Average Error: 31.9 → 8.4
Time: 28.3s
Precision: binary64
Cost: 40396
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{-2}\\ t_2 := \frac{\sqrt[3]{\ell \cdot 2}}{t}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-46}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot t_3}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{{t_2}^{2}}{t_3 \cdot \sin k} \cdot \frac{t_2}{\frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \end{array} \]
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt k) -2.0))
        (t_2 (/ (cbrt (* l 2.0)) t))
        (t_3 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -6.8e-46)
     (pow
      (* (cbrt (/ l (* (tan k) t_3))) (/ (cbrt (/ 2.0 (/ (sin k) l))) t))
      3.0)
     (if (<= t 0.00192)
       (* 2.0 (/ (* l (/ (cos k) k)) (* k (* t (/ (pow (sin k) 2.0) l)))))
       (if (<= t 5e+151)
         (* (/ (pow t_2 2.0) (* t_3 (sin k))) (/ t_2 (/ (tan k) l)))
         (* (* (/ l t) (* t_1 (/ t_1 t))) (/ l (* t (pow (cbrt k) 2.0)))))))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}
↓
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(k), -2.0);
	double t_2 = cbrt((l * 2.0)) / t;
	double t_3 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -6.8e-46) {
		tmp = pow((cbrt((l / (tan(k) * t_3))) * (cbrt((2.0 / (sin(k) / l))) / t)), 3.0);
	} else if (t <= 0.00192) {
		tmp = 2.0 * ((l * (cos(k) / k)) / (k * (t * (pow(sin(k), 2.0) / l))));
	} else if (t <= 5e+151) {
		tmp = (pow(t_2, 2.0) / (t_3 * sin(k))) * (t_2 / (tan(k) / l));
	} else {
		tmp = ((l / t) * (t_1 * (t_1 / t))) * (l / (t * pow(cbrt(k), 2.0)));
	}
	return tmp;
}

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}
↓
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(k), -2.0);
	double t_2 = Math.cbrt((l * 2.0)) / t;
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -6.8e-46) {
		tmp = Math.pow((Math.cbrt((l / (Math.tan(k) * t_3))) * (Math.cbrt((2.0 / (Math.sin(k) / l))) / t)), 3.0);
	} else if (t <= 0.00192) {
		tmp = 2.0 * ((l * (Math.cos(k) / k)) / (k * (t * (Math.pow(Math.sin(k), 2.0) / l))));
	} else if (t <= 5e+151) {
		tmp = (Math.pow(t_2, 2.0) / (t_3 * Math.sin(k))) * (t_2 / (Math.tan(k) / l));
	} else {
		tmp = ((l / t) * (t_1 * (t_1 / t))) * (l / (t * Math.pow(Math.cbrt(k), 2.0)));
	}
	return tmp;
}

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 = cbrt(k) ^ -2.0
	t_2 = Float64(cbrt(Float64(l * 2.0)) / t)
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -6.8e-46)
		tmp = Float64(cbrt(Float64(l / Float64(tan(k) * t_3))) * Float64(cbrt(Float64(2.0 / Float64(sin(k) / l))) / t)) ^ 3.0;
	elseif (t <= 0.00192)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(cos(k) / k)) / Float64(k * Float64(t * Float64((sin(k) ^ 2.0) / l)))));
	elseif (t <= 5e+151)
		tmp = Float64(Float64((t_2 ^ 2.0) / Float64(t_3 * sin(k))) * Float64(t_2 / Float64(tan(k) / l)));
	else
		tmp = Float64(Float64(Float64(l / t) * Float64(t_1 * Float64(t_1 / t))) * Float64(l / Float64(t * (cbrt(k) ^ 2.0))));
	end
	return tmp
end

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[k, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e-46], N[Power[N[(N[Power[N[(l / N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[(2.0 / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t, 0.00192], N[(2.0 * N[(N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+151], N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] / N[(t$95$3 * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\begin{array}{l}
t_1 := {\left(\sqrt[3]{k}\right)}^{-2}\\
t_2 := \frac{\sqrt[3]{\ell \cdot 2}}{t}\\
t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -6.8 \cdot 10^{-46}:\\
\;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot t_3}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell}}}}{t}\right)}^{3}\\

