Average Error: 32.1 → 3.6
Time: 1.1min
Precision: binary64
Cost: 52680
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
\[\begin{array}{l} t_1 := {\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
         (pow
          (/
           (*
            (/ (cbrt l) (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
            (cbrt (* l 2.0)))
           (* t (cbrt (sin k))))
          3.0)))
   (if (<= t -1.7e-23)
     t_1
     (if (<= t 4.5e-26)
       (/ (/ (/ l (tan k)) (sin k)) (* (* t (/ k l)) (* k 0.5)))
       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 = pow((((cbrt(l) / cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))) * cbrt((l * 2.0))) / (t * cbrt(sin(k)))), 3.0);
	double tmp;
	if (t <= -1.7e-23) {
		tmp = t_1;
	} else if (t <= 4.5e-26) {
		tmp = ((l / tan(k)) / sin(k)) / ((t * (k / l)) * (k * 0.5));
	} 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 = Math.pow((((Math.cbrt(l) / Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))) * Math.cbrt((l * 2.0))) / (t * Math.cbrt(Math.sin(k)))), 3.0);
	double tmp;
	if (t <= -1.7e-23) {
		tmp = t_1;
	} else if (t <= 4.5e-26) {
		tmp = ((l / Math.tan(k)) / Math.sin(k)) / ((t * (k / l)) * (k * 0.5));
	} 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(Float64(Float64(cbrt(l) / cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) * cbrt(Float64(l * 2.0))) / Float64(t * cbrt(sin(k)))) ^ 3.0
	tmp = 0.0
	if (t <= -1.7e-23)
		tmp = t_1;
	elseif (t <= 4.5e-26)
		tmp = Float64(Float64(Float64(l / tan(k)) / sin(k)) / Float64(Float64(t * Float64(k / l)) * Float64(k * 0.5)));
	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[Power[N[(N[(N[(N[Power[l, 1/3], $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] * N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[t, -1.7e-23], t$95$1, If[LessEqual[t, 4.5e-26], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * 0.5), $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 := {\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\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}\\
\mathbf{if}\;t \leq -1.7 \cdot 10^{-23}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 4.5 \cdot 10^{-26}:\\
\;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\

\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 < -1.7e-23 or 4.4999999999999999e-26 < t

    1. Initial program 21.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    2. Simplified18.5

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      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))))): 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<= +-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)))): 2 points increase in error, 4 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))))): 23 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)))))): 2 points increase in error, 4 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)))): 7 points increase in error, 13 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, 10 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)))): 1 points increase in error, 0 points decrease in error
    3. Applied egg-rr14.6

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}} \]
    4. Applied egg-rr5.8

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{2 \cdot \ell} \cdot \sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
    5. Simplified5.8

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      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))): 1 points increase in error, 1 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))): 161 points increase in error, 32 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))))): 32 points increase in error, 161 points decrease in error
    6. Applied egg-rr2.5

      \[\leadsto {\left(\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \frac{1}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right)} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
    7. Simplified2.5

      \[\leadsto {\left(\frac{\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      Proof
      (/.f64 (cbrt.f64 l) (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (cbrt.f64 l) 1)) (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (cbrt.f64 l) (/.f64 1 (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))))): 17 points increase in error, 19 points decrease in error

    if -1.7e-23 < t < 4.4999999999999999e-26

    1. Initial program 52.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    2. Simplified49.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      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))): 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))))): 19 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))))): 18 points increase in error, 22 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)))): 0 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)))): 34 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 2 points increase in error, 0 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, 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, 33 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 4 points increase in error, 1 points decrease in error
    3. Taylor expanded in 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. Simplified21.4

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) t) l)): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 l) (*.f64 (pow.f64 k 2) t))): 16 points increase in error, 19 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 0 points increase in error, 0 points decrease in error
    5. Applied egg-rr7.2

      \[\leadsto \frac{2}{\color{blue}{k \cdot \left(\left(k \cdot t\right) \cdot \frac{1}{\ell}\right)}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
    6. Applied egg-rr15.1

