Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 10.3

Time: 54.4s

Precision: binary64

Cost: 27280

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq -2.164062791118162 \cdot 10^{+170}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\frac{\cos k}{k} \cdot \frac{1}{k \cdot t}\right)}{\frac{t_1}{\ell}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\left(\ell \cdot {\sin k}^{-2}\right) \cdot \left(\left(2 \cdot \frac{\ell}{t}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)\right)\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{t_2}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.1578262085078074 \cdot 10^{+111}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot t_2}{\sin k \cdot \left(\sin k \cdot {\ell}^{-1}\right)}\\ \end{array} \]

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 (pow (sin k) 2.0)) (t_2 (/ (cos k) (* k (* k t)))))
   (if (<= k -2.164062791118162e+170)
     (* l (/ (* 2.0 (* (/ (cos k) k) (/ 1.0 (* k t)))) (/ t_1 l)))
     (if (<= k -1e-80)
       (*
        (* l (pow (sin k) -2.0))
        (* (* 2.0 (/ l t)) (* (cos k) (pow k -2.0))))
       (if (<= k 1e-85)
         (* 2.0 (/ t_2 (pow (/ (sin k) l) 2.0)))
         (if (<= k 3.1578262085078074e+111)
           (* l (* (/ (cos k) (* k k)) (* 2.0 (/ (/ l t) t_1))))
           (* l (/ (* 2.0 t_2) (* (sin k) (* (sin k) (pow l -1.0)))))))))))

