Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 2.0

Time: 31.9s

Precision: binary64

Cost: 20548

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\cos k}{t}\\ \mathbf{if}\;k \leq 0:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sin k \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right)}{t_1}}\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{t_1}}\\ \end{array} \]

FPCore

(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 k) (/ (cos k) t))))
   (if (<= k 0.0)
     (/ 2.0 (/ (* k (* (sin k) (* (sin k) (/ 1.0 l)))) t_1))
     (if (<= k 1e-140)
       (/ 2.0 (* (* k (/ k l)) (/ (* k (* k t)) (* l (cos k)))))
       (/ 2.0 (/ (* k (/ (pow (sin k) 2.0) l)) t_1))))))

C

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 / k) * (cos(k) / t);
	double tmp;
	if (k <= 0.0) {
		tmp = 2.0 / ((k * (sin(k) * (sin(k) * (1.0 / l)))) / t_1);
	} else if (k <= 1e-140) {
		tmp = 2.0 / ((k * (k / l)) * ((k * (k * t)) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((k * (pow(sin(k), 2.0) / l)) / t_1);
	}
	return tmp;
}

Fortran

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 / k) * (cos(k) / t)
    if (k <= 0.0d0) then
        tmp = 2.0d0 / ((k * (sin(k) * (sin(k) * (1.0d0 / l)))) / t_1)
    else if (k <= 1d-140) then
        tmp = 2.0d0 / ((k * (k / l)) * ((k * (k * t)) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((k * ((sin(k) ** 2.0d0) / l)) / t_1)
    end if
    code = tmp
end function

Java

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 / k) * (Math.cos(k) / t);
	double tmp;
	if (k <= 0.0) {
		tmp = 2.0 / ((k * (Math.sin(k) * (Math.sin(k) * (1.0 / l)))) / t_1);
	} else if (k <= 1e-140) {
		tmp = 2.0 / ((k * (k / l)) * ((k * (k * t)) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((k * (Math.pow(Math.sin(k), 2.0) / l)) / t_1);
	}
	return tmp;
}

Python

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 / k) * (math.cos(k) / t)
	tmp = 0
	if k <= 0.0:
		tmp = 2.0 / ((k * (math.sin(k) * (math.sin(k) * (1.0 / l)))) / t_1)
	elif k <= 1e-140:
		tmp = 2.0 / ((k * (k / l)) * ((k * (k * t)) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((k * (math.pow(math.sin(k), 2.0) / l)) / t_1)
	return tmp

Julia

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(l / k) * Float64(cos(k) / t))
	tmp = 0.0
	if (k <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(sin(k) * Float64(sin(k) * Float64(1.0 / l)))) / t_1));
	elseif (k <= 1e-140)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k / l)) * Float64(Float64(k * Float64(k * t)) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64((sin(k) ^ 2.0) / l)) / t_1));
	end
	return tmp
end

MATLAB

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 / k) * (cos(k) / t);
	tmp = 0.0;
	if (k <= 0.0)
		tmp = 2.0 / ((k * (sin(k) * (sin(k) * (1.0 / l)))) / t_1);
	elseif (k <= 1e-140)
		tmp = 2.0 / ((k * (k / l)) * ((k * (k * t)) / (l * cos(k))));
	else
		tmp = 2.0 / ((k * ((sin(k) ^ 2.0) / l)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

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[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.0], N[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1e-140], N[(2.0 / N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0

   1. Initial program 47.4

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

   2. Applied egg-rr 42.8

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

   3. Taylor expanded in t around 0 22.4

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

   4. Simplified 12.0

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k (*.f64 k t)) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t)) (*.f64 (cos.f64 k) l))): 34 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (pow.f64 k 2))) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 t (pow.f64 k 2)) (Rewrite=> *-commutative_binary64 (*.f64 l (cos.f64 k))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 t (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k))))): 50 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k)))): 8 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 l (*.f64 l (cos.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) (cos.f64 k)))): 8 points increase in error, 9 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 2.5

