Average Error: 32.1 → 9.2
Time: 33.7s
Precision: binary64
Cost: 33552
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\\ t_2 := \frac{{t_1}^{2}}{k} \cdot \frac{t_1}{k}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.0316794519469151 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{{t}^{3}}}{t_3}}{\tan k \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{elif}\;t \leq 2.8559432303904032 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(t_3 \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (cbrt l) (/ t (cbrt l))))
        (t_2 (* (/ (pow t_1 2.0) k) (/ t_1 k)))
        (t_3 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -1.0316794519469151e+87)
     t_2
     (if (<= t -1.5e-27)
       (/ (* 2.0 (/ (/ l (pow t 3.0)) t_3)) (* (tan k) (/ (sin k) l)))
       (if (<= t 1e-65)
         (* (/ 2.0 (* (/ (tan k) l) (* k (sin k)))) (/ (/ l k) t))
         (if (<= t 2.8559432303904032e+87)
           (/ (/ l (sin k)) (* (tan k) (* t_3 (* (/ (pow t 3.0) l) 0.5))))
           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 = cbrt(l) / (t / cbrt(l));
	double t_2 = (pow(t_1, 2.0) / k) * (t_1 / k);
	double t_3 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -1.0316794519469151e+87) {
		tmp = t_2;
	} else if (t <= -1.5e-27) {
		tmp = (2.0 * ((l / pow(t, 3.0)) / t_3)) / (tan(k) * (sin(k) / l));
	} else if (t <= 1e-65) {
		tmp = (2.0 / ((tan(k) / l) * (k * sin(k)))) * ((l / k) / t);
	} else if (t <= 2.8559432303904032e+87) {
		tmp = (l / sin(k)) / (tan(k) * (t_3 * ((pow(t, 3.0) / l) * 0.5)));
	} else {
		tmp = 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.cbrt(l) / (t / Math.cbrt(l));
	double t_2 = (Math.pow(t_1, 2.0) / k) * (t_1 / k);
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.0316794519469151e+87) {
		tmp = t_2;
	} else if (t <= -1.5e-27) {
		tmp = (2.0 * ((l / Math.pow(t, 3.0)) / t_3)) / (Math.tan(k) * (Math.sin(k) / l));
	} else if (t <= 1e-65) {
		tmp = (2.0 / ((Math.tan(k) / l) * (k * Math.sin(k)))) * ((l / k) / t);
	} else if (t <= 2.8559432303904032e+87) {
		tmp = (l / Math.sin(k)) / (Math.tan(k) * (t_3 * ((Math.pow(t, 3.0) / l) * 0.5)));
	} else {
		tmp = 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(cbrt(l) / Float64(t / cbrt(l)))
	t_2 = Float64(Float64((t_1 ^ 2.0) / k) * Float64(t_1 / k))
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -1.0316794519469151e+87)
		tmp = t_2;
	elseif (t <= -1.5e-27)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / (t ^ 3.0)) / t_3)) / Float64(tan(k) * Float64(sin(k) / l)));
	elseif (t <= 1e-65)
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(k * sin(k)))) * Float64(Float64(l / k) / t));
	elseif (t <= 2.8559432303904032e+87)
		tmp = Float64(Float64(l / sin(k)) / Float64(tan(k) * Float64(t_3 * Float64(Float64((t ^ 3.0) / l) * 0.5))));
	else
		tmp = 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[(N[Power[l, 1/3], $MachinePrecision] / N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.0316794519469151e+87], t$95$2, If[LessEqual[t, -1.5e-27], N[(N[(2.0 * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-65], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8559432303904032e+87], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(t$95$3 * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\begin{array}{l}
t_1 := \frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\\
t_2 := \frac{{t_1}^{2}}{k} \cdot \frac{t_1}{k}\\
t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -1.0316794519469151 \cdot 10^{+87}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t \leq -1.5 \cdot 10^{-27}:\\
\;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{{t}^{3}}}{t_3}}{\tan k \cdot \frac{\sin k}{\ell}}\\

\mathbf{elif}\;t \leq 10^{-65}:\\
\;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\

\mathbf{elif}\;t \leq 2.8559432303904032 \cdot 10^{+87}:\\
\;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(t_3 \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -1.0316794519469151e87 or 2.8559432303904032e87 < t

    1. Initial program 23.0

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

    2. Simplified22.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 21 points increase in error, 4 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, 14 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)))): 1 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)))): 31 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)): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 3 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)))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in k around 0 29.1

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

    4. Simplified26.2

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k \cdot k}} \]
      Proof
      (/.f64 (/.f64 l (/.f64 (pow.f64 t 3) l)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (pow.f64 t 3))) (*.f64 k k)): 32 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 t 3)) (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 t 3)) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 k 2) (pow.f64 t 3)))): 6 points increase in error, 8 points decrease in error

    5. Applied egg-rr12.1

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

    if -1.0316794519469151e87 < t < -1.5000000000000001e-27

    1. Initial program 19.5

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

    2. Simplified12.4

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 21 points increase in error, 4 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, 14 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)))): 1 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)))): 31 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)): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 3 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)))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr11.8

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

    if -1.5000000000000001e-27 < t < 9.99999999999999923e-66

    1. Initial program 54.5

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

    2. Simplified51.4

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 21 points increase in error, 4 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, 14 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)))): 1 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)))): 31 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)): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 3 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)))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in t around 0 21.3

