Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 4.1

Time: 43.0s

Precision: binary64

Cost: 20752

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{t}{\cos k}\\ t_3 := {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}\\ \mathbf{if}\;k \leq -5.88830357579077 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{t_3}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{k \cdot k} \cdot \frac{t_1}{t_2}\right)\\ \mathbf{elif}\;k \leq 2.562542258717251 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{\frac{k}{\cos k}} \cdot \frac{t_1}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 3.5877754439980213 \cdot 10^{+269}:\\ \;\;\;\;2 \cdot \frac{1}{t_3 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \end{array} \]

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 (sin k)))
        (t_2 (/ t (cos k)))
        (t_3 (pow (/ (* k (sin k)) l) 2.0)))
   (if (<= k -5.88830357579077e+151)
     (* 2.0 (/ (/ (cos k) t) t_3))
     (if (<= k -1e-145)
       (* 2.0 (* (/ t_1 (* k k)) (/ t_1 t_2)))
       (if (<= k 2.562542258717251e+100)
         (* 2.0 (* (/ t_1 (/ k (cos k))) (/ t_1 (* k t))))
         (if (<= k 3.5877754439980213e+269)
           (* 2.0 (/ 1.0 (* t_3 t_2)))
           (*
            2.0
            (/ (* (/ (cos k) (pow (sin k) 2.0)) (* l (/ l k))) (* k t)))))))))

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 / sin(k);
	double t_2 = t / cos(k);
	double t_3 = pow(((k * sin(k)) / l), 2.0);
	double tmp;
	if (k <= -5.88830357579077e+151) {
		tmp = 2.0 * ((cos(k) / t) / t_3);
	} else if (k <= -1e-145) {
		tmp = 2.0 * ((t_1 / (k * k)) * (t_1 / t_2));
	} else if (k <= 2.562542258717251e+100) {
		tmp = 2.0 * ((t_1 / (k / cos(k))) * (t_1 / (k * t)));
	} else if (k <= 3.5877754439980213e+269) {
		tmp = 2.0 * (1.0 / (t_3 * t_2));
	} else {
		tmp = 2.0 * (((cos(k) / pow(sin(k), 2.0)) * (l * (l / k))) / (k * t));
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = l / sin(k)
    t_2 = t / cos(k)
    t_3 = ((k * sin(k)) / l) ** 2.0d0
    if (k <= (-5.88830357579077d+151)) then
        tmp = 2.0d0 * ((cos(k) / t) / t_3)
    else if (k <= (-1d-145)) then
        tmp = 2.0d0 * ((t_1 / (k * k)) * (t_1 / t_2))
    else if (k <= 2.562542258717251d+100) then
        tmp = 2.0d0 * ((t_1 / (k / cos(k))) * (t_1 / (k * t)))
    else if (k <= 3.5877754439980213d+269) then
        tmp = 2.0d0 * (1.0d0 / (t_3 * t_2))
    else
        tmp = 2.0d0 * (((cos(k) / (sin(k) ** 2.0d0)) * (l * (l / k))) / (k * t))
    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 / Math.sin(k);
	double t_2 = t / Math.cos(k);
	double t_3 = Math.pow(((k * Math.sin(k)) / l), 2.0);
	double tmp;
	if (k <= -5.88830357579077e+151) {
		tmp = 2.0 * ((Math.cos(k) / t) / t_3);
	} else if (k <= -1e-145) {
		tmp = 2.0 * ((t_1 / (k * k)) * (t_1 / t_2));
	} else if (k <= 2.562542258717251e+100) {
		tmp = 2.0 * ((t_1 / (k / Math.cos(k))) * (t_1 / (k * t)));
	} else if (k <= 3.5877754439980213e+269) {
		tmp = 2.0 * (1.0 / (t_3 * t_2));
	} else {
		tmp = 2.0 * (((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (l * (l / k))) / (k * t));
	}
	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 / math.sin(k)
	t_2 = t / math.cos(k)
	t_3 = math.pow(((k * math.sin(k)) / l), 2.0)
	tmp = 0
	if k <= -5.88830357579077e+151:
		tmp = 2.0 * ((math.cos(k) / t) / t_3)
	elif k <= -1e-145:
		tmp = 2.0 * ((t_1 / (k * k)) * (t_1 / t_2))
	elif k <= 2.562542258717251e+100:
		tmp = 2.0 * ((t_1 / (k / math.cos(k))) * (t_1 / (k * t)))
	elif k <= 3.5877754439980213e+269:
		tmp = 2.0 * (1.0 / (t_3 * t_2))
	else:
		tmp = 2.0 * (((math.cos(k) / math.pow(math.sin(k), 2.0)) * (l * (l / k))) / (k * t))
	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(l / sin(k))
	t_2 = Float64(t / cos(k))
	t_3 = Float64(Float64(k * sin(k)) / l) ^ 2.0
	tmp = 0.0
	if (k <= -5.88830357579077e+151)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) / t_3));
	elseif (k <= -1e-145)
		tmp = Float64(2.0 * Float64(Float64(t_1 / Float64(k * k)) * Float64(t_1 / t_2)));
	elseif (k <= 2.562542258717251e+100)
		tmp = Float64(2.0 * Float64(Float64(t_1 / Float64(k / cos(k))) * Float64(t_1 / Float64(k * t))));
	elseif (k <= 3.5877754439980213e+269)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t_3 * t_2)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(l * Float64(l / k))) / Float64(k * t)));
	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 / sin(k);
	t_2 = t / cos(k);
	t_3 = ((k * sin(k)) / l) ^ 2.0;
	tmp = 0.0;
	if (k <= -5.88830357579077e+151)
		tmp = 2.0 * ((cos(k) / t) / t_3);
	elseif (k <= -1e-145)
		tmp = 2.0 * ((t_1 / (k * k)) * (t_1 / t_2));
	elseif (k <= 2.562542258717251e+100)
		tmp = 2.0 * ((t_1 / (k / cos(k))) * (t_1 / (k * t)));
	elseif (k <= 3.5877754439980213e+269)
		tmp = 2.0 * (1.0 / (t_3 * t_2));
	else
		tmp = 2.0 * (((cos(k) / (sin(k) ^ 2.0)) * (l * (l / k))) / (k * t));
	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[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, -5.88830357579077e+151], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-145], N[(2.0 * N[(N[(t$95$1 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.562542258717251e+100], N[(2.0 * N[(N[(t$95$1 / N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5877754439980213e+269], N[(2.0 * N[(1.0 / N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -5.8883035757907706e151