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 = pow(sin(k), 2.0);
	double t_2 = cos(k) / (k * (k * t));
	double tmp;
	if (k <= -2.164062791118162e+170) {
		tmp = l * ((2.0 * ((cos(k) / k) * (1.0 / (k * t)))) / (t_1 / l));
	} else if (k <= -1e-80) {
		tmp = (l * pow(sin(k), -2.0)) * ((2.0 * (l / t)) * (cos(k) * pow(k, -2.0)));
	} else if (k <= 1e-85) {
		tmp = 2.0 * (t_2 / pow((sin(k) / l), 2.0));
	} else if (k <= 3.1578262085078074e+111) {
		tmp = l * ((cos(k) / (k * k)) * (2.0 * ((l / t) / t_1)));
	} else {
		tmp = l * ((2.0 * t_2) / (sin(k) * (sin(k) * pow(l, -1.0))));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = cos(k) / (k * (k * t))
    if (k <= (-2.164062791118162d+170)) then
        tmp = l * ((2.0d0 * ((cos(k) / k) * (1.0d0 / (k * t)))) / (t_1 / l))
    else if (k <= (-1d-80)) then
        tmp = (l * (sin(k) ** (-2.0d0))) * ((2.0d0 * (l / t)) * (cos(k) * (k ** (-2.0d0))))
    else if (k <= 1d-85) then
        tmp = 2.0d0 * (t_2 / ((sin(k) / l) ** 2.0d0))
    else if (k <= 3.1578262085078074d+111) then
        tmp = l * ((cos(k) / (k * k)) * (2.0d0 * ((l / t) / t_1)))
    else
        tmp = l * ((2.0d0 * t_2) / (sin(k) * (sin(k) * (l ** (-1.0d0)))))
    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 = Math.pow(Math.sin(k), 2.0);
	double t_2 = Math.cos(k) / (k * (k * t));
	double tmp;
	if (k <= -2.164062791118162e+170) {
		tmp = l * ((2.0 * ((Math.cos(k) / k) * (1.0 / (k * t)))) / (t_1 / l));
	} else if (k <= -1e-80) {
		tmp = (l * Math.pow(Math.sin(k), -2.0)) * ((2.0 * (l / t)) * (Math.cos(k) * Math.pow(k, -2.0)));
	} else if (k <= 1e-85) {
		tmp = 2.0 * (t_2 / Math.pow((Math.sin(k) / l), 2.0));
	} else if (k <= 3.1578262085078074e+111) {
		tmp = l * ((Math.cos(k) / (k * k)) * (2.0 * ((l / t) / t_1)));
	} else {
		tmp = l * ((2.0 * t_2) / (Math.sin(k) * (Math.sin(k) * Math.pow(l, -1.0))));
	}
	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 = math.pow(math.sin(k), 2.0)
	t_2 = math.cos(k) / (k * (k * t))
	tmp = 0
	if k <= -2.164062791118162e+170:
		tmp = l * ((2.0 * ((math.cos(k) / k) * (1.0 / (k * t)))) / (t_1 / l))
	elif k <= -1e-80:
		tmp = (l * math.pow(math.sin(k), -2.0)) * ((2.0 * (l / t)) * (math.cos(k) * math.pow(k, -2.0)))
	elif k <= 1e-85:
		tmp = 2.0 * (t_2 / math.pow((math.sin(k) / l), 2.0))
	elif k <= 3.1578262085078074e+111:
		tmp = l * ((math.cos(k) / (k * k)) * (2.0 * ((l / t) / t_1)))
	else:
		tmp = l * ((2.0 * t_2) / (math.sin(k) * (math.sin(k) * math.pow(l, -1.0))))
	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 = sin(k) ^ 2.0
	t_2 = Float64(cos(k) / Float64(k * Float64(k * t)))
	tmp = 0.0
	if (k <= -2.164062791118162e+170)
		tmp = Float64(l * Float64(Float64(2.0 * Float64(Float64(cos(k) / k) * Float64(1.0 / Float64(k * t)))) / Float64(t_1 / l)));
	elseif (k <= -1e-80)
		tmp = Float64(Float64(l * (sin(k) ^ -2.0)) * Float64(Float64(2.0 * Float64(l / t)) * Float64(cos(k) * (k ^ -2.0))));
	elseif (k <= 1e-85)
		tmp = Float64(2.0 * Float64(t_2 / (Float64(sin(k) / l) ^ 2.0)));
	elseif (k <= 3.1578262085078074e+111)
		tmp = Float64(l * Float64(Float64(cos(k) / Float64(k * k)) * Float64(2.0 * Float64(Float64(l / t) / t_1))));
	else
		tmp = Float64(l * Float64(Float64(2.0 * t_2) / Float64(sin(k) * Float64(sin(k) * (l ^ -1.0)))));
	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 = sin(k) ^ 2.0;
	t_2 = cos(k) / (k * (k * t));
	tmp = 0.0;
	if (k <= -2.164062791118162e+170)
		tmp = l * ((2.0 * ((cos(k) / k) * (1.0 / (k * t)))) / (t_1 / l));
	elseif (k <= -1e-80)
		tmp = (l * (sin(k) ^ -2.0)) * ((2.0 * (l / t)) * (cos(k) * (k ^ -2.0)));
	elseif (k <= 1e-85)
		tmp = 2.0 * (t_2 / ((sin(k) / l) ^ 2.0));
	elseif (k <= 3.1578262085078074e+111)
		tmp = l * ((cos(k) / (k * k)) * (2.0 * ((l / t) / t_1)));
	else
		tmp = l * ((2.0 * t_2) / (sin(k) * (sin(k) * (l ^ -1.0))));
	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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.164062791118162e+170], N[(l * N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-80], N[(N[(l * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1e-85], N[(2.0 * N[(t$95$2 / N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1578262085078074e+111], N[(l * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -2.1640627911181619e170

   1. Initial program 38.8

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

   2. Taylor expanded in t around 0 23.6

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

   3. Simplified 23.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (*.f64 (*.f64 k k) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 k 2) t) (cos.f64 k)) (/.f64 (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 (*.f64 (pow.f64 k 2) t) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 22 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 8 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 15.2

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

   5. Applied egg-rr 14.8

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

   if -2.1640627911181619e170 < k < -9.99999999999999961e-81

   1. Initial program 51.4

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

   2. Simplified 41.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 1 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))): 1 points increase in error, 3 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))): 4 points increase in error, 3 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))): 2 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)))): 28 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)))): 4 points increase in error, 2 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))): 0 points increase in error, 1 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))))): 4 points increase in error, 2 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)))): 0 points increase in error, 2 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. Applied egg-rr 34.8

