Average Error: 32.4 → 8.4
Time: 1.4min
Precision: binary64
Cost: 46220
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := {\left(\left(\frac{1}{t} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\ t_2 := {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{t_2}}\right)}^{-2} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot t_2}\right)\\ \end{array} \]
(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
          (*
           (* (/ 1.0 t) (cbrt (/ l (* (tan k) (+ 2.0 (pow (/ k t) 2.0))))))
           (cbrt (/ (* l 2.0) (sin k))))
          3.0))
        (t_2 (pow (cbrt k) 2.0)))
   (if (<= t -6.5e-6)
     t_1
     (if (<= t 6.9e-124)
       (* (* 2.0 (/ (* (/ l k) (/ 1.0 k)) t)) (/ (/ l (sin k)) (tan k)))
       (if (<= t 7.2e+229)
         t_1
         (*
          (pow (/ t (/ (cbrt l) t_2)) -2.0)
          (* (cbrt l) (/ l (* t t_2)))))))))
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((((1.0 / t) * cbrt((l / (tan(k) * (2.0 + pow((k / t), 2.0)))))) * cbrt(((l * 2.0) / sin(k)))), 3.0);
	double t_2 = pow(cbrt(k), 2.0);
	double tmp;
	if (t <= -6.5e-6) {
		tmp = t_1;
	} else if (t <= 6.9e-124) {
		tmp = (2.0 * (((l / k) * (1.0 / k)) / t)) * ((l / sin(k)) / tan(k));
	} else if (t <= 7.2e+229) {
		tmp = t_1;
	} else {
		tmp = pow((t / (cbrt(l) / t_2)), -2.0) * (cbrt(l) * (l / (t * t_2)));
	}
	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((((1.0 / t) * Math.cbrt((l / (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0)))))) * Math.cbrt(((l * 2.0) / Math.sin(k)))), 3.0);
	double t_2 = Math.pow(Math.cbrt(k), 2.0);
	double tmp;
	if (t <= -6.5e-6) {
		tmp = t_1;
	} else if (t <= 6.9e-124) {
		tmp = (2.0 * (((l / k) * (1.0 / k)) / t)) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (t <= 7.2e+229) {
		tmp = t_1;
	} else {
		tmp = Math.pow((t / (Math.cbrt(l) / t_2)), -2.0) * (Math.cbrt(l) * (l / (t * t_2)));
	}
	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(Float64(Float64(1.0 / t) * cbrt(Float64(l / Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))))) * cbrt(Float64(Float64(l * 2.0) / sin(k)))) ^ 3.0
	t_2 = cbrt(k) ^ 2.0
	tmp = 0.0
	if (t <= -6.5e-6)
		tmp = t_1;
	elseif (t <= 6.9e-124)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(1.0 / k)) / t)) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (t <= 7.2e+229)
		tmp = t_1;
	else
		tmp = Float64((Float64(t / Float64(cbrt(l) / t_2)) ^ -2.0) * Float64(cbrt(l) * Float64(l / Float64(t * t_2))));
	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[(N[(N[(1.0 / t), $MachinePrecision] * N[Power[N[(l / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -6.5e-6], t$95$1, If[LessEqual[t, 6.9e-124], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e+229], t$95$1, N[(N[Power[N[(t / N[(N[Power[l, 1/3], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(l / N[(t * t$95$2), $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(\left(\frac{1}{t} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\
t_2 := {\left(\sqrt[3]{k}\right)}^{2}\\
\mathbf{if}\;t \leq -6.5 \cdot 10^{-6}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 6.9 \cdot 10^{-124}:\\
\;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\

\mathbf{elif}\;t \leq 7.2 \cdot 10^{+229}:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{t_2}}\right)}^{-2} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot t_2}\right)\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -6.4999999999999996e-6 or 6.9e-124 < t < 7.19999999999999973e229

    1. Initial program 24.4

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

    2. Simplified21.0

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      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))))): 1 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)))): 3 points increase in error, 2 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))))): 15 points increase in error, 3 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)))))): 3 points increase in error, 5 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)))): 12 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))))): 9 points increase in error, 12 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)))): 6 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, 3 points decrease in error