   1. Initial program 40.3

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

   2. Simplified 35.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 3 points increase in error, 2 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))): 4 points increase in error, 1 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))): 1 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, 3 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)))): 22 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, 4 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))): 3 points increase in error, 0 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, 3 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, 3 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, 1 points decrease in error

   3. Taylor expanded in k around inf 26.1

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

   4. Applied egg-rr 26.0

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

   5. Applied egg-rr 4.3

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

   if -5.8883035757907706e151 < k < -9.99999999999999915e-146

   1. Initial program 51.3

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

   2. Taylor expanded in t around 0 17.8

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

   3. Simplified 16.7

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{\frac{k \cdot k}{\frac{\cos k}{t}}}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 l l) (pow.f64 (sin.f64 k) 2)) (/.f64 (*.f64 k k) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 (sin.f64 k) 2)) (/.f64 (*.f64 k k) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 (sin.f64 k) 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (pow.f64 l 2) (pow.f64 (sin.f64 k) 2)) (/.f64 (cos.f64 k) t)) (pow.f64 k 2)))): 22 points increase in error, 21 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (pow.f64 k 2))): 21 points increase in error, 8 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 20 points increase in error, 22 points decrease in error

   4. Applied egg-rr 3.6

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

   if -9.99999999999999915e-146 < k < 2.56254225871725087e100

   1. Initial program 57.3

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

   2. Taylor expanded in t around 0 26.1

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

   3. Simplified 24.2

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{{\sin k}^{2}}}{\frac{k \cdot k}{\frac{\cos k}{t}}}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 l l) (pow.f64 (sin.f64 k) 2)) (/.f64 (*.f64 k k) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (pow.f64 (sin.f64 k) 2)) (/.f64 (*.f64 k k) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (pow.f64 l 2) (pow.f64 (sin.f64 k) 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (/.f64 (cos.f64 k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (pow.f64 l 2) (pow.f64 (sin.f64 k) 2)) (/.f64 (cos.f64 k) t)) (pow.f64 k 2)))): 22 points increase in error, 21 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (pow.f64 k 2))): 21 points increase in error, 8 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 20 points increase in error, 22 points decrease in error

   4. Applied egg-rr 24.2

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

   5. Applied egg-rr 2.8

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

   if 2.56254225871725087e100 < k < 3.5877754439980213e269

   1. Initial program 43.2

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

   2. Simplified 36.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 3 points increase in error, 2 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))): 4 points increase in error, 1 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))): 1 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, 3 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)))): 22 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, 4 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))): 3 points increase in error, 0 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, 3 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, 3 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, 1 points decrease in error

   3. Taylor expanded in k around inf 23.0

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

   4. Applied egg-rr 23.1

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

   5. Applied egg-rr 5.4

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

   if 3.5877754439980213e269 < k

   1. Initial program 36.9

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

   2. Simplified 35.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 3 points increase in error, 2 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))): 4 points increase in error, 1 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))): 1 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, 3 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)))): 22 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, 4 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))): 3 points increase in error, 0 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, 3 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, 3 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, 1 points decrease in error

   3. Applied egg-rr 36.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

   4. Taylor expanded in l around 0 22.1

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

   5. Simplified 2.8

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 62 points increase in error, 12 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 25 points increase in error, 16 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.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 5.9

