}}{t_5}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_2 \cdot t}\right)\\ \end{array} \]

Alternative 4
Error	8.6
Cost	46480

\[\begin{array}{l} t_1 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ t_2 := {\sin k}^{2}\\ t_3 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ \mathbf{if}\;k \leq -3.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4.45 \cdot 10^{-150}:\\ \;\;\;\;{\left(\frac{t_3 \cdot t_3}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_2 \cdot t}\right)\\ \end{array} \]

Alternative 5
Error	8.7
Cost	33292

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ t_2 := \sqrt[3]{\frac{1}{k}}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.36 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -1.2 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;{\left(\frac{t_2 \cdot \left(\sqrt[3]{\ell} \cdot \left(\sqrt[3]{\ell} \cdot t_2\right)\right)}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_3 \cdot t}\right)\\ \end{array} \]

Alternative 6
Error	8.8
Cost	33292

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ t_2 := \sqrt[3]{\ell} \cdot \sqrt[3]{\frac{1}{k}}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.65 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -1.25 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{-105}:\\ \;\;\;\;{\left(\frac{t_2 \cdot t_2}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_3 \cdot t}\right)\\ \end{array} \]

Alternative 7
Error	10.3
Cost	21136

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.02 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -4.4 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_2 \cdot t}\right)\\ \end{array} \]

Alternative 8
Error	13.1
Cost	21004

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -245000:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;k \leq 10000000000000:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \end{array} \]

Alternative 9
Error	13.1
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -245000 \lor \neg \left(k \leq 245000\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]

Alternative 10
Error	14.6
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -245000:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot \frac{t_1}{\cos k}}\\ \mathbf{elif}\;k \leq 245000:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \end{array} \]

Alternative 11
Error	13.0
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -245000:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 46000000:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \end{array} \]

Alternative 12
Error	20.1
Cost	7304

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot t_1}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k} + \frac{\ell}{t} \cdot \left(\ell \cdot -0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot {t_1}^{2}}\\ \end{array} \]

Alternative 13
Error	20.9
Cost	1736

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k} + \frac{\ell}{t} \cdot \left(\ell \cdot -0.16666666666666666\right)}}\\ \mathbf{elif}\;t \leq 10^{+103}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 14
Error	24.0
Cost	1360

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{\left(\frac{k}{\ell} \cdot t\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}}{t}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 15
Error	21.1
Cost	1228

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}}\\ \mathbf{elif}\;t \leq 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 16
Error	25.1
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -5.1 \cdot 10^{-95} \lor \neg \left(k \leq 8.5 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 17
Error	25.2
Cost	1096

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{\left(\frac{k}{\ell} \cdot t\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t_1}\\ \end{array} \]

Alternative 18
Error	25.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{\left(\frac{k}{\ell} \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 19
Error	25.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell}{\left(\frac{k}{\ell} \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 20
Error	23.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}}{t}\\ \end{array} \]

Alternative 21
Error	29.7
Cost	832

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

Alternative 22
Error	29.4
Cost	832

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

Reproduce

herbie shell --seed 2022343 
(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))))