Average Error: 32.4 → 8.5
Time: 53.0s
Precision: binary64
Cost: 52680
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := {\left(t \cdot k\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.545798596712696 \cdot 10^{+176}:\\ \;\;\;\;\frac{\ell}{t_1} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{t_2}}\right)}^{3}\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{{t}^{3}}}{t_2 \cdot \frac{\sin k}{\ell}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
(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 (* t k) 2.0)) (t_2 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -5.545798596712696e+176)
     (* (/ l t_1) (/ l t))
     (if (<= t -1e-12)
       (pow
        (/
         (/ (cbrt (* l 2.0)) (* (cbrt (* (sin k) (tan k))) (/ t (cbrt l))))
         (cbrt t_2))
        3.0)
       (if (<= t 1e-68)
         (* (/ (* (/ l k) (/ 2.0 t)) k) (/ (/ l (sin k)) (tan k)))
         (if (<= t 1.0591470171781518e+102)
           (/ (/ (* l (/ 2.0 (pow t 3.0))) (* t_2 (/ (sin k) l))) (tan k))
           (/ 1.0 (* (/ t_1 l) (/ t l)))))))))
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((t * k), 2.0);
	double t_2 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -5.545798596712696e+176) {
		tmp = (l / t_1) * (l / t);
	} else if (t <= -1e-12) {
		tmp = pow(((cbrt((l * 2.0)) / (cbrt((sin(k) * tan(k))) * (t / cbrt(l)))) / cbrt(t_2)), 3.0);
	} else if (t <= 1e-68) {
		tmp = (((l / k) * (2.0 / t)) / k) * ((l / sin(k)) / tan(k));
	} else if (t <= 1.0591470171781518e+102) {
		tmp = ((l * (2.0 / pow(t, 3.0))) / (t_2 * (sin(k) / l))) / tan(k);
	} else {
		tmp = 1.0 / ((t_1 / l) * (t / l));
	}
	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((t * k), 2.0);
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -5.545798596712696e+176) {
		tmp = (l / t_1) * (l / t);
	} else if (t <= -1e-12) {
		tmp = Math.pow(((Math.cbrt((l * 2.0)) / (Math.cbrt((Math.sin(k) * Math.tan(k))) * (t / Math.cbrt(l)))) / Math.cbrt(t_2)), 3.0);
	} else if (t <= 1e-68) {
		tmp = (((l / k) * (2.0 / t)) / k) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (t <= 1.0591470171781518e+102) {
		tmp = ((l * (2.0 / Math.pow(t, 3.0))) / (t_2 * (Math.sin(k) / l))) / Math.tan(k);
	} else {
		tmp = 1.0 / ((t_1 / l) * (t / l));
	}
	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 = Float64(t * k) ^ 2.0
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -5.545798596712696e+176)
		tmp = Float64(Float64(l / t_1) * Float64(l / t));
	elseif (t <= -1e-12)
		tmp = Float64(Float64(cbrt(Float64(l * 2.0)) / Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t / cbrt(l)))) / cbrt(t_2)) ^ 3.0;
	elseif (t <= 1e-68)
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(2.0 / t)) / k) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (t <= 1.0591470171781518e+102)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(t_2 * Float64(sin(k) / l))) / tan(k));
	else
		tmp = Float64(1.0 / Float64(Float64(t_1 / l) * Float64(t / l)));
	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[(t * k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.545798596712696e+176], N[(N[(l / t$95$1), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-12], N[Power[N[(N[(N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t, 1e-68], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0591470171781518e+102], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$1 / l), $MachinePrecision] * N[(t / l), $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(t \cdot k\right)}^{2}\\
t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -5.545798596712696 \cdot 10^{+176}:\\
\;\;\;\;\frac{\ell}{t_1} \cdot \frac{\ell}{t}\\

\mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\
\;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{t_2}}\right)}^{3}\\

\mathbf{elif}\;t \leq 10^{-68}:\\
\;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\

\mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\
\;\;\;\;\frac{\frac{\ell \cdot \frac{2}{{t}^{3}}}{t_2 \cdot \frac{\sin k}{\ell}}}{\tan k}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{t_1}{\ell} \cdot \frac{t}{\ell}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if t < -5.5457985967126961e176

    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. Simplified20.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 1 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))))): 12 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 30 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 5 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)): 3 points increase in error, 28 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)))): 4 points increase in error, 1 points decrease in error

    3. Taylor expanded in k around 0 29.7

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

    4. Simplified24.7

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
      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)))): 1 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)))): 30 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-rr17.5

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

    6. Applied egg-rr6.4

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

    if -5.5457985967126961e176 < t < -9.9999999999999998e-13

    1. Initial program 22.9

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

    2. Simplified20.2

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 1 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))))): 12 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 30 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 5 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)): 3 points increase in error, 28 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)))): 4 points increase in error, 1 points decrease in error

    3. Applied egg-rr16.7

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

    4. Applied egg-rr23.6

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

    5. Applied egg-rr14.7

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

    6. Applied egg-rr13.8

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

    if -9.9999999999999998e-13 < t < 1.00000000000000007e-68

    1. Initial program 53.7

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

    2. Simplified50.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 1 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))))): 12 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 30 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 5 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)): 3 points increase in error, 28 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)))): 4 points increase in error, 1 points decrease in error

