Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 10.0

Time: 1.0min

Precision: binary64

Cost: 33028

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\sin k}{\ell}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.292046717466389 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{t_2}} \cdot \frac{1}{\sqrt[3]{\frac{\tan k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{+38}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{t}{\ell}}}{k \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{\left(2 + t_3\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

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
 (let* ((t_1 (/ l (sin k)))
        (t_2 (/ (sin k) l))
        (t_3 (pow (/ k t) 2.0))
        (t_4
         (/
          2.0
          (*
           (+ 1.0 (+ t_3 1.0))
           (* (tan k) (* (sin k) (* (/ (* t t) l) (/ t l))))))))
   (if (<= t -4.292046717466389e+154)
     (pow (/ (* (/ 1.0 (cbrt t_2)) (/ 1.0 (cbrt (/ (tan k) l)))) t) 3.0)
     (if (<= t -3.6e+38)
       t_4
       (if (<= t -9.8e+27)
         (/ (/ l (* (* t t) (/ k (/ l k)))) t)
         (if (<= t -1e-30)
           (/ (/ 2.0 (* k (/ t l))) (* k (* (tan k) t_2)))
           (if (<= t -1e-60)
             (/ (/ t_1 (* (+ 2.0 t_3) (* (/ (pow t 3.0) l) 0.5))) (tan k))
             (if (<= t 1.0)
               (* (* (/ l k) (/ 2.0 (* t k))) (/ t_1 (tan k)))
               (if (<= t 1.0579812393352837e+108)
                 t_4
                 (/ (/ l t) (/ (pow (* t k) 2.0) l)))))))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = l / sin(k);
	double t_2 = sin(k) / l;
	double t_3 = pow((k / t), 2.0);
	double t_4 = 2.0 / ((1.0 + (t_3 + 1.0)) * (tan(k) * (sin(k) * (((t * t) / l) * (t / l)))));
	double tmp;
	if (t <= -4.292046717466389e+154) {
		tmp = pow((((1.0 / cbrt(t_2)) * (1.0 / cbrt((tan(k) / l)))) / t), 3.0);
	} else if (t <= -3.6e+38) {
		tmp = t_4;
	} else if (t <= -9.8e+27) {
		tmp = (l / ((t * t) * (k / (l / k)))) / t;
	} else if (t <= -1e-30) {
		tmp = (2.0 / (k * (t / l))) / (k * (tan(k) * t_2));
	} else if (t <= -1e-60) {
		tmp = (t_1 / ((2.0 + t_3) * ((pow(t, 3.0) / l) * 0.5))) / tan(k);
	} else if (t <= 1.0) {
		tmp = ((l / k) * (2.0 / (t * k))) * (t_1 / tan(k));
	} else if (t <= 1.0579812393352837e+108) {
		tmp = t_4;
	} else {
		tmp = (l / t) / (pow((t * k), 2.0) / l);
	}
	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 = l / Math.sin(k);
	double t_2 = Math.sin(k) / l;
	double t_3 = Math.pow((k / t), 2.0);
	double t_4 = 2.0 / ((1.0 + (t_3 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (((t * t) / l) * (t / l)))));
	double tmp;
	if (t <= -4.292046717466389e+154) {
		tmp = Math.pow((((1.0 / Math.cbrt(t_2)) * (1.0 / Math.cbrt((Math.tan(k) / l)))) / t), 3.0);
	} else if (t <= -3.6e+38) {
		tmp = t_4;
	} else if (t <= -9.8e+27) {
		tmp = (l / ((t * t) * (k / (l / k)))) / t;
	} else if (t <= -1e-30) {
		tmp = (2.0 / (k * (t / l))) / (k * (Math.tan(k) * t_2));
	} else if (t <= -1e-60) {
		tmp = (t_1 / ((2.0 + t_3) * ((Math.pow(t, 3.0) / l) * 0.5))) / Math.tan(k);
	} else if (t <= 1.0) {
		tmp = ((l / k) * (2.0 / (t * k))) * (t_1 / Math.tan(k));
	} else if (t <= 1.0579812393352837e+108) {
		tmp = t_4;
	} else {
		tmp = (l / t) / (Math.pow((t * k), 2.0) / l);
	}
	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(l / sin(k))
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(k / t) ^ 2.0
	t_4 = Float64(2.0 / Float64(Float64(1.0 + Float64(t_3 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(t * t) / l) * Float64(t / l))))))
	tmp = 0.0
	if (t <= -4.292046717466389e+154)
		tmp = Float64(Float64(Float64(1.0 / cbrt(t_2)) * Float64(1.0 / cbrt(Float64(tan(k) / l)))) / t) ^ 3.0;
	elseif (t <= -3.6e+38)
		tmp = t_4;
	elseif (t <= -9.8e+27)
		tmp = Float64(Float64(l / Float64(Float64(t * t) * Float64(k / Float64(l / k)))) / t);
	elseif (t <= -1e-30)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t / l))) / Float64(k * Float64(tan(k) * t_2)));
	elseif (t <= -1e-60)
		tmp = Float64(Float64(t_1 / Float64(Float64(2.0 + t_3) * Float64(Float64((t ^ 3.0) / l) * 0.5))) / tan(k));
	elseif (t <= 1.0)
		tmp = Float64(Float64(Float64(l / k) * Float64(2.0 / Float64(t * k))) * Float64(t_1 / tan(k)));
	elseif (t <= 1.0579812393352837e+108)
		tmp = t_4;
	else
		tmp = Float64(Float64(l / t) / Float64((Float64(t * k) ^ 2.0) / l));
	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[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(2.0 / N[(N[(1.0 + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.292046717466389e+154], N[Power[N[(N[(N[(1.0 / N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t, -3.6e+38], t$95$4, If[LessEqual[t, -9.8e+27], N[(N[(l / N[(N[(t * t), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, -1e-30], N[(N[(2.0 / N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-60], N[(N[(t$95$1 / N[(N[(2.0 + t$95$3), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0], N[(N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0579812393352837e+108], t$95$4, N[(N[(l / t), $MachinePrecision] / N[(N[Power[N[(t * k), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -4.29204671746638875e154

