Average Error: 32.7 → 6.1
Time: 34.2s
Precision: binary64
Cost: 52616
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
\[\begin{array}{l} t_1 := \frac{2}{{\left(\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
           (*
            (* (pow (cbrt l) -2.0) (* t (cbrt (sin k))))
            (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
           3.0))))
   (if (<= t -1e-32)
     t_1
     (if (<= t 5e-64)
       (* (* 2.0 (/ l (* k (* t k)))) (/ (/ 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(((pow(cbrt(l), -2.0) * (t * cbrt(sin(k)))) * cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))), 3.0);
	double tmp;
	if (t <= -1e-32) {
		tmp = t_1;
	} else if (t <= 5e-64) {
		tmp = (2.0 * (l / (k * (t * k)))) * ((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.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt(Math.sin(k)))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))), 3.0);
	double tmp;
	if (t <= -1e-32) {
		tmp = t_1;
	} else if (t <= 5e-64) {
		tmp = (2.0 * (l / (k * (t * k)))) * ((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((cbrt(l) ^ -2.0) * Float64(t * cbrt(sin(k)))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) ^ 3.0))
	tmp = 0.0
	if (t <= -1e-32)
		tmp = t_1;
	elseif (t <= 5e-64)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(k * Float64(t * k)))) * 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[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e-32], t$95$1, If[LessEqual[t, 5e-64], N[(N[(2.0 * N[(l / N[(k * N[(t * k), $MachinePrecision]), $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({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\
\mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\
\;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\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 2 regimes
  2. if t < -1.00000000000000006e-32 or 5.00000000000000033e-64 < t

    1. Initial program 23.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    2. Applied egg-rr18.6

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    3. Applied egg-rr7.9

      \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    4. Applied egg-rr7.8

      \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{\ell}}\right)}}^{2}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    5. Applied egg-rr3.5

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

    if -1.00000000000000006e-32 < t < 5.00000000000000033e-64

    1. Initial program 55.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    2. Simplified52.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))))): 20 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))))): 17 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)))): 7 points increase in error, 3 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)))): 36 points increase in error, 2 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)): 2 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, 34 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)))): 3 points increase in error, 0 points decrease in error
    3. Taylor expanded in t around 0 20.5

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
    4. Simplified12.1

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (*.f64 2 (/.f64 l (*.f64 k (*.f64 k t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 l (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t)))): 49 points increase in error, 18 points decrease in error
      (*.f64 2 (/.f64 l (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t))): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.
  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{{\left(\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error9.2
Cost46604
\[\begin{array}{l} t_1 := \frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\frac{1}{\sqrt[3]{\ell}}\right)}^{2}\right)\right)}^{3}\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.2391371801762941 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
Alternative 2
Error9.2
Cost46604
\[\begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\frac{1}{\ell}}\right)}^{2}\right)\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.2391371801762941 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
Alternative 3
Error10.8
Cost33808
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + t_3}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 1.2391371801762941 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error10.9
Cost27080
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-34}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + t_3}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 3.139074227889673 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error11.7
Cost21396
\[\begin{array}{l} t_1 := \frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ t_2 := \frac{\ell}{t \cdot k}\\ t_3 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot t_2\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+73}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_2 \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 3.139074227889673 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 6
Error10.9
Cost21264
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{2 + t_3}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot t_1\\ \mathbf{elif}\;t \leq 3.139074227889673 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 7
Error11.4
Cost20620
\[\begin{array}{l} t_1 := \left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{elif}\;k \leq 2.134777612818594 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot {\sin k}^{2}}}{t}\\ \end{array} \]
Alternative 8
Error11.9
Cost14024
\[\begin{array}{l} t_1 := \left(2 \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error17.9
Cost1480
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(\left(-0.5 + \frac{1}{k \cdot k}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error18.1
Cost1352
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error26.9
Cost832
\[\frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{t \cdot k}}{t \cdot k} \]
Alternative 12
Error24.0
Cost832
\[\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k} \]

Error

Reproduce

herbie shell --seed 2022300 
(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))))