    3. Applied egg-rr43.9

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

    4. Taylor expanded in t around 0 22.0

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

    5. Simplified19.7

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{\ell}{k \cdot k}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 (/.f64 2 t) (/.f64 l (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 t) (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 l) (*.f64 t (pow.f64 k 2)))): 30 points increase in error, 30 points decrease in error
      (/.f64 (*.f64 2 l) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 0 points increase in error, 1 points decrease in error

    6. Applied egg-rr4.9

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

    if 1.00000000000000007e-68 < t < 1.0591470171781518e102

    1. Initial program 20.6

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

    2. Simplified13.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 1 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))))): 12 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 30 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 5 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)): 3 points increase in error, 28 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)))): 4 points increase in error, 1 points decrease in error

    3. Applied egg-rr13.8

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

    4. Applied egg-rr10.8

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

    if 1.0591470171781518e102 < t

    1. Initial program 23.5

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

    2. Simplified22.8

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 1 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))))): 12 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 30 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 5 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)): 3 points increase in error, 28 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)))): 4 points increase in error, 1 points decrease in error

    3. Taylor expanded in k around 0 29.2

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

    4. Simplified26.4

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
      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)))): 1 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)))): 30 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-rr18.3

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

    6. Applied egg-rr9.3

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(t \cdot k\right)}^{2}}{\ell} \cdot \frac{t}{\ell}}} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.545798596712696 \cdot 10^{+176}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{{t}^{3}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot k\right)}^{2}}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error8.7
Cost27344
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {\left(t \cdot k\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.058305558025961 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{t \cdot \frac{t \cdot t}{\ell}}}{t_3}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{{t}^{3}}}} \cdot \frac{\ell}{\tan k}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 2
Error8.7
Cost27344
\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -1.0615760383848548 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{t_3} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t_1}{\tan k \cdot \left(t_2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{{t}^{3}}}} \cdot \frac{\ell}{\tan k}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_3}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 3
Error8.5
Cost27344
\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -1.0615760383848548 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{t_3} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{t_1}{\tan k \cdot \left(t_2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.0591470171781518 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{{t}^{3}}}{t_2 \cdot \frac{\sin k}{\ell}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_3}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 4
Error9.4
Cost21136
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := t_1 \cdot \frac{\frac{\ell \cdot 2}{{t}^{3}}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ t_3 := {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -6.058305558025961 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{t_3} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_3}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 5
Error8.9
Cost21136
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -6.058305558025961 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{t \cdot \frac{t \cdot t}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\ell \cdot 2}{{t}^{3}}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 6
Error10.4
Cost20884
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {\left(t \cdot k\right)}^{2}\\ t_3 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -6.058305558025961 \cdot 10^{+139}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{{t}^{3} \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)}\right)\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 3.06961944074906 \cdot 10^{+66}:\\ \;\;\;\;t_3 \cdot \left(t_3 \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 7
Error10.5
Cost20752
\[\begin{array}{l} t_1 := {\left(t \cdot k\right)}^{2}\\ t_2 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\ell}{t_1} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 3.06961944074906 \cdot 10^{+66}:\\ \;\;\;\;t_2 \cdot \left(t_2 \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 8
Error13.1
Cost14288
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right)\\ t_2 := {\left(t \cdot k\right)}^{2}\\ t_3 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.06961944074906 \cdot 10^{+66}:\\ \;\;\;\;t_3 \cdot \left(t_3 \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 9
Error10.5
Cost14288
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {\left(t \cdot k\right)}^{2}\\ t_3 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 3.06961944074906 \cdot 10^{+66}:\\ \;\;\;\;t_3 \cdot \left(t_3 \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 10
Error10.5
Cost14288
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {\left(t \cdot k\right)}^{2}\\ t_3 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k} \cdot t_1\\ \mathbf{elif}\;t \leq 3.06961944074906 \cdot 10^{+66}:\\ \;\;\;\;t_3 \cdot \left(t_3 \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.0755893373830081 \cdot 10^{+95}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\ell \cdot \frac{2}{t}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_2}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 11
Error13.8
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-24}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \end{array} \]
Alternative 12
Error10.5
Cost14024
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot k}\right)\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10000000000:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\frac{\ell}{t \cdot k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error19.8
Cost7752
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-46}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 10^{-120}:\\ \;\;\;\;\left(\frac{2}{t} \cdot t_1\right) \cdot \mathsf{fma}\left(\ell, -0.16666666666666666, t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \end{array} \]
Alternative 14
Error20.0
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := \frac{1}{t} \cdot {\left(\frac{\ell}{t \cdot k}\right)}^{2}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 15
Error20.2
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\frac{\ell}{t \cdot k}\right)}^{2}\\ \end{array} \]
Alternative 16
Error20.2
Cost7176
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \end{array} \]
Alternative 17
Error26.9
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := \frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 18
Error21.5
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := \frac{\frac{\ell}{k}}{t}\\ t_3 := \frac{t_2 \cdot t_2}{t}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 19
Error20.7
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := \frac{\frac{\ell}{k}}{t}\\ t_3 := t_2 \cdot \left(t_2 \cdot \frac{1}{t}\right)\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 20
Error20.7
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_2 \cdot \frac{1}{t}\right)\\ \end{array} \]
Alternative 21
Error33.9
Cost1092
\[\begin{array}{l} \mathbf{if}\;t \leq 10^{-110}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)}\\ \end{array} \]
Alternative 22
Error36.2
Cost832
\[\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)} \]
Alternative 23
Error33.5
Cost832
\[\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t \cdot \left(t \cdot t\right)} \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))