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

   4. Taylor expanded in k around inf 16.2

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

   5. Simplified 9.3

      \[\leadsto \color{blue}{\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot 2\right)\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 (/.f64 l (pow.f64 (sin.f64 k) 2)) (*.f64 (/.f64 l t) 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 l (pow.f64 (sin.f64 k) 2)) (*.f64 (/.f64 l t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l (pow.f64 (sin.f64 k) 2)) (/.f64 l t)) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 39 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (pow.f64 (sin.f64 k) 2) t)) 2)): 0 points increase in error, 0 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)): 1 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): 27 points increase in error, 26 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

   6. Applied egg-rr 4.7

      \[\leadsto \color{blue}{0 + \left(\ell \cdot {\sin k}^{-2}\right) \cdot \left(\left(2 \cdot \frac{\ell}{t}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)\right)} \]

   if -9.99999999999999961e-81 < k < 9.9999999999999998e-86

   1. Initial program 63.9

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

   2. Taylor expanded in t around 0 51.3

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

   3. Simplified 45.0

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (*.f64 (*.f64 k k) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 k 2) t) (cos.f64 k)) (/.f64 (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 (*.f64 (pow.f64 k 2) t) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 22 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 8 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 19.8

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

   if 9.9999999999999998e-86 < k < 3.15782620850780739e111

   1. Initial program 53.9

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

   2. Taylor expanded in t around 0 16.3

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

   3. Simplified 15.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (*.f64 (*.f64 k k) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 k 2) t) (cos.f64 k)) (/.f64 (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 (*.f64 (pow.f64 k 2) t) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 22 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 8 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 6.2

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

   5. Taylor expanded in k around inf 8.4

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

   6. Simplified 2.9

      \[\leadsto \color{blue}{\left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{\sin k}^{2}}\right)\right)} \cdot \ell \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 2 (/.f64 (/.f64 l t) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (*.f64 2 (/.f64 (/.f64 l t) (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 2 (Rewrite=> associate-/l/_binary64 (/.f64 l (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 19 points increase in error, 18 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l (*.f64 (pow.f64 (sin.f64 k) 2) t)) 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 l (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 0 points increase in error, 2 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) l) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 32 points increase in error, 36 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 (cos.f64 k) l) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error

   if 3.15782620850780739e111 < k

   1. Initial program 41.9

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

   2. Taylor expanded in t around 0 23.5

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

   3. Simplified 23.5

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
      Proof
      (*.f64 (/.f64 (*.f64 (*.f64 k k) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (pow.f64 k 2) t) (cos.f64 k)) (/.f64 (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 (*.f64 (pow.f64 k 2) t) (pow.f64 (sin.f64 k) 2)) (*.f64 (cos.f64 k) (pow.f64 l 2)))): 22 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 8 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 14.8

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

   5. Applied egg-rr 14.8

      \[\leadsto \frac{2 \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\color{blue}{\sin k \cdot \left(\sin k \cdot {\ell}^{-1}\right)}} \cdot \ell \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.164062791118162 \cdot 10^{+170}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\frac{\cos k}{k} \cdot \frac{1}{k \cdot t}\right)}{\frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\left(\ell \cdot {\sin k}^{-2}\right) \cdot \left(\left(2 \cdot \frac{\ell}{t}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)\right)\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot \left(k \cdot t\right)}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.1578262085078074 \cdot 10^{+111}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\sin k \cdot \left(\sin k \cdot {\ell}^{-1}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	26824

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ t_3 := \frac{\cos k}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq -2.164062791118162 \cdot 10^{+170}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\frac{\cos k}{k} \cdot \frac{1}{k \cdot t}\right)}{t_2}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\left(\ell \cdot {\sin k}^{-2}\right) \cdot \left(\left(2 \cdot \frac{\ell}{t}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)\right)\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{t_3}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot t_3}{t_2}\\ \end{array} \]

Alternative 2
Error	12.1
Cost	21004

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2 \cdot \ell}{\tan k \cdot \frac{t_1}{\frac{1}{t \cdot \left(k \cdot k\right)}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+86}:\\ \;\;\;\;\cos k \cdot \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {t_1}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+299}:\\ \;\;\;\;2 \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t \cdot \frac{k}{\frac{\ell \cdot \ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k} \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \]