   1. Initial program 21.7

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

   2. Simplified 21.0

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Taylor expanded in t around inf 21.0

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

   4. Applied egg-rr 28.6

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

   5. Applied egg-rr 22.5

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

   6. Applied egg-rr 8.4

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

   if -4.29204671746638875e154 < t < -3.59999999999999969e38 or 1 < t < 1.057981239335284e108

   1. Initial program 22.8

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

   2. Applied egg-rr 13.1

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

   if -3.59999999999999969e38 < t < -9.8000000000000003e27

   1. Initial program 14.8

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

   2. Simplified 14.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Taylor expanded in k around 0 26.1

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

   4. Simplified 24.3

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
      Proof
      (*.f64 (/.f64 l (pow.f64 t 3)) (/.f64 l (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l (pow.f64 t 3)) (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 (pow.f64 t 3) (pow.f64 k 2)))): 41 points increase in error, 11 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 t 3) (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 24.3

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

   6. Applied egg-rr 22.2

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

   if -9.8000000000000003e27 < t < -1e-30

   1. Initial program 18.8

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

   2. Simplified 10.2

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Applied egg-rr 10.2

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

   4. Taylor expanded in t around 0 28.7

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

   5. Simplified 28.7

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{t}}{k \cdot k}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (/.f64 (/.f64 (*.f64 2 l) t) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 2 l) t) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 2 l) (*.f64 (pow.f64 k 2) t))): 31 points increase in error, 34 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 0 points increase in error, 2 points decrease in error

   6. Applied egg-rr 22.2

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

   if -1e-30 < t < -9.9999999999999997e-61

   1. Initial program 24.3

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

   2. Simplified 9.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Applied egg-rr 9.2

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

   if -9.9999999999999997e-61 < t < 1

   1. Initial program 52.3

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

   2. Simplified 48.7

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Applied egg-rr 42.9

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

   4. Taylor expanded in t around 0 21.2

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

   5. Simplified 19.3

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{t}}{k \cdot k}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (/.f64 (/.f64 (*.f64 2 l) t) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 2 l) t) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 2 l) (*.f64 (pow.f64 k 2) t))): 31 points increase in error, 34 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 0 points increase in error, 2 points decrease in error

   6. Taylor expanded in l around 0 21.2

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

   7. Simplified 7.6

      \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \frac{2}{k \cdot t}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 (/.f64 l k) (/.f64 2 (*.f64 k t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l 2) (*.f64 k (*.f64 k t)))): 44 points increase in error, 27 points decrease in error
      (/.f64 (*.f64 l 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t))): 31 points increase in error, 23 points decrease in error
      (/.f64 (*.f64 l 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l (*.f64 (pow.f64 k 2) t)) 2)): 0 points increase in error, 2 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 0 points increase in error, 0 points decrease in error

   if 1.057981239335284e108 < t

   1. Initial program 23.2

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

   2. Simplified 23.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 14 points increase in error, 5 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))))): 14 points increase in error, 11 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)))): 4 points increase in error, 3 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)))): 32 points increase in error, 2 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)): 3 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, 29 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)))): 1 points increase in error, 2 points decrease in error