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

    4. Simplified6.9

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 l k) (*.f64 t k))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 l k) k) t))): 47 points increase in error, 37 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 k k))) t)): 43 points increase in error, 25 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, 31 points decrease in error

    5. Applied egg-rr6.1

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

    6. Applied egg-rr4.0

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

    if 9.99999999999999923e-66 < t < 2.8559432303904032e87

    1. Initial program 19.6

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

    2. Simplified12.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 21 points increase in error, 4 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, 14 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)))): 1 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)))): 31 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)): 5 points increase in error, 2 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)): 3 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)))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr9.3

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k}}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right) \cdot \tan k}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0316794519469151 \cdot 10^{+87}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\right)}^{2}}{k} \cdot \frac{\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}}{k}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{elif}\;t \leq 2.8559432303904032 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}\right)}^{2}}{k} \cdot \frac{\frac{\sqrt[3]{\ell}}{\frac{t}{\sqrt[3]{\ell}}}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error13.4
Cost21132
\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{1}{t} \cdot \frac{\ell}{t \cdot t}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \frac{\sin k}{\ell}}\\ \mathbf{if}\;t \leq -1.0316794519469151 \cdot 10^{+87}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell \cdot {t}^{-3}}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error13.4
Cost21132
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.0316794519469151 \cdot 10^{+87}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell \cdot {t}^{-3}}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{t_1} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{1}{t} \cdot \frac{\ell}{t \cdot t}}{t_1}}{\tan k \cdot \frac{\sin k}{\ell}}\\ \end{array} \]
Alternative 3
Error14.4
Cost21004
\[\begin{array}{l} t_1 := \tan k \cdot \sin k\\ \mathbf{if}\;k \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right) \cdot \left(\frac{\ell}{\sin k} \cdot \frac{1}{\tan k}\right)\\ \mathbf{elif}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot {t}^{-3}}{k}\\ \mathbf{elif}\;k \leq 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{2}{t} \cdot \frac{\frac{\ell}{t_1}}{\frac{t}{\ell}}}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot \left(0.5 \cdot \left(k \cdot \frac{t_1}{\ell}\right)\right)}\\ \end{array} \]
Alternative 4
Error14.1
Cost21004
\[\begin{array}{l} t_1 := \tan k \cdot \sin k\\ \mathbf{if}\;k \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right) \cdot \left(\frac{\ell}{\sin k} \cdot \frac{1}{\tan k}\right)\\ \mathbf{elif}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot {t}^{-3}}{k}\\ \mathbf{elif}\;k \leq 10^{+14}:\\ \;\;\;\;\frac{\frac{2}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\frac{\ell}{t_1}}{\frac{t}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot \left(0.5 \cdot \left(k \cdot \frac{t_1}{\ell}\right)\right)}\\ \end{array} \]
Alternative 5
Error11.8
Cost21000
\[\begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error16.0
Cost14288
\[\begin{array}{l} t_1 := \frac{2}{\left(k \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ t_2 := \ell \cdot {t}^{-3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t_2}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{t_2}{k}\\ \mathbf{elif}\;t \leq 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}}{t \cdot t}}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
Alternative 7
Error16.0
Cost14288
\[\begin{array}{l} t_1 := k \cdot \frac{\tan k \cdot \sin k}{\ell}\\ t_2 := \ell \cdot {t}^{-3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t_2}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{t_2}{k}\\ \mathbf{elif}\;t \leq 10^{+55}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot \left(0.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}}{t \cdot t}}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
Alternative 8
Error15.9
Cost14288
\[\begin{array}{l} t_1 := \ell \cdot {t}^{-3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t_1}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+31}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{t_1}{k}\\ \mathbf{elif}\;t \leq 10^{+55}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot \left(0.5 \cdot \left(k \cdot \frac{\tan k \cdot \sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}}{t \cdot t}}{t}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
Alternative 9
Error23.4
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell \cdot {t}^{-3}}{k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error23.5
Cost7304
\[\begin{array}{l} t_1 := \ell \cdot {t}^{-3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t_1}} \cdot \frac{1}{k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{t_1}{k}\\ \end{array} \]
Alternative 11
Error27.4
Cost1352
\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \frac{t \cdot t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{k \cdot k}\\ \end{array} \]
Alternative 12
Error27.9
Cost1224
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t \cdot \frac{t \cdot t}{\ell}}}{k \cdot k}\\ \mathbf{if}\;t \leq -6.3 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k}}{t}}{k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error27.9
Cost1224
\[\begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \frac{t \cdot t}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k}}{t}}{k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{k \cdot k}\\ \end{array} \]
Alternative 14
Error35.6
Cost960
\[\frac{\ell}{k \cdot k} \cdot \frac{2}{\frac{k}{\ell} \cdot \left(t \cdot k\right)} \]
Alternative 15
Error35.0
Cost960
\[\frac{\frac{\frac{\ell \cdot 2}{k}}{t}}{k} \cdot \frac{\ell}{k \cdot k} \]
Alternative 16
Error35.0
Cost960
\[\frac{\frac{\frac{\ell \cdot \left(2 \cdot \frac{\ell}{k}\right)}{t \cdot k}}{k}}{k} \]

Error

Reproduce

herbie shell --seed 2022310 
(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))))