Average Error: 47.7 → 5.6
Time: 32.0s
Precision: binary64
Cost: 33096
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}}\\ \mathbf{if}\;t \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-72}:\\ \;\;\;\;2 \cdot {\left(\frac{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}{\cos k}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
         (* (/ (* l (* 2.0 (cos k))) k) (/ (/ (/ l t) k) (pow (sin k) 2.0)))))
   (if (<= t 0.0)
     t_1
     (if (<= t 1e-72)
       (*
        2.0
        (pow (/ (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0) (cos k)) -1.0))
       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 = ((l * (2.0 * cos(k))) / k) * (((l / t) / k) / pow(sin(k), 2.0));
	double tmp;
	if (t <= 0.0) {
		tmp = t_1;
	} else if (t <= 1e-72) {
		tmp = 2.0 * pow((pow(((k * (sin(k) * sqrt(t))) / l), 2.0) / cos(k)), -1.0);
	} 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 = ((l * (2.0d0 * cos(k))) / k) * (((l / t) / k) / (sin(k) ** 2.0d0))
    if (t <= 0.0d0) then
        tmp = t_1
    else if (t <= 1d-72) then
        tmp = 2.0d0 * (((((k * (sin(k) * sqrt(t))) / l) ** 2.0d0) / cos(k)) ** (-1.0d0))
    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 = ((l * (2.0 * Math.cos(k))) / k) * (((l / t) / k) / Math.pow(Math.sin(k), 2.0));
	double tmp;
	if (t <= 0.0) {
		tmp = t_1;
	} else if (t <= 1e-72) {
		tmp = 2.0 * Math.pow((Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0) / Math.cos(k)), -1.0);
	} 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 = ((l * (2.0 * math.cos(k))) / k) * (((l / t) / k) / math.pow(math.sin(k), 2.0))
	tmp = 0
	if t <= 0.0:
		tmp = t_1
	elif t <= 1e-72:
		tmp = 2.0 * math.pow((math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0) / math.cos(k)), -1.0)
	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(l * Float64(2.0 * cos(k))) / k) * Float64(Float64(Float64(l / t) / k) / (sin(k) ^ 2.0)))
	tmp = 0.0
	if (t <= 0.0)
		tmp = t_1;
	elseif (t <= 1e-72)
		tmp = Float64(2.0 * (Float64((Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0) / cos(k)) ^ -1.0));
	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 = ((l * (2.0 * cos(k))) / k) * (((l / t) / k) / (sin(k) ^ 2.0));
	tmp = 0.0;
	if (t <= 0.0)
		tmp = t_1;
	elseif (t <= 1e-72)
		tmp = 2.0 * (((((k * (sin(k) * sqrt(t))) / l) ^ 2.0) / cos(k)) ^ -1.0);
	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[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 0.0], t$95$1, If[LessEqual[t, 1e-72], N[(2.0 * N[Power[N[(N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision], -1.0], $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{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}}\\
\mathbf{if}\;t \leq 0:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 10^{-72}:\\
\;\;\;\;2 \cdot {\left(\frac{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}{\cos k}\right)}^{-1}\\

\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 < 0.0 or 9.9999999999999997e-73 < t

    1. Initial program 45.7

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

    2. Simplified37.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 3 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 29 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 5 points increase in error, 7 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 2 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (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) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (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, 0 points decrease in error

    3. Taylor expanded in k around inf 22.5

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

    4. Simplified21.8

      \[\leadsto \color{blue}{\frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}} \cdot 2\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 (/.f64 (/.f64 (*.f64 l l) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (*.f64 (/.f64 (/.f64 (*.f64 l l) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 11 points increase in error, 10 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 2 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 20 points increase in error, 16 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error

    5. Applied egg-rr15.3

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

    6. Applied egg-rr9.7

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

    7. Applied egg-rr6.2

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

    if 0.0 < t < 9.9999999999999997e-73

    1. Initial program 59.0

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

    2. Simplified59.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 3 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 29 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 5 points increase in error, 7 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 2 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (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) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (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, 0 points decrease in error

    3. Taylor expanded in k around inf 25.9

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

    4. Applied egg-rr2.2

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 10^{-72}:\\ \;\;\;\;2 \cdot {\left(\frac{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}{\cos k}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error5.8
Cost26760
\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}}\\ \mathbf{if}\;t \leq 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-72}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{1}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error6.5
Cost20488
\[\begin{array}{l} t_1 := \frac{\ell \cdot 2}{k \cdot {\sin k}^{2}} \cdot \frac{\cos k}{k \cdot \frac{t}{\ell}}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-19}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error7.2
Cost20224
\[\frac{\ell \cdot \left(2 \cdot \cos k\right)}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{{\sin k}^{2}} \]
Alternative 4
Error19.3
Cost14408
\[\begin{array}{l} t_1 := \frac{2 \cdot \cos k}{\left(0.5 + \cos \left(k + k\right) \cdot -0.5\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-10}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error13.3
Cost14408
\[\begin{array}{l} t_1 := \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-10}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error8.2
Cost14408
\[\begin{array}{l} t_1 := \frac{\frac{\cos k \cdot \left(\ell \cdot 2\right)}{k \cdot \frac{t}{\ell}}}{k \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-10}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error25.1
Cost13956
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k \cdot \frac{t}{\ell}}}{k \cdot {\sin k}^{2}}\\ \end{array} \]
Alternative 8
Error27.4
Cost960
\[\frac{2}{k \cdot k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \]
Alternative 9
Error26.0
Cost960
\[\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k} \]

Error

Reproduce

herbie shell --seed 2022306 
(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))))