    3. Applied egg-rr15.8

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

    4. Applied egg-rr7.0

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

    5. Simplified7.0

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      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))): 7 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))): 169 points increase in error, 38 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))))): 38 points increase in error, 169 points decrease in error

    6. Applied egg-rr22.5

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

    7. Simplified7.0

      \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{t} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}}^{3} \]
      Proof
      (*.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)))): 0 points increase in error, 0 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))))))): 33 points increase in error, 9 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)): 28 points increase in error, 80 points decrease in error

    8. Applied egg-rr7.0

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

    if -6.4999999999999996e-6 < t < 6.9e-124

    1. Initial program 54.7

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

    2. Simplified52.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 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))))): 11 points increase in error, 2 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))))): 10 points increase in error, 17 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)))): 3 points increase in error, 1 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)))): 37 points increase in error, 4 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, 2 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, 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)): 5 points increase in error, 36 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 5 points decrease in error

    3. Taylor expanded in t around 0 20.8

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

    4. Simplified18.9

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 l (*.f64 k k)) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 k 2))) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 31 points increase in error, 30 points decrease in error

    5. Applied egg-rr12.9

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

    if 7.19999999999999973e229 < t

    1. Initial program 18.6

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

    2. Simplified17.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 11 points increase in error, 2 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))))): 10 points increase in error, 17 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)))): 3 points increase in error, 1 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)))): 37 points increase in error, 4 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, 2 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, 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)): 5 points increase in error, 36 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 2 points increase in error, 5 points decrease in error

    3. Taylor expanded in k around 0 26.6

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

    4. Simplified20.1

      \[\leadsto \color{blue}{\frac{\ell}{\frac{\left(k \cdot k\right) \cdot {t}^{3}}{\ell}}} \]
      Proof
      (/.f64 l (/.f64 (*.f64 (*.f64 k k) (pow.f64 t 3)) l)): 0 points increase in error, 0 points decrease in error
      (/.f64 l (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (pow.f64 t 3)) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 46 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (pow.f64 t 3))): 0 points increase in error, 0 points decrease in error