Alternative 3
Error	14.0
Cost	20752

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ t_2 := \frac{\ell}{\frac{k}{t}}\\ t_3 := \frac{\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \left({\sin k}^{-2} \cdot \frac{\ell}{t}\right)\right)\right)}{k}}{k}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -7.685428593217671 \cdot 10^{-227}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot \left(k \cdot t\right)}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 10^{+145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 10^{+220}:\\ \;\;\;\;t_1 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	14.0
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{k}{t}}\\ t_2 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq -7.685428593217671 \cdot 10^{-227}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot \left(k \cdot t\right)}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 10^{+145}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \left({\sin k}^{-2} \cdot \frac{\ell}{t}\right)\right)\right)}{k}}{k}\\ \mathbf{elif}\;\ell \leq 10^{+220}:\\ \;\;\;\;t_2 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{k} \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{t}\right)\right)}{k \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 5
Error	13.2
Cost	20752

\[\begin{array}{l} t_1 := \ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{\sin k}^{2}}\right)\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -7.685428593217671 \cdot 10^{-227}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot \left(k \cdot t\right)}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 10^{+145}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \left({\sin k}^{-2} \cdot \frac{\ell}{t}\right)\right)\right)}{k}}{k}\\ \mathbf{elif}\;\ell \leq 10^{+220}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k} \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	10.5
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := k \cdot \left(k \cdot t\right)\\ t_3 := \frac{2}{\frac{\frac{t_1}{\ell} \cdot t_2}{\cos k \cdot \ell}}\\ \mathbf{if}\;k \leq -2.900716830405538 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{\frac{t_1 \cdot \left(k \cdot k\right)}{\ell}}{\frac{\cos k}{\frac{t}{\ell}}}}\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t_2}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	10.4
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ t_3 := k \cdot \left(k \cdot t\right)\\ t_4 := \frac{2}{\frac{t_2 \cdot t_3}{\cos k \cdot \ell}}\\ \mathbf{if}\;k \leq -2.900716830405538 \cdot 10^{+135}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(k \cdot k\right)}{\ell \cdot \frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t_3}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Error	10.6
Cost	20752

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{t_2}{\ell}\\ t_4 := \frac{\cos k}{t_1}\\ \mathbf{if}\;k \leq -6.811906495958277 \cdot 10^{+118}:\\ \;\;\;\;\frac{2}{\frac{t_3}{\ell \cdot t_4}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t_3 \cdot \left(k \cdot k\right)}{\ell \cdot \frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{t_4}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_3 \cdot t_1}{\cos k \cdot \ell}}\\ \end{array} \]

Alternative 9
Error	10.5
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ t_3 := \frac{\cos k}{k \cdot \left(k \cdot t\right)}\\ t_4 := \ell \cdot \frac{2 \cdot t_3}{t_2}\\ \mathbf{if}\;k \leq -6.811906495958277 \cdot 10^{+118}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(k \cdot k\right)}{\ell \cdot \frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{t_3}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 10
Error	10.4
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t_1}{\ell}\\ t_3 := \frac{\cos k}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq -6.811906495958277 \cdot 10^{+118}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\frac{\cos k}{k} \cdot \frac{1}{k \cdot t}\right)}{t_2}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(k \cdot k\right)}{\ell \cdot \frac{\cos k}{t}}}\\ \mathbf{elif}\;k \leq 10^{-85}:\\ \;\;\;\;2 \cdot \frac{t_3}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.096956749131518 \cdot 10^{+116}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot t_3}{t_2}\\ \end{array} \]

Alternative 11
Error	13.7
Cost	20624

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{\ell}{\frac{k}{t}}\\ t_3 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ t_4 := \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\cos k}{{t_1}^{2}}}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -7.493061948223018 \cdot 10^{-162}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 7.305938612869355 \cdot 10^{-155}:\\ \;\;\;\;\frac{2 \cdot \ell}{\tan k \cdot \frac{t_1}{\frac{1}{t \cdot \left(k \cdot k\right)}}}\\ \mathbf{elif}\;\ell \leq 10^{+155}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \]

