Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 0.9

Time: 38.8s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \ell \cdot \frac{\cos k}{k}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{\frac{-t}{\frac{-\ell}{k \cdot t_1}}}{t_2}}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \frac{1}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{t_1}}}{t_2}}\\ \end{array} \]

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 (* l (/ (cos k) k))) (t_3 (* k (/ k l))))
   (if (<= k -4.3e-100)
     (/ 2.0 (/ (/ (- t) (/ (- l) (* k t_1))) t_2))
     (if (<= k 6.8e-84)
       (* 2.0 (/ 1.0 (* t_3 (* t t_3))))
       (/ 2.0 (/ (* t (/ k (/ l t_1))) t_2))))))

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 = l * (cos(k) / k);
	double t_3 = k * (k / l);
	double tmp;
	if (k <= -4.3e-100) {
		tmp = 2.0 / ((-t / (-l / (k * t_1))) / t_2);
	} else if (k <= 6.8e-84) {
		tmp = 2.0 * (1.0 / (t_3 * (t * t_3)));
	} else {
		tmp = 2.0 / ((t * (k / (l / t_1))) / t_2);
	}
	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 = sin(k) ** 2.0d0
    t_2 = l * (cos(k) / k)
    t_3 = k * (k / l)
    if (k <= (-4.3d-100)) then
        tmp = 2.0d0 / ((-t / (-l / (k * t_1))) / t_2)
    else if (k <= 6.8d-84) then
        tmp = 2.0d0 * (1.0d0 / (t_3 * (t * t_3)))
    else
        tmp = 2.0d0 / ((t * (k / (l / t_1))) / t_2)
    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 = l * (Math.cos(k) / k);
	double t_3 = k * (k / l);
	double tmp;
	if (k <= -4.3e-100) {
		tmp = 2.0 / ((-t / (-l / (k * t_1))) / t_2);
	} else if (k <= 6.8e-84) {
		tmp = 2.0 * (1.0 / (t_3 * (t * t_3)));
	} else {
		tmp = 2.0 / ((t * (k / (l / t_1))) / t_2);
	}
	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 = l * (math.cos(k) / k)
	t_3 = k * (k / l)
	tmp = 0
	if k <= -4.3e-100:
		tmp = 2.0 / ((-t / (-l / (k * t_1))) / t_2)
	elif k <= 6.8e-84:
		tmp = 2.0 * (1.0 / (t_3 * (t * t_3)))
	else:
		tmp = 2.0 / ((t * (k / (l / t_1))) / t_2)
	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(l * Float64(cos(k) / k))
	t_3 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= -4.3e-100)
		tmp = Float64(2.0 / Float64(Float64(Float64(-t) / Float64(Float64(-l) / Float64(k * t_1))) / t_2));
	elseif (k <= 6.8e-84)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t_3 * Float64(t * t_3))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / Float64(l / t_1))) / t_2));
	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 = l * (cos(k) / k);
	t_3 = k * (k / l);
	tmp = 0.0;
	if (k <= -4.3e-100)
		tmp = 2.0 / ((-t / (-l / (k * t_1))) / t_2);
	elseif (k <= 6.8e-84)
		tmp = 2.0 * (1.0 / (t_3 * (t * t_3)));
	else
		tmp = 2.0 / ((t * (k / (l / t_1))) / t_2);
	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[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.3e-100], N[(2.0 / N[(N[((-t) / N[((-l) / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.8e-84], N[(2.0 * N[(1.0 / N[(t$95$3 * N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.29999999999999998e-100

   1. Initial program 45.6

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

   2. Simplified 37.7

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 45.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 45.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 45.6	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 45.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 37.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 37.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.6

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

   4. Simplified 14.5

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

      [Start] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.2	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.2	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.2	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 14.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 4.5

