Average Error: 47.5 → 1.0
Time: 36.2s
Precision: binary64
Cost: 20552
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
\[\begin{array}{l} t_1 := \frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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 (* (/ (cos k) t) (/ l k))) (* k (/ (pow (sin k) 2.0) l)))))
   (if (<= k -3e-9)
     t_1
     (if (<= k 2.6e-70)
       (/ 2.0 (/ (* k (* (/ t (cos k)) (* (sin k) (/ (sin k) l)))) (/ l 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 * ((cos(k) / t) * (l / k))) / (k * (pow(sin(k), 2.0) / l));
	double tmp;
	if (k <= -3e-9) {
		tmp = t_1;
	} else if (k <= 2.6e-70) {
		tmp = 2.0 / ((k * ((t / cos(k)) * (sin(k) * (sin(k) / l)))) / (l / k));
	} else {
		tmp = t_1;
	}
	return tmp;
}
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * ((cos(k) / t) * (l / k))) / (k * ((sin(k) ** 2.0d0) / l))
    if (k <= (-3d-9)) then
        tmp = t_1
    else if (k <= 2.6d-70) then
        tmp = 2.0d0 / ((k * ((t / cos(k)) * (sin(k) * (sin(k) / l)))) / (l / k))
    else
        tmp = t_1
    end if
    code = tmp
end function
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.cos(k) / t) * (l / k))) / (k * (Math.pow(Math.sin(k), 2.0) / l));
	double tmp;
	if (k <= -3e-9) {
		tmp = t_1;
	} else if (k <= 2.6e-70) {
		tmp = 2.0 / ((k * ((t / Math.cos(k)) * (Math.sin(k) * (Math.sin(k) / l)))) / (l / k));
	} else {
		tmp = t_1;
	}
	return tmp;
}
def code(t, l, 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))
def code(t, l, k):
	t_1 = (2.0 * ((math.cos(k) / t) * (l / k))) / (k * (math.pow(math.sin(k), 2.0) / l))
	tmp = 0
	if k <= -3e-9:
		tmp = t_1
	elif k <= 2.6e-70:
		tmp = 2.0 / ((k * ((t / math.cos(k)) * (math.sin(k) * (math.sin(k) / l)))) / (l / 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(Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(l / k))) / Float64(k * Float64((sin(k) ^ 2.0) / l)))
	tmp = 0.0
	if (k <= -3e-9)
		tmp = t_1;
	elseif (k <= 2.6e-70)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(t / cos(k)) * Float64(sin(k) * Float64(sin(k) / l)))) / Float64(l / k)));
	else
		tmp = t_1;
	end
	return tmp
end
function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end
function tmp_2 = code(t, l, k)
	t_1 = (2.0 * ((cos(k) / t) * (l / k))) / (k * ((sin(k) ^ 2.0) / l));
	tmp = 0.0;
	if (k <= -3e-9)
		tmp = t_1;
	elseif (k <= 2.6e-70)
		tmp = 2.0 / ((k * ((t / cos(k)) * (sin(k) * (sin(k) / l)))) / (l / k));
	else
		tmp = t_1;
	end
	tmp_2 = 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[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -3e-9], t$95$1, If[LessEqual[k, 2.6e-70], N[(2.0 / N[(N[(k * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / 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 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\
\mathbf{if}\;k \leq -3 \cdot 10^{-9}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;k \leq 2.6 \cdot 10^{-70}:\\
\;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)\right)}{\frac{\ell}{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 k < -2.99999999999999998e-9 or 2.60000000000000002e-70 < k

    1. Initial program 45.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
    2. Simplified36.8

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 33 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.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))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0)))): 42 points increase in error, 0 points decrease in error
      (/.f64 2 (+.f64 (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))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0))): 4 points increase in error, 4 points decrease in error
      (/.f64 2 (+.f64 (*.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)) (Rewrite=> mul0-rgt_binary64 0))): 0 points increase in error, 42 points decrease in error
      (/.f64 2 (Rewrite=> +-rgt-identity_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)))): 0 points increase in error, 0 points decrease in error
    3. Taylor expanded in t around 0 19.8

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}} \]
    4. Simplified15.6

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]
      Proof
      (*.f64 (/.f64 t (cos.f64 k)) (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 k 2)) (*.f64 l l)))): 45 points increase in error, 13 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2))) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2))) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 25 points increase in error, 22 points decrease in error
    5. Applied egg-rr4.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}\right) \cdot k}{\frac{\ell}{k}}}} \]
    6. Applied egg-rr0.8

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\frac{\cos k}{t}}}}{\frac{\ell}{k}}} \]
    7. Applied egg-rr0.4

      \[\leadsto \color{blue}{\frac{2}{k \cdot \frac{{\sin k}^{2}}{\ell}} \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)} \]
    8. Simplified0.4