    5. Applied egg-rr3.4

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

    6. Simplified3.4

      \[\leadsto \color{blue}{{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{k}\right)}^{2}}}\right)}^{-2} \cdot \left(\frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}} \cdot \sqrt[3]{\ell}\right)} \]
      Proof
      (*.f64 (pow.f64 (/.f64 t (/.f64 (cbrt.f64 l) (pow.f64 (cbrt.f64 k) 2))) -2) (*.f64 (/.f64 l (*.f64 t (pow.f64 (cbrt.f64 k) 2))) (cbrt.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t (pow.f64 (cbrt.f64 k) 2)) (cbrt.f64 l))) -2) (*.f64 (/.f64 l (*.f64 t (pow.f64 (cbrt.f64 k) 2))) (cbrt.f64 l))): 22 points increase in error, 17 points decrease in error
      (*.f64 (pow.f64 (/.f64 (*.f64 t (pow.f64 (cbrt.f64 k) 2)) (cbrt.f64 l)) -2) (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 (*.f64 t (pow.f64 (cbrt.f64 k) 2)) (cbrt.f64 l))))): 10 points increase in error, 20 points decrease in error
  3. Recombined 3 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{1}{t} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+229}:\\ \;\;\;\;{\left(\left(\frac{1}{t} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{k}\right)}^{2}}}\right)}^{-2} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error8.5
Cost40140
\[\begin{array}{l} t_1 := {\left(\left(\frac{1}{t} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-127}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 2
Error10.9
Cost40084
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\frac{\ell \cdot 2}{\sin k}} \cdot \frac{\sqrt[3]{0.5 \cdot \left(\ell \cdot \frac{\cos k}{\sin k}\right)}}{t}\right)}^{3}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \frac{\frac{t_2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(0.5 \cdot \frac{{t}^{3}}{\ell}\right)}}{\tan k}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{t_2}{\tan k}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 3
Error10.9
Cost40084
\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \frac{\frac{t_2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(0.5 \cdot \frac{{t}^{3}}{\ell}\right)}}{\tan k}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(t_1 \cdot \frac{\sqrt[3]{0.5 \cdot \frac{\ell \cdot \cos k}{\sin k}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-88}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{t_2}{\tan k}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+226}:\\ \;\;\;\;{\left(t_1 \cdot \frac{\sqrt[3]{0.5 \cdot \left(\ell \cdot \frac{\cos k}{\sin k}\right)}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 4
Error8.5
Cost40012
\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\sqrt[3]{2 \cdot t_1}}{t}\right)}^{3}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 5
Error8.4
Cost40012
\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell}{\tan k \cdot t_2}} \cdot \frac{\sqrt[3]{2 \cdot t_1}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-124}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+229}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell \cdot 2}{\sin k}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{\tan k}}{t_2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 6
Error11.3
Cost27344
\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\frac{t_1}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(0.5 \cdot \frac{{t}^{3}}{\ell}\right)}}{\tan k}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-90}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{t_1}{\tan k}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 7
Error12.1
Cost27212
\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 8
Error12.7
Cost27020
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{t}{\frac{1}{t} \cdot \frac{\ell}{t}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot \left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)\right)}^{2}}\\ \end{array} \]
Alternative 9
Error12.6
Cost26572
\[\begin{array}{l} t_1 := \frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-120}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot t_2\\ \mathbf{elif}\;t \leq 10^{+56}:\\ \;\;\;\;t_2 \cdot \frac{\frac{2}{\frac{t}{\frac{1}{t} \cdot \frac{\ell}{t}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error14.4
Cost21132
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-120}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{t}{\frac{1}{t} \cdot \frac{\ell}{t}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \end{array} \]
Alternative 11
Error14.4
Cost21004
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-120}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{t \cdot t}{\frac{\ell}{t}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \end{array} \]
Alternative 12
Error14.4
Cost21004
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-120}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{t \cdot \frac{t \cdot t}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \end{array} \]
Alternative 13
Error16.0
Cost20236
\[\begin{array}{l} t_1 := \left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;k \leq -1.45 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-147}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot \sqrt[3]{k \cdot k}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error14.5
Cost14152
\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-36}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 15
Error17.1
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -64000000000000:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 16
Error16.6
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 17
Error14.8
Cost14024
\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-36}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(t \cdot k\right)} \cdot \left(2 \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 18
Error20.8
Cost13512
\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot {\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
Alternative 19
Error20.7
Cost7436
\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t_1}\\ \end{array} \]
Alternative 20
Error21.4
Cost7304
\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-40}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 21
Error21.4
Cost7304
\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k} \cdot \frac{1}{k}}{t}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t_1}\\ \end{array} \]
Alternative 22
Error32.8
Cost1480
\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}}\\ \mathbf{if}\;k \leq -3.7 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 23
Error32.9
Cost1480
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := t_1 + \ell \cdot -0.16666666666666666\\ \mathbf{if}\;k \leq -8.8 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{k \cdot \left(t \cdot k\right)}{t_2}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{t_1}{t}\right) \cdot t_2\\ \end{array} \]
Alternative 24
Error32.6
Cost1480
\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
Alternative 25
Error33.5
Cost1092
\[\begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot 0.3333333333333333}{t \cdot \left(-k\right)}\\ \end{array} \]
Alternative 26
Error35.0
Cost1032
\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{-112}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot 0.3333333333333333}{t \cdot \left(-k\right)}\\ \end{array} \]
Alternative 27
Error35.3
Cost968
\[\begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{-112}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+80}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t_1}\right)\\ \end{array} \]
Alternative 28
Error34.8
Cost968
\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{-112}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 10^{-138}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]
Alternative 29
Error35.0
Cost968
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.92 \cdot 10^{-110}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-172}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{\frac{k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]
Alternative 30
Error34.6
Cost968
\[\begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-110}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{\frac{k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]
Alternative 31
Error34.8
Cost964
\[\begin{array}{l} t_1 := k \cdot \left(t \cdot k\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-281}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t_1}\right)\\ \end{array} \]
Alternative 32
Error37.4
Cost704
\[-0.3333333333333333 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \]
Alternative 33
Error36.1
Cost704
\[-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \]

Error

Reproduce

herbie shell --seed 2022339 
(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))))