Average Error: 32.5 → 7.3
Time: 30.9s
Precision: binary64
Cost: 59404
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{2}{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right) \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_2 := k \cdot \sqrt{t}\\ \mathbf{if}\;t \leq -10000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-296}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\frac{1}{t_2} \cdot \frac{\ell}{t_2}\right)\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
         (/
          2.0
          (*
           (*
            (pow
             (*
              (log1p (expm1 (cbrt (sin k))))
              (* t (pow (cbrt (/ 1.0 l)) 2.0)))
             3.0)
            (tan k))
           (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))))
        (t_2 (* k (sqrt t))))
   (if (<= t -10000000000.0)
     t_1
     (if (<= t 1e-296)
       (* (/ (* 2.0 l) k) (/ l (* k (* t (* (sin k) (tan k))))))
       (if (<= t 1e-40)
         (* (* 2.0 (* (/ 1.0 t_2) (/ l t_2))) (/ (/ l (sin k)) (tan k)))
         t_1)))))
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 = 2.0 / ((pow((log1p(expm1(cbrt(sin(k)))) * (t * pow(cbrt((1.0 / l)), 2.0))), 3.0) * tan(k)) * (1.0 + (1.0 + pow((k / t), 2.0))));
	double t_2 = k * sqrt(t);
	double tmp;
	if (t <= -10000000000.0) {
		tmp = t_1;
	} else if (t <= 1e-296) {
		tmp = ((2.0 * l) / k) * (l / (k * (t * (sin(k) * tan(k)))));
	} else if (t <= 1e-40) {
		tmp = (2.0 * ((1.0 / t_2) * (l / t_2))) * ((l / sin(k)) / tan(k));
	} else {
		tmp = t_1;
	}
	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 = 2.0 / ((Math.pow((Math.log1p(Math.expm1(Math.cbrt(Math.sin(k)))) * (t * Math.pow(Math.cbrt((1.0 / l)), 2.0))), 3.0) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t), 2.0))));
	double t_2 = k * Math.sqrt(t);
	double tmp;
	if (t <= -10000000000.0) {
		tmp = t_1;
	} else if (t <= 1e-296) {
		tmp = ((2.0 * l) / k) * (l / (k * (t * (Math.sin(k) * Math.tan(k)))));
	} else if (t <= 1e-40) {
		tmp = (2.0 * ((1.0 / t_2) * (l / t_2))) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = t_1;
	}
	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(2.0 / Float64(Float64((Float64(log1p(expm1(cbrt(sin(k)))) * Float64(t * (cbrt(Float64(1.0 / l)) ^ 2.0))) ^ 3.0) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))))
	t_2 = Float64(k * sqrt(t))
	tmp = 0.0
	if (t <= -10000000000.0)
		tmp = t_1;
	elseif (t <= 1e-296)
		tmp = Float64(Float64(Float64(2.0 * l) / k) * Float64(l / Float64(k * Float64(t * Float64(sin(k) * tan(k))))));
	elseif (t <= 1e-40)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 / t_2) * Float64(l / t_2))) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = t_1;
	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[(2.0 / N[(N[(N[Power[N[(N[Log[1 + N[(Exp[N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(t * N[Power[N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -10000000000.0], t$95$1, If[LessEqual[t, 1e-296], N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-40], N[(N[(2.0 * N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\begin{array}{l}
t_1 := \frac{2}{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right) \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\
t_2 := k \cdot \sqrt{t}\\
\mathbf{if}\;t \leq -10000000000:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 10^{-296}:\\
\;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\

\mathbf{elif}\;t \leq 10^{-40}:\\
\;\;\;\;\left(2 \cdot \left(\frac{1}{t_2} \cdot \frac{\ell}{t_2}\right)\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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 < -1e10 or 9.9999999999999993e-41 < t

    1. Initial program 22.1

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

    2. Applied egg-rr17.3

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

    3. Applied egg-rr7.0

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

    4. Applied egg-rr7.0

      \[\leadsto \frac{2}{\left({\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right)} \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

    if -1e10 < t < 1e-296

    1. Initial program 48.6

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

    2. Simplified43.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 27 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))))): 8 points increase in error, 16 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)))): 2 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)))): 28 points increase in error, 0 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))): 0 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)): 1 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 35 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, 0 points decrease in error