Alternative 12
Error	13.6
Cost	20624

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{\ell}{\frac{k}{t}}\\ t_3 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ t_4 := \cos k \cdot \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {t_1}^{2}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq -7.493061948223018 \cdot 10^{-162}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 7.305938612869355 \cdot 10^{-155}:\\ \;\;\;\;\frac{2 \cdot \ell}{\tan k \cdot \frac{t_1}{\frac{1}{t \cdot \left(k \cdot k\right)}}}\\ \mathbf{elif}\;\ell \leq 10^{+155}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \]

Alternative 13
Error	11.7
Cost	20624

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ t_2 := k \cdot \left(k \cdot t\right)\\ t_3 := {\sin k}^{2}\\ t_4 := \frac{\ell}{\frac{k}{t}}\\ \mathbf{if}\;t \leq -4.217581214812091 \cdot 10^{+68}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{t_3}\right)\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-35}:\\ \;\;\;\;t_1 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;t \leq 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{\frac{t_3}{\ell} \cdot t_2}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t \leq 5.1674682709165216 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot t_4\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t_2}}{{\left(\frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 14
Error	14.7
Cost	14928

\[\begin{array}{l} t_1 := \frac{1}{\tan k}\\ t_2 := \tan k \cdot \frac{\frac{\sin k}{\ell}}{\frac{1}{t \cdot \left(k \cdot k\right)}}\\ t_3 := \frac{2 \cdot \ell}{t_2}\\ t_4 := \frac{\ell}{\frac{k}{t}}\\ \mathbf{if}\;t \leq -4.217581214812091 \cdot 10^{+68}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{1}{t_2}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t} \cdot \frac{t_1}{t}}{\sin k} \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;t \leq 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.1674682709165216 \cdot 10^{+63}:\\ \;\;\;\;\left(t_4 \cdot t_4\right) \cdot \frac{\frac{2}{t} \cdot \frac{t_1}{t \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	14.7
Cost	14928

\[\begin{array}{l} t_1 := \frac{\frac{2}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{\sin k}\\ t_2 := \tan k \cdot \frac{\frac{\sin k}{\ell}}{\frac{1}{t \cdot \left(k \cdot k\right)}}\\ t_3 := \frac{2 \cdot \ell}{t_2}\\ t_4 := \frac{\ell}{\frac{k}{t}}\\ \mathbf{if}\;t \leq -4.217581214812091 \cdot 10^{+68}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{1}{t_2}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-35}:\\ \;\;\;\;t_1 \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;t \leq 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.1674682709165216 \cdot 10^{+63}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot t_4\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	15.4
Cost	14664

\[\begin{array}{l} t_1 := \frac{2 \cdot \ell}{\tan k \cdot \frac{\frac{\sin k}{\ell}}{\frac{1}{t \cdot \left(k \cdot k\right)}}}\\ t_2 := \frac{\ell}{\frac{k}{t}}\\ \mathbf{if}\;t \leq 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.1674682709165216 \cdot 10^{+63}:\\ \;\;\;\;\left(t_2 \cdot t_2\right) \cdot \frac{\frac{2}{t} \cdot \frac{\frac{1}{\tan k}}{t \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	15.5
Cost	14152

\[\begin{array}{l} t_1 := \frac{2 \cdot \ell}{\tan k \cdot \frac{\frac{\sin k}{\ell}}{\frac{1}{t \cdot \left(k \cdot k\right)}}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-159}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	24.1
Cost	13572

\[\begin{array}{l} \mathbf{if}\;t \leq 1.9789043471616536 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 19
Error	24.1
Cost	13572

\[\begin{array}{l} \mathbf{if}\;t \leq 1.9789043471616536 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{k}{\frac{\ell}{k}}\right)}^{2}}\\ \end{array} \]

Alternative 20
Error	24.4
Cost	7620

\[\begin{array}{l} \mathbf{if}\;t \leq 1.9789043471616536 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{k \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 21
Error	24.7
Cost	1088

\[\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.3333333333333333\right) \]

Alternative 22
Error	26.3
Cost	960

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

Alternative 23
Error	27.2
Cost	960

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

Alternative 24
Error	26.3
Cost	960

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

Alternative 25
Error	26.2
Cost	960

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

Alternative 26
Error	25.8
Cost	960

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

Reproduce

herbie shell --seed 2022298 
(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))))