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.88830357579077 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k \cdot k} \cdot \frac{\frac{\ell}{\sin k}}{\frac{t}{\cos k}}\right)\\ \mathbf{elif}\;k \leq 2.562542258717251 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 3.5877754439980213 \cdot 10^{+269}:\\ \;\;\;\;2 \cdot \frac{1}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2} \cdot \frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.5
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\cos k \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t}}{{\sin k}^{2}}\right)\right)\\ t_2 := {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}\\ \mathbf{if}\;k \leq -5.88830357579077 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{t_2}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-135}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{\sin k}\right)}^{2}}{\frac{k}{\cos k}}}{k \cdot t}\\ \mathbf{elif}\;k \leq 2.562542258717251 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_2 \cdot \frac{t}{\cos k}}\\ \end{array} \]

Alternative 2
Error	4.3
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\sin k}^{2}\\ t_3 := {\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}\\ \mathbf{if}\;k \leq -5.88830357579077 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{t_3}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t}}{t_2}\right)\right)\\ \mathbf{elif}\;k \leq 2.562542258717251 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{\frac{k}{\cos k}} \cdot \frac{t_1}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 3.5877754439980213 \cdot 10^{+269}:\\ \;\;\;\;2 \cdot \frac{1}{t_3 \cdot \frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t_2} \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \end{array} \]

Alternative 3
Error	6.0
Cost	20624

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := 2 \cdot \frac{\frac{{\left(\frac{\ell}{\sin k}\right)}^{2}}{\frac{k}{\cos k}}}{k \cdot t}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\frac{t_1}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq -8.117232362823506 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7.909016025548328 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{t}}{k \cdot \frac{k}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 10^{+28}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{t_1}\right)}^{2}}{t}\\ \end{array} \]

Alternative 4
Error	6.0
Cost	20624

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := {\left(\frac{\ell}{\sin k}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\frac{t_1}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq -8.117232362823506 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \frac{\frac{t_2}{\frac{k}{\cos k}}}{k \cdot t}\\ \mathbf{elif}\;\ell \leq 7.909016025548328 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{t}}{k \cdot \frac{k}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 10^{+20}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{t_2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{t_1}\right)}^{2}}{t}\\ \end{array} \]

Alternative 5
Error	5.6
Cost	20624

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := {\left(\frac{\ell}{\sin k}\right)}^{2}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\frac{t_1}{\ell}\right)}^{2}}\\ \mathbf{elif}\;\ell \leq -8.117232362823506 \cdot 10^{-283}:\\ \;\;\;\;2 \cdot \frac{\frac{t_2}{\frac{k}{\cos k}}}{k \cdot t}\\ \mathbf{elif}\;\ell \leq 7.909016025548328 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{t}}{k \cdot \frac{k}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 10^{+50}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{t_2}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{t_1}\right)}^{2}}{t}\\ \end{array} \]

Alternative 6
Error	5.4
Cost	20364

\[\begin{array}{l} t_1 := k \cdot \sin k\\ \mathbf{if}\;k \leq -1.1104007812121505 \cdot 10^{+114}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{t_1}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq -1:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \ell}{t \cdot \frac{{t_1}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-132}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{t_1}\right)}^{2}}{t}\\ \end{array} \]

Alternative 7
Error	5.3
Cost	20364

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ t_2 := k \cdot \sin k\\ \mathbf{if}\;k \leq -1.1104007812121505 \cdot 10^{+114}:\\ \;\;\;\;2 \cdot \frac{t_1}{{\left(\frac{t_2}{\ell}\right)}^{2}}\\ \mathbf{elif}\;k \leq -1:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot \ell}{\frac{{t_2}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-132}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{t_2}\right)}^{2}}{t}\\ \end{array} \]

Alternative 8
Error	6.5
Cost	20232

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-132}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	6.6
Cost	20232

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{t_1}{\frac{t}{\cos k}}\\ \mathbf{elif}\;k \leq 10^{-132}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot t_1}{t}\\ \end{array} \]

Alternative 10
Error	5.7
Cost	14408

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-5}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	23.2
Cost	14340

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{{k}^{4}} + \frac{\ell \cdot 0.3333333333333333}{k \cdot k}\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	22.7
Cost	14148

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-321}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\cos k}} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\ell \cdot \left(\frac{\frac{\ell}{k}}{k} + \ell \cdot 0.3333333333333333\right)}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t_1} + \frac{-0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 13
Error	23.8
Cost	8264

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+254}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\ell \cdot \left(\frac{\frac{\ell}{k}}{k} + \ell \cdot 0.3333333333333333\right)}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t_1} + \frac{-0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 14
Error	23.9
Cost	7752

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-200}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot \frac{k}{\ell}}}{k} \cdot \frac{\cos k}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	24.1
Cost	7560

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{t_1}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot 0.5\right)}\\ \mathbf{elif}\;k \leq 10^{-200}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \end{array} \]

Alternative 16
Error	25.6
Cost	960

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

Alternative 17
Error	25.2
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ 2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right) \end{array} \]

Alternative 18
Error	25.2
Cost	960

\[\frac{\frac{\ell}{k \cdot k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot 0.5\right)} \]

Reproduce

herbie shell --seed 2022295 
(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))))