    3. Taylor expanded in t around 0 22.2

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

    4. Applied egg-rr22.2

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

    5. Applied egg-rr9.8

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

    if 1e-296 < t < 9.9999999999999993e-41

    1. Initial program 53.8

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

    2. Simplified50.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 27 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))))): 8 points increase in error, 16 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)))): 2 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)))): 28 points increase in error, 0 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))): 0 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)): 1 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 2 points increase in error, 35 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, 0 points decrease in error

    3. Taylor expanded in t around 0 20.6

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

    4. Applied egg-rr5.1

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -10000000000:\\ \;\;\;\;\frac{2}{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right) \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-296}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\frac{1}{k \cdot \sqrt{t}} \cdot \frac{\ell}{k \cdot \sqrt{t}}\right)\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right) \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error7.3
Cost46604
\[\begin{array}{l} t_1 := \frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3}\right)}\\ t_2 := k \cdot \sqrt{t}\\ \mathbf{if}\;t \leq -10000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-296}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\frac{1}{t_2} \cdot \frac{\ell}{t_2}\right)\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error9.8
Cost27924
\[\begin{array}{l} t_1 := k \cdot \sqrt{t}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \frac{t_2}{\tan k}\\ t_4 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.890186470054164 \cdot 10^{+112}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_3 \cdot \frac{\frac{2}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{t_5}\\ \mathbf{elif}\;t \leq 10^{-296}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\frac{1}{t_1} \cdot \frac{\ell}{t_1}\right)\right) \cdot t_3\\ \mathbf{elif}\;t \leq 5.42152087124875 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\frac{\ell}{t} \cdot \frac{1}{t}}}}{t_5} \cdot \left(\frac{t_2}{\sin k} \cdot \cos k\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 3
Error9.8
Cost27344
\[\begin{array}{l} t_1 := k \cdot \sqrt{t}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_3 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.890186470054164 \cdot 10^{+112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_2 \cdot \frac{\frac{2}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 10^{-296}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\frac{1}{t_1} \cdot \frac{\ell}{t_1}\right)\right) \cdot t_2\\ \mathbf{elif}\;t \leq 5.42152087124875 \cdot 10^{+157}:\\ \;\;\;\;t_2 \cdot \frac{\frac{2}{\frac{t}{\frac{\ell}{t} \cdot \frac{1}{t}}}}{2 + \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 4
Error11.4
Cost20872
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ t_3 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.890186470054164 \cdot 10^{+112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-305}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot t_2\\ \mathbf{elif}\;t \leq 10^{-191}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2 \cdot \ell}{t}}{k \cdot k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k}{t_2}}\\ \mathbf{elif}\;t \leq 5.42152087124875 \cdot 10^{+157}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{t}{\frac{\ell}{t} \cdot \frac{1}{t}}}}{2 + \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 5
Error11.4
Cost15320
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := t_1 \cdot \frac{\frac{2}{\frac{t}{\frac{\ell}{t} \cdot \frac{1}{t}}}}{2 + \frac{k}{t \cdot \frac{t}{k}}}\\ t_3 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_4 := \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{if}\;t \leq -1.890186470054164 \cdot 10^{+112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-305}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot t_4\\ \mathbf{elif}\;t \leq 10^{-191}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2 \cdot \ell}{t}}{k \cdot k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k}{t_4}}\\ \mathbf{elif}\;t \leq 5.42152087124875 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 6
Error10.5
Cost14156
\[\begin{array}{l} t_1 := \frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\ell}{\frac{t}{{k}^{-2}}}}{t}\\ \mathbf{elif}\;k \leq 10^{-30}:\\ \;\;\;\;\frac{{\left(\frac{\ell \cdot {k}^{-1}}{t}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error20.6
Cost13640
\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell \cdot {k}^{-1}}{t}\right)}^{2}}{t}\\ \end{array} \]
Alternative 8
Error19.8
Cost7436
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{elif}\;t \leq 3.8495645867799676 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error22.2
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t}}{t}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-36}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error25.0
Cost1096
\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t}}{t}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-280}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error28.7
Cost832
\[\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}{t} \]

Error

Reproduce

herbie shell --seed 2022290 
(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))))