\mathbf{elif}\;t \leq 0.00192:\\
\;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\

\mathbf{elif}\;t \leq 5 \cdot 10^{+151}:\\
\;\;\;\;\frac{{t_2}^{2}}{t_3 \cdot \sin k} \cdot \frac{t_2}{\frac{\tan k}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\ell}{t} \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -6.79999999999999992e-46

    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. Simplified18.6

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      Proof
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (tan.f64 k)) l)))): 1 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 18 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))))): 1 points increase in error, 7 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 l l)))): 8 points increase in error, 10 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 6 points increase in error, 8 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k))) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 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)))): 2 points increase in error, 2 points decrease in error

    3. Applied egg-rr14.5

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

    4. Applied egg-rr6.3

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

    5. Simplified6.3

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      Proof
      (*.f64 (cbrt.f64 (/.f64 l (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (cbrt.f64 (*.f64 l 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (cbrt.f64 (*.f64 l 2))): 5 points increase in error, 2 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (Rewrite<= unpow1/3_binary64 (pow.f64 (*.f64 2 l) 1/3))): 168 points increase in error, 29 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 2 l) 1/3) (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/3_binary64 (cbrt.f64 (*.f64 2 l))) (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 29 points increase in error, 168 points decrease in error

    6. Applied egg-rr23.8

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{t} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)} - 1\right)}}^{3} \]

    7. Simplified6.3

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell}}}}{t}\right)}}^{3} \]
      Proof
      (*.f64 (cbrt.f64 (/.f64 l (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 (cbrt.f64 (/.f64 2 (/.f64 (sin.f64 k) l))) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 (cbrt.f64 (/.f64 2 (/.f64 (sin.f64 k) l))) t)): 4 points increase in error, 4 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (/.f64 (cbrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 l) (sin.f64 k)))) t)): 12 points increase in error, 8 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (/.f64 (cbrt.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 l 2)) (sin.f64 k))) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))) 1)) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 1 (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))))) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 t) (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k)))))): 13 points increase in error, 19 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 t (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))))))): 22 points increase in error, 15 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) 1) (/.f64 t (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k)))))): 9 points increase in error, 26 points decrease in error
      (/.f64 (Rewrite=> *-rgt-identity_binary64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (/.f64 t (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) t) (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))))): 27 points increase in error, 13 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (/.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) t) (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k))))))): 13 points increase in error, 7 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (/.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) t) (cbrt.f64 (/.f64 (*.f64 l 2) (sin.f64 k)))))) 1)): 17 points increase in error, 102 points decrease in error

    if -6.79999999999999992e-46 < t < 0.00192000000000000005

    1. Initial program 51.3

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

    2. Simplified51.7

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      Proof
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (tan.f64 k)) l)))): 1 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 18 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))))): 1 points increase in error, 7 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 l l)))): 8 points increase in error, 10 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 6 points increase in error, 8 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k))) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 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)))): 2 points increase in error, 2 points decrease in error

    3. Taylor expanded in t around 0 25.6

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

    4. Simplified25.1

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{{\sin k}^{2} \cdot t}{\ell}}\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 l (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 l (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 32 points increase in error, 8 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 22 points increase in error, 24 points decrease in error

    5. Applied egg-rr12.6

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

    if 0.00192000000000000005 < t < 5.0000000000000002e151

    1. Initial program 22.5

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

    2. Simplified15.7

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      Proof
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (tan.f64 k)) l)))): 1 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 18 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))))): 1 points increase in error, 7 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 l l)))): 8 points increase in error, 10 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 6 points increase in error, 8 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k))) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 7 points increase in error, 3 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)))): 2 points increase in error, 2 points decrease in error

    3. Applied egg-rr17.5

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

    4. Applied egg-rr6.3

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

    if 5.0000000000000002e151 < t

    1. Initial program 21.6

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

    2. Simplified21.6

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (*.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 (Rewrite<= associate-*l/_binary64 (*.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)))))): 7 points increase in error, 3 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)))): 2 points increase in error, 2 points decrease in error