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

   6. Applied egg-rr 2.1

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right)} \cdot k}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}} \]

   if 0.0 < k < 9.9999999999999998e-141

   1. Initial program 64.0

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

   2. Applied egg-rr 63.9

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

   3. Taylor expanded in t around 0 63.1

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

   4. Simplified 49.3

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k (*.f64 k t)) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t)) (*.f64 (cos.f64 k) l))): 34 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (pow.f64 k 2))) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 t (pow.f64 k 2)) (Rewrite=> *-commutative_binary64 (*.f64 l (cos.f64 k))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 t (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k))))): 50 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k)))): 8 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 l (*.f64 l (cos.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) (cos.f64 k)))): 8 points increase in error, 9 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in k around 0 49.3

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

   6. Simplified 14.1

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot \ell}} \]
      Proof
      (*.f64 (/.f64 k l) k): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 k (/.f64 l k))): 24 points increase in error, 26 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 k k) l)): 47 points increase in error, 24 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l): 0 points increase in error, 0 points decrease in error

   if 9.9999999999999998e-141 < k

   1. Initial program 47.7

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

   2. Applied egg-rr 43.0

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

   3. Taylor expanded in t around 0 21.7

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

   4. Simplified 10.8

      \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k (*.f64 k t)) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t)) (*.f64 (cos.f64 k) l))): 34 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (pow.f64 k 2))) (*.f64 (cos.f64 k) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 t (pow.f64 k 2)) (Rewrite=> *-commutative_binary64 (*.f64 l (cos.f64 k))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (*.f64 t (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k))))): 50 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2))) (*.f64 l (*.f64 l (cos.f64 k)))): 8 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 l (*.f64 l (cos.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) (cos.f64 k)))): 8 points increase in error, 9 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{\sin k}^{2}}{\ell} \cdot k}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sin k \cdot \left(\sin k \cdot \frac{1}{\ell}\right)\right)}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.7
Cost	20488

\[\begin{array}{l} t_1 := \frac{2}{\frac{k \cdot \frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell}}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{if}\;k \leq -1000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.0337882780495886 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\ell}}{\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	2.0
Cost	20488

\[\begin{array}{l} t_1 := \frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.9
Cost	14540

\[\begin{array}{l} t_1 := \frac{2}{\frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\cos k}\right)\right)}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;k \leq 0.01:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_2}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	6.8
Cost	14540

\[\begin{array}{l} t_1 := \ell \cdot \cos k\\ t_2 := \frac{2}{\frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell} \cdot \left(\left(k \cdot t\right) \cdot \frac{k}{t_1}\right)}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{k \cdot \left(k \cdot t\right)}{t_1}}\\ \mathbf{elif}\;k \leq 0.01:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_3}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	6.8
Cost	14540

\[\begin{array}{l} t_1 := \ell \cdot \cos k\\ t_2 := \frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{k \cdot \left(k \cdot t\right)}{t_1}}\\ \mathbf{elif}\;k \leq 0.01:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_3}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\left(k \cdot t\right) \cdot \frac{k}{t_1}\right)}\\ \end{array} \]

Alternative 6
Error	2.5
Cost	14540

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\cos k}{t}\\ t_2 := \frac{2}{\frac{k \cdot \frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell}}{t_1}}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_3 \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;k \leq 0.01:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_3}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	23.2
Cost	13960

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{elif}\;k \leq 100000:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_1}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 8
Error	23.8
Cost	7620

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 10^{-140}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_1}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}}\\ \end{array} \]

Alternative 9
Error	24.3
Cost	7488

\[\frac{2}{\frac{k \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}} \]

Alternative 10
Error	26.4
Cost	960

\[\frac{\frac{\frac{2}{k}}{k}}{\frac{\left(k \cdot k\right) \cdot \frac{t}{\ell}}{\ell}} \]

Alternative 11
Error	25.9
Cost	960

\[\frac{\ell \cdot \frac{\frac{2}{k}}{k}}{\left(k \cdot k\right) \cdot \frac{t}{\ell}} \]

Alternative 12
Error	25.7
Cost	960

\[\frac{\ell \cdot \frac{2}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k} \]

Reproduce

herbie shell --seed 2022313 
(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))))