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right) \cdot 0.5\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
    7. Simplified5.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}} \]
      Proof
      (/.f64 (/.f64 (/.f64 l (tan.f64 k)) (sin.f64 k)) (*.f64 (*.f64 t (/.f64 k l)) (*.f64 k 1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (tan.f64 k) (sin.f64 k)))) (*.f64 (*.f64 t (/.f64 k l)) (*.f64 k 1/2))): 20 points increase in error, 15 points decrease in error
      (/.f64 (/.f64 l (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 k) (tan.f64 k)))) (*.f64 (*.f64 t (/.f64 k l)) (*.f64 k 1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 k l) t)) (*.f64 k 1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))) (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 k t) l)) (*.f64 k 1/2))): 13 points increase in error, 20 points decrease in error
      (/.f64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))) (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 k (/.f64 t l))) (*.f64 k 1/2))): 24 points increase in error, 13 points decrease in error
      (/.f64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 k (/.f64 t l)) k) 1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 k (*.f64 k (/.f64 t l)))) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 l (*.f64 (*.f64 (*.f64 k (*.f64 k (/.f64 t l))) 1/2) (*.f64 (sin.f64 k) (tan.f64 k))))): 10 points increase in error, 17 points decrease in error
  3. Recombined 2 regimes into one program.
  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-23}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Error5.1
Cost46544
\[\begin{array}{l} t_1 := {\left(\frac{t}{\ell}\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := {\left(\frac{\sqrt[3]{\ell \cdot 2} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot t_2}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(t \cdot \left(\sin k \cdot t_1\right)\right)}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(t_1 \cdot \left(\sin k \cdot \left(t + t\right)\right)\right)}{\cos k}}\\ \end{array} \]
Alternative 2
Error5.1
Cost40144
\[\begin{array}{l} t_1 := {\left(\frac{t}{\ell}\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\ell}{\tan k}\\ t_4 := {\left(\frac{\sqrt[3]{\frac{t_3}{t_2}}}{t} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\sin k}}\right)}^{3}\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{+175}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(t \cdot \left(\sin k \cdot t_1\right)\right)}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-23}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_3}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+202}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(t_1 \cdot \left(\sin k \cdot \left(t + t\right)\right)\right)}{\cos k}}\\ \end{array} \]
Alternative 3
Error11.7
Cost27548
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \sin k\right)}\\ t_3 := \frac{\ell}{\sin k}\\ t_4 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t_3}{\tan k}\\ \mathbf{if}\;k \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -1.65 \cdot 10^{+20}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -4.2 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.1 \cdot 10^{-263}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-257}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(t + t\right)\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 58000000000000:\\ \;\;\;\;\frac{t_3 \cdot \left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right)}{\tan k}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 4
Error11.7
Cost27484
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \tan k \cdot \sin k\\ t_3 := \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}} \cdot t_2}\\ t_4 := \frac{\ell}{\sin k}\\ t_5 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t_4}{\tan k}\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq -1.56 \cdot 10^{-158}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-260}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-258}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-17}:\\ \;\;\;\;{\left(\frac{{\left(\frac{t}{\ell}\right)}^{2}}{\frac{2}{t \cdot \left(2 \cdot t_2\right)}}\right)}^{-1}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{t_4 \cdot \left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right)}{\tan k}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+96}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]
Alternative 5
Error7.9
Cost27344
\[\begin{array}{l} t_1 := \frac{2}{\tan k \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.52 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{{t}^{-3} \cdot \frac{\ell}{k}}{\frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error6.0
Cost27344
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k \cdot \left(t \cdot \left(t_1 \cdot \left(\sin k \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)}\\ t_3 := \frac{2}{\frac{\left(\sin k \cdot {t}^{3}\right) \cdot \left(t_1 \cdot \frac{\tan k}{\ell}\right)}{\ell}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9.6 \cdot 10^{-24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+100}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 7
Error5.6
Cost27344
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k \cdot \left(t \cdot \left(t_1 \cdot \left(\sin k \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)}\\ t_3 := \frac{\frac{\frac{\ell}{\frac{{t}^{3}}{\frac{2}{\sin k}}}}{t_1}}{\frac{\tan k}{\ell}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\tan k}}{\sin k}}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 8
Error11.8
Cost21796
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \tan k \cdot \sin k\\ t_3 := \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}} \cdot t_2}\\ t_4 := \frac{\ell}{\sin k}\\ t_5 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{t_4}{\tan k}\\ \mathbf{if}\;k \leq -8.5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -3.6 \cdot 10^{+19}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq -2.4 \cdot 10^{-158}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-266}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\ell}\right)}^{2}}}{t \cdot \left(2 \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 58000000000000:\\ \;\;\;\;\frac{t_4 \cdot \left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right)}{\tan k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+97}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]
Alternative 9
Error12.6
Cost21020
\[\begin{array}{l} t_1 := \frac{2}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot 2\right)\right)\right)}\\ t_2 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_3 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -11.5:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.9 \cdot 10^{-263}:\\ \;\;\;\;\frac{t_3}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-258}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_3\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 10
Error12.5
Cost21020
\[\begin{array}{l} t_1 := \frac{2}{{\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot \left(2 \cdot \left(\tan k \cdot \sin k\right)\right)\right)}\\ t_2 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_3 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -3.5:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -4.2 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_3}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-268}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_3\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 11
Error12.4
Cost21020