    3. Taylor expanded in k around 0 29.0

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

    4. Simplified29.0

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
      Proof
      (/.f64 (*.f64 l l) (*.f64 (pow.f64 t 3) (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 t 3) (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 t 3) (Rewrite<= unpow2_binary64 (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-rr8.6

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

    6. Applied egg-rr5.7

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

    7. Simplified5.7

      \[\leadsto \color{blue}{\left(\frac{\ell}{t} \cdot \left({\left(\sqrt[3]{k}\right)}^{-2} \cdot \frac{{\left(\sqrt[3]{k}\right)}^{-2}}{t}\right)\right)} \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}} \]
      Proof
      (*.f64 (/.f64 l t) (*.f64 (pow.f64 (cbrt.f64 k) -2) (/.f64 (pow.f64 (cbrt.f64 k) -2) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l t) (pow.f64 (cbrt.f64 k) -2)) (/.f64 (pow.f64 (cbrt.f64 k) -2) t))): 23 points increase in error, 21 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 (cbrt.f64 k) -2) t) (*.f64 (/.f64 l t) (pow.f64 (cbrt.f64 k) -2)))): 0 points increase in error, 0 points decrease in error
  3. Recombined 4 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-46}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell \cdot 2}}{t}\right)}^{2}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \frac{\frac{\sqrt[3]{\ell \cdot 2}}{t}}{\frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left({\left(\sqrt[3]{k}\right)}^{-2} \cdot \frac{{\left(\sqrt[3]{k}\right)}^{-2}}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error11.3
Cost40012
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ t_2 := {\left(\sqrt[3]{k}\right)}^{-2}\\ \mathbf{if}\;k \leq -1.15 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.26 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(t_2 \cdot \frac{t_2}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 10^{+32}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{t}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{\tan k}{\ell}\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error12.3
Cost39816
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ t_2 := {\left(\sqrt[3]{k}\right)}^{-2}\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(t_2 \cdot \frac{t_2}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error9.9
Cost39816
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{-2}\\ \mathbf{if}\;t \leq -1.56 \cdot 10^{-46}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell}}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 0.0021:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\right) \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \end{array} \]
Alternative 4
Error12.5
Cost33292
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_3 := \frac{\ell}{t_2}\\ \mathbf{if}\;k \leq -2.95 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -5.4 \cdot 10^{-216}:\\ \;\;\;\;t_3 \cdot \frac{\frac{\ell}{t} \cdot {\left(\sqrt[3]{k}\right)}^{-4}}{t}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;t_3 \cdot \frac{\ell}{{t_2}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error13.3
Cost27088
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ t_2 := \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}} \cdot \frac{\frac{\ell}{t} \cdot {\left(\sqrt[3]{k}\right)}^{-4}}{t}\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 8.6 \cdot 10^{-166}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error14.7
Cost26956
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-164}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}} \cdot \frac{\ell}{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error14.4
Cost20488
\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-32}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 8
Error17.2
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 9
Error17.2
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-34}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 10
Error20.3
Cost13644
\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-47}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-53}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t \cdot \frac{k}{\frac{\ell}{k}}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 11
Error20.8
Cost13644
\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t \cdot \frac{k}{\frac{\ell}{k}}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{{t}^{1.5}}}{k}\right)}^{2}\\ \end{array} \]
Alternative 12
Error20.2
Cost7884
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ t_2 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.06 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{t \cdot \frac{k}{\frac{\ell}{k}}}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error20.3
Cost7692
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ t_2 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{k} \cdot {k}^{-2}}{k}\right)\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 14
Error20.3
Cost7568
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ t_2 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 15
Error23.6
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 16
Error33.5
Cost1608
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{1}{\frac{t}{\ell \cdot \ell}}}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t \cdot k}}{k}\right)\\ \end{array} \]
Alternative 17
Error35.5
Cost960
\[2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t \cdot k}}{k}\right) \]
Alternative 18
Error34.6
Cost960
\[2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right) \]
Alternative 19
Error34.4
Cost960
\[2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{k \cdot k} \]

Error

Reproduce

herbie shell --seed 2022330 
(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))))