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

   6. Applied egg-rr 5.5

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

   7. Simplified 0.9

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

      [Start] 5.5	\[ \frac{2}{\frac{\frac{\left(-t\right) \cdot \left(k \cdot {\sin k}^{2}\right)}{-\ell}}{\ell \cdot \frac{\cos k}{k}}} \]
      associate-/l* [=>] 0.9	\[ \frac{2}{\frac{\color{blue}{\frac{-t}{\frac{-\ell}{k \cdot {\sin k}^{2}}}}}{\ell \cdot \frac{\cos k}{k}}} \]

   if -4.29999999999999998e-100 < k < 6.80000000000000042e-84

   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. Simplified 56.2

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

      [Start] 64.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 64.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 64.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 64.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 63.8	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 63.8	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 63.8	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 63.8	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 63.8	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 63.8	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 64.0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 64.0

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 64.0	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 64.0	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 64.0	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 30.3

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

   6. Applied egg-rr 0.7

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

   if 6.80000000000000042e-84 < k

   1. Initial program 46.0

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

   2. Simplified 37.5

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 46.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 46.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 45.9	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 45.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 37.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 37.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.6

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

   4. Simplified 14.2

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

      [Start] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.2	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.2	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.2	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 14.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 4.5

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

   6. Taylor expanded in t around 0 5.3

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

   7. Simplified 0.8

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

      [Start] 5.3	\[ \frac{2}{\frac{\frac{k \cdot \left({\sin k}^{2} \cdot t\right)}{\ell}}{\ell \cdot \frac{\cos k}{k}}} \]
      associate-*r* [=>] 5.3	\[ \frac{2}{\frac{\frac{\color{blue}{\left(k \cdot {\sin k}^{2}\right) \cdot t}}{\ell}}{\ell \cdot \frac{\cos k}{k}}} \]
      associate-*l/ [<=] 0.8	\[ \frac{2}{\frac{\color{blue}{\frac{k \cdot {\sin k}^{2}}{\ell} \cdot t}}{\ell \cdot \frac{\cos k}{k}}} \]
      associate-/l* [=>] 0.8	\[ \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}}} \cdot t}{\ell \cdot \frac{\cos k}{k}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{\frac{-t}{\frac{-\ell}{k \cdot {\sin k}^{2}}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{{\sin k}^{2}}}}{\ell \cdot \frac{\cos k}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000 \lor \neg \left(k \leq 1.4 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{-46} \lor \neg \left(k \leq 1.46 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{{\sin k}^{2}}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	12.3
Cost	14672

\[\begin{array}{l} t_1 := \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}\\ t_2 := k \cdot \frac{k}{\ell}\\ t_3 := \frac{2}{\frac{k \cdot \frac{k}{\frac{\cos k}{t}}}{\frac{\ell}{t_1}}}\\ t_4 := \frac{\cos k}{k}\\ \mathbf{if}\;k \leq -16500000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{1}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{\frac{k}{t_4} \cdot \left(\frac{t}{\ell} \cdot t_1\right)}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+206}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333}}{\ell \cdot t_4}}\\ \end{array} \]

Alternative 4
Error	12.6
Cost	14540

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := \frac{2}{\frac{k}{t_1} \cdot \left(\frac{t}{\ell} \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}\right)}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 0.000225:\\ \;\;\;\;2 \cdot \frac{1}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333}}{\ell \cdot t_1}}\\ \end{array} \]

Alternative 5
Error	4.5
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000 \lor \neg \left(k \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\left(0.5 - \frac{\cos \left(k + k\right)}{2}\right) \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Error	1.4
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000 \lor \neg \left(k \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{0.5 - \frac{\cos \left(k + k\right)}{2}}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 7
Error	13.2
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000 \lor \neg \left(k \leq 7.8 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Error	20.1
Cost	8009

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -16500000 \lor \neg \left(k \leq 1.65 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	1088

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ 2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Alternative 10
Error	25.7
Cost	960

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

Alternative 11
Error	24.4
Cost	960

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

Alternative 12
Error	24.6
Cost	960

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

Alternative 13
Error	23.4
Cost	960

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

Reproduce

herbie shell --seed 2023237 
(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))))