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}} \]
      Proof
      (/.f64 (*.f64 2 (*.f64 (/.f64 (cos.f64 k) t) (/.f64 l k))) (*.f64 k (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 2 (*.f64 k (/.f64 (pow.f64 (sin.f64 k) 2) l))) (*.f64 (/.f64 (cos.f64 k) t) (/.f64 l k)))): 23 points increase in error, 11 points decrease in error

    if -2.99999999999999998e-9 < k < 2.60000000000000002e-70

    1. Initial program 62.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
    2. Simplified58.5

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 33 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.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))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0)))): 42 points increase in error, 0 points decrease in error
      (/.f64 2 (+.f64 (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))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0))): 4 points increase in error, 4 points decrease in error
      (/.f64 2 (+.f64 (*.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)) (Rewrite=> mul0-rgt_binary64 0))): 0 points increase in error, 42 points decrease in error
      (/.f64 2 (Rewrite=> +-rgt-identity_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)))): 0 points increase in error, 0 points decrease in error
    3. Taylor expanded in t around 0 50.0

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}} \]
    4. Simplified20.8

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}} \]
      Proof
      (*.f64 (/.f64 t (cos.f64 k)) (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 k 2)) (*.f64 l l)))): 45 points increase in error, 13 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2))) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2))) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 25 points increase in error, 22 points decrease in error
    5. Applied egg-rr15.1

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}\right) \cdot k}{\frac{\ell}{k}}}} \]
    6. Applied egg-rr4.9

      \[\leadsto \frac{2}{\frac{\left(\frac{t}{\cos k} \cdot \color{blue}{\left(\frac{\sin k}{1} \cdot \frac{\sin k}{\ell}\right)}\right) \cdot k}{\frac{\ell}{k}}} \]
  3. Recombined 2 regimes into one program.
  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error1.0
Cost20680
\[\begin{array}{l} t_1 := \frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \left(\sin k \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right)\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error2.7
Cost20488
\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{k \cdot \frac{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}{\ell}}{\frac{\cos k}{t}}}{\frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -2.8 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\ell} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error2.2
Cost20488
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \frac{\cos k}{\frac{\frac{k}{\frac{\ell}{t_1}}}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{-42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_1}{\ell} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error3.3
Cost20488
\[\begin{array}{l} t_1 := \frac{{\sin k}^{2}}{\ell}\\ t_2 := \frac{2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell}{k}\right)}{k \cdot t_1}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{t}{\cos k}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error3.1
Cost20224
\[\frac{2}{\frac{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\frac{\cos k}{t}}}{\frac{\ell}{k}}} \]
Alternative 6
Error15.9
Cost14408
\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \ell}}\\ \mathbf{if}\;k \leq -0.00079:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.00094:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error17.1
Cost14408
\[\begin{array}{l} t_1 := 0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\\ \mathbf{if}\;k \leq -0.00075:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_1 \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;k \leq 0.00014:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \ell}}\\ \end{array} \]
Alternative 8
Error6.0
Cost14408
\[\begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -0.00062:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error3.0
Cost14408
\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{k \cdot \frac{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}{\ell}}{\frac{\cos k}{t}}}{\frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -0.00062:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error24.6
Cost7488
\[\frac{2}{\frac{k \cdot \left(\frac{t}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{k}}} \]
Alternative 11
Error25.9
Cost7432
\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{3}}{\frac{\ell}{k}}}\\ t_2 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -7 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_2} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error26.5
Cost7304
\[\begin{array}{l} t_1 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-69}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]
Alternative 13
Error26.5
Cost7304
\[\begin{array}{l} t_1 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -4.7 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left({k}^{-4} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]
Alternative 14
Error26.5
Cost7304
\[\begin{array}{l} t_1 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left({k}^{-4} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot {k}^{-4}}{\frac{t}{\ell}}\\ \end{array} \]
Alternative 15
Error26.5
Cost7304
\[\begin{array}{l} t_1 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -5.3 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{{k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-69}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot {k}^{-4}}{\frac{t}{\ell}}\\ \end{array} \]
Alternative 16
Error26.1
Cost7304
\[\begin{array}{l} t_1 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{-69}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot {k}^{-4}}{\frac{t}{\ell}}\\ \end{array} \]
Alternative 17
Error26.9
Cost1736
\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}}{k \cdot k}\\ t_2 := \frac{t}{\frac{1}{k \cdot k}}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \left(\frac{t \cdot \ell}{t_2} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 18
Error27.9
Cost960
\[2 \cdot \frac{\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}}{k \cdot k} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))