\[\begin{array}{l} t_1 := \frac{\frac{2}{{\left(\frac{t}{\ell}\right)}^{2}}}{t \cdot \left(2 \cdot \left(\tan k \cdot \sin k\right)\right)}\\ t_2 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_3 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -4 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -3.8:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.15 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.85 \cdot 10^{-260}:\\ \;\;\;\;\frac{t_3}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-258}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-163}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_3\\ \mathbf{elif}\;k \leq 6.3 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 12
Error12.7
Cost14944
\[\begin{array}{l} t_1 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ t_2 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -100:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;k \leq -2.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -9.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{-274}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_2\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 13
Error12.7
Cost14816
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{2}{k \cdot \frac{t \cdot k}{\ell}}\\ t_3 := \frac{2}{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{if}\;k \leq -1.15 \cdot 10^{+245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -100:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -7.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{-256}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 14
Error12.8
Cost14816
\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{2}{k \cdot \frac{t \cdot k}{\ell}}\\ t_2 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -4 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell \cdot 2}{\sin k}}{\tan k}}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq -19.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -1.02 \cdot 10^{-263}:\\ \;\;\;\;\frac{t_2}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{-271}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{-135}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_2\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 15
Error17.7
Cost14552
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \left(2 \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{if}\;k \leq -6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -2.7 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-272}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 16
Error15.4
Cost14552
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{2}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -3.8 \cdot 10^{-261}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-277}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 3.05 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 17
Error12.3
Cost14552
\[\begin{array}{l} t_1 := {t}^{-3} \cdot \frac{\ell}{k}\\ t_2 := \frac{\frac{\ell}{\sin k}}{\tan k} \cdot \frac{2}{k \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{if}\;k \leq -8:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ \mathbf{elif}\;k \leq -1.4 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-275}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 18
Error22.3
Cost8600
\[\begin{array}{l} t_1 := \frac{2}{\left(k \cdot k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}\\ t_2 := \frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ t_3 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -4 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.6 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.4 \cdot 10^{-260}:\\ \;\;\;\;\frac{t_3}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-256}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-138}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_3\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 19
Error24.5
Cost8088
\[\begin{array}{l} t_1 := \frac{2}{\left(k \cdot k\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(t \cdot 2\right)\right)}\\ t_2 := \frac{\ell}{k \cdot k}\\ t_3 := {t}^{-3} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -8.8 \cdot 10^{+21}:\\ \;\;\;\;t_2 \cdot \frac{\ell \cdot \frac{2}{t \cdot k}}{k}\\ \mathbf{elif}\;k \leq -9.6 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-265}:\\ \;\;\;\;\frac{t_3}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-261}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-137}:\\ \;\;\;\;\frac{\ell}{k} \cdot t_3\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(t_2 + \ell \cdot -0.16666666666666666\right)\\ \end{array} \]
Alternative 20
Error21.6
Cost7436
\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left({t}^{-3} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 21
Error21.6
Cost7436
\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{{t}^{-3} \cdot \frac{\ell}{k}}{\frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 22
Error24.1
Cost7304
\[\begin{array}{l} t_1 := \ell \cdot \left(\frac{\ell}{k} \cdot \frac{{t}^{-3}}{k}\right)\\ \mathbf{if}\;t \leq -4 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 23
Error23.9
Cost7304
\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left({t}^{-3} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \frac{{t}^{-3}}{k}\right)\\ \end{array} \]
Alternative 24
Error31.3
Cost1608
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{-82}:\\ \;\;\;\;t_1 \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-44}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot \left(t_1 + \ell \cdot -0.16666666666666666\right)\\ \end{array} \]
Alternative 25
Error31.5
Cost1480
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.25 \cdot 10^{-74}:\\ \;\;\;\;t_1 \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.68 \cdot 10^{-43}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \ell \cdot -0.16666666666666666\right) \cdot \frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \]
Alternative 26
Error31.7
Cost1352
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.15 \cdot 10^{-74}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-44}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot t_1\\ \end{array} \]
Alternative 27
Error31.6
Cost1352
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;t_1 \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot k\right) \cdot \frac{1}{\ell}\right)} \cdot t_1\\ \end{array} \]
Alternative 28
Error31.9
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k} \cdot \left(\ell \cdot \frac{\frac{2}{t}}{k \cdot k}\right)\\ \mathbf{if}\;k \leq -3.6 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{-44}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 29
Error31.7
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.32 \cdot 10^{-79}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;k \leq 6.7 \cdot 10^{-44}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \frac{\frac{2}{t}}{k \cdot k}\right)\\ \end{array} \]
Alternative 30
Error31.7
Cost1224
\[\begin{array}{l} t_1 := \frac{2}{k \cdot k}\\ t_2 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot t_1\right) \cdot t_2\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{-43}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\ell \cdot \frac{t_1}{t}\right)\\ \end{array} \]
Alternative 31
Error31.7
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.8 \cdot 10^{-73}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{2}{k \cdot k}\right) \cdot t_1\\ \mathbf{elif}\;k \leq 6.1 \cdot 10^{-44}:\\ \;\;\;\;-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \frac{2}{t \cdot k}}{k}\\ \end{array} \]
Alternative 32
Error45.0
Cost448
\[-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}} \]
Alternative 33
Error43.0
Cost448
\[-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

Error

Reproduce

herbie shell --seed 2022321 
(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))))