   3. Taylor expanded in k around 0 28.6

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

   4. Simplified 26.5

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
      Proof
      (*.f64 (/.f64 l (pow.f64 t 3)) (/.f64 l (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l (pow.f64 t 3)) (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 (pow.f64 t 3) (pow.f64 k 2)))): 41 points increase in error, 11 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 t 3) (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 26.0

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

   6. Applied egg-rr 26.0

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

   7. Applied egg-rr 8.8

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.292046717466389 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{1}{\sqrt[3]{\frac{\tan k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{t}{\ell}}}{k \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\sin k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	46280

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 + t_2\\ \mathbf{if}\;t \leq -2.170971498792176 \cdot 10^{+200}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq -2.6213798646244923 \cdot 10^{+104}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt[3]{\ell \cdot 2}}{t}}{\sqrt[3]{t_3}}\right)}^{3}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{\ell}{\frac{\tan k \cdot {t}^{3}}{t_1}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{t_3 \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(t_2 + 1\right)\right)}\\ \end{array} \]

Alternative 2
Error	10.8
Cost	34072

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\sin k}{\ell}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := 1 + \left(t_3 + 1\right)\\ \mathbf{if}\;t \leq -4.292046717466389 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{t_2}} \cdot \frac{1}{\sqrt[3]{\frac{\tan k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{t_4 \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{t}{\ell}}}{k \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{\left(2 + t_3\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot t_4}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	32900

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\sin k}{\ell}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.292046717466389 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{t_2} \cdot \sqrt[3]{\frac{\tan k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{+38}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{t}{\ell}}}{k \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{\left(2 + t_3\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4
Error	10.1
Cost	32772

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{\left(1 + \left(t_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.292046717466389 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{t_1} \cdot \sqrt[3]{\frac{\ell}{\tan k}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{+38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{t}{\ell}}}{k \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{\left(2 + t_2\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	27080

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.708583569691195 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{t_1}{\left(2 + t_3\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	13.6
Cost	21268

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot t_1\\ \mathbf{if}\;k \leq -1 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-188}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-26}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell \cdot \frac{1}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.614521061543967 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{k \cdot \frac{t}{\ell}}}{k}\\ \mathbf{elif}\;k \leq 6.992882560752865 \cdot 10^{+106}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{{t}^{3} \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)}\right)\\ \mathbf{elif}\;k \leq 1.8079941985246638 \cdot 10^{+300}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	9.6
Cost	21264

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.576063180147204 \cdot 10^{+98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{t \cdot \left(t \cdot \frac{t}{\ell}\right)}}{2 + t_3}\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	9.3
Cost	21136

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := t_1 \cdot \frac{\frac{2}{t \cdot \left(t \cdot \frac{t}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ t_3 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -6.576063180147204 \cdot 10^{+98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-60}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	17.5
Cost	14292

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{k}\\ \mathbf{elif}\;t \leq 10^{-8}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell + \ell\right)}{t_2 \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 2.7589645087748444 \cdot 10^{+80}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t_2}}{t}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	11.2
Cost	14288

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 10^{+74}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	11.3
Cost	14160

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 10^{+74}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.0579812393352837 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}{t}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	13.6
Cost	14024

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;\frac{\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2}{k \cdot \frac{t}{\ell}}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	19.2
Cost	7884

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{k}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \mathsf{fma}\left(\ell, -0.16666666666666666, \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	19.4
Cost	7756

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{k}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \frac{\frac{\ell}{k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	20.7
Cost	7436

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\ell}{t}}{{\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.6930170943671406 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	19.4
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	19.4
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	22.7
Cost	1224

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 19
Error	28.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2847196052605388 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;\ell \leq 8.662408435442512 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 20
Error	35.9
Cost	832

\[\frac{\frac{\frac{\ell \cdot \ell}{k \cdot k}}{t \cdot t}}{t} \]

Alternative 21
Error	31.4
Cost	832

\[\frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{k} \]

Alternative 22
Error	30.3
Cost	832

\[\frac{\frac{\ell}{t \cdot t}}{t \cdot \frac{k}{\frac{\ell}{k}}} \]

Alternative 23
Error	29.0
Cost	832

\[\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{t \cdot k} \]

Alternative 24
Error	28.9
Cost	832

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

Reproduce

herbie shell --seed 2022298 
(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))))