Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 94.6%

Time: 19.9s

Alternatives: 9

Speedup: 28.1×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Specification

Math

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

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))))

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));
}

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

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));
}

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))

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

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

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]

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

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))))

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));
}

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

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));
}

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))

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

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

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]

Alternative 1: 94.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := \frac{\ell}{\sin k}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{t_2}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+149}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{t_1}{k}\right)\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+184}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (/ l (sin k))))
   (if (<= k 5e-22)
     (* (/ 2.0 k) (* (/ t_2 t) (/ (/ l k) k)))
     (if (<= k 1.3e+149)
       (* (/ l (* (sin k) t)) (* (/ 2.0 k) (/ t_1 k)))
       (if (<= k 1.9e+184)
         (* t_1 (* t_2 (/ 2.0 (* k (* k t)))))
         (* (/ 2.0 k) (* (/ l k) (/ (/ (/ l t) (sin k)) (tan k)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = l / sin(k);
	double tmp;
	if (k <= 5e-22) {
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	} else if (k <= 1.3e+149) {
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	} else if (k <= 1.9e+184) {
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 / k) * ((l / k) * (((l / t) / sin(k)) / tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this 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 = l / tan(k)
    t_2 = l / sin(k)
    if (k <= 5d-22) then
        tmp = (2.0d0 / k) * ((t_2 / t) * ((l / k) / k))
    else if (k <= 1.3d+149) then
        tmp = (l / (sin(k) * t)) * ((2.0d0 / k) * (t_1 / k))
    else if (k <= 1.9d+184) then
        tmp = t_1 * (t_2 * (2.0d0 / (k * (k * t))))
    else
        tmp = (2.0d0 / k) * ((l / k) * (((l / t) / sin(k)) / tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = l / Math.sin(k);
	double tmp;
	if (k <= 5e-22) {
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	} else if (k <= 1.3e+149) {
		tmp = (l / (Math.sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	} else if (k <= 1.9e+184) {
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 / k) * ((l / k) * (((l / t) / Math.sin(k)) / Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = l / math.tan(k)
	t_2 = l / math.sin(k)
	tmp = 0
	if k <= 5e-22:
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k))
	elif k <= 1.3e+149:
		tmp = (l / (math.sin(k) * t)) * ((2.0 / k) * (t_1 / k))
	elif k <= 1.9e+184:
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))))
	else:
		tmp = (2.0 / k) * ((l / k) * (((l / t) / math.sin(k)) / math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(l / sin(k))
	tmp = 0.0
	if (k <= 5e-22)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(t_2 / t) * Float64(Float64(l / k) / k)));
	elseif (k <= 1.3e+149)
		tmp = Float64(Float64(l / Float64(sin(k) * t)) * Float64(Float64(2.0 / k) * Float64(t_1 / k)));
	elseif (k <= 1.9e+184)
		tmp = Float64(t_1 * Float64(t_2 * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / k) * Float64(Float64(Float64(l / t) / sin(k)) / tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	t_2 = l / sin(k);
	tmp = 0.0;
	if (k <= 5e-22)
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	elseif (k <= 1.3e+149)
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	elseif (k <= 1.9e+184)
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	else
		tmp = (2.0 / k) * ((l / k) * (((l / t) / sin(k)) / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5e-22], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(t$95$2 / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.3e+149], N[(N[(l / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+184], N[(t$95$1 * N[(t$95$2 * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 4.99999999999999954e-22

   1. Initial program 42.2%

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

      1. associate-*l* 42.2%

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

      2. associate-*l* 42.2%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 43.7%

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

      7. +-commutative 43.7%

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

      8. associate--l+ 47.1%

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

      9. metadata-eval 47.1%

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

      10. +-rgt-identity 47.1%

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

      11. times-frac 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in t around 0 83.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.1%

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

   6. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 83.1%

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

      2. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

      3. times-frac 96.1%

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

   10. Simplified 96.1%

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

   11. Taylor expanded in k around 0 78.2%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   12. Step-by-step derivation

       1. unpow2 78.2%

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

       2. associate-/r* 82.4%

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

   13. Simplified 82.4%

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

   if 4.99999999999999954e-22 < k < 1.29999999999999989e149

   1. Initial program 28.4%

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

      1. associate-*l* 28.4%

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

      2. associate-*l* 28.4%

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

      3. associate-/r* 28.4%

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

      4. associate-/r/ 28.4%

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

      5. *-commutative 28.4%

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

      6. times-frac 31.0%

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

      7. +-commutative 31.0%

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

      8. associate--l+ 41.7%

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

      9. metadata-eval 41.7%

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

      10. +-rgt-identity 41.7%

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

      11. times-frac 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in t around 0 86.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.4%

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

   6. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 86.5%

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

      2. associate-*l* 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 84.1%

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

      2. *-commutative 84.1%

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

      3. frac-times 92.1%

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

      4. associate-*r* 97.2%

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

      5. associate-/l/ 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 1.29999999999999989e149 < k < 1.9000000000000001e184

   1. Initial program 40.0%

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

      1. associate-*l* 40.0%

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

      2. associate-*l* 40.0%

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

      3. associate-/r* 40.0%

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

      4. associate-/r/ 40.0%

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

      5. *-commutative 40.0%

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

      6. times-frac 40.0%

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

      7. +-commutative 40.0%

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

      8. associate--l+ 60.0%

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

      9. metadata-eval 60.0%

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

      10. +-rgt-identity 60.0%

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

      11. times-frac 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in t around 0 71.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 71.1%

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

   6. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 71.1%

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

      2. associate-*l* 91.0%

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

   8. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 90.8%

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

      2. associate-*r* 99.7%

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

   10. Simplified 99.7%

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

   if 1.9000000000000001e184 < k

   1. Initial program 26.7%

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

      1. associate-*l* 26.7%

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

      2. associate-*l* 26.7%

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

      3. associate-/r* 26.7%

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

      4. associate-/r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. times-frac 23.7%

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

      7. +-commutative 23.7%

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

      8. associate--l+ 40.4%

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

      9. metadata-eval 40.4%

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

      10. +-rgt-identity 40.4%

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

      11. times-frac 40.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in t around 0 56.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 56.8%

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

   6. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 56.8%

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

      2. associate-*l* 63.2%

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

   8. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 80.1%

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

      2. *-commutative 80.1%

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

      3. times-frac 90.0%

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

   10. Simplified 90.0%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)} \]
   11. Step-by-step derivation

       1. div-inv 90.0%

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

   12. Applied egg-rr 90.0%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{1}{k}\right)} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 90.1%

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

       2. associate-*r/ 90.2%

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

       3. *-rgt-identity 90.2%

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

   14. Simplified 90.2%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{\tan k}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   15. Step-by-step derivation

       1. expm1-log1p-u 69.8%

          \[\leadsto \frac{2}{k} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)\right)} \]

       2. expm1-udef 62.6%

          \[\leadsto \frac{2}{k} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)} - 1\right)} \]

       3. associate-/l/ 62.6%

          \[\leadsto \frac{2}{k} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\tan k \cdot k}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)} - 1\right) \]

       4. associate-/l/ 62.6%

          \[\leadsto \frac{2}{k} \cdot \left(e^{\mathsf{log1p}\left(\frac{\ell}{\tan k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \sin k}}\right)} - 1\right) \]

   16. Applied egg-rr 62.6%

       \[\leadsto \frac{2}{k} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\tan k \cdot k} \cdot \frac{\ell}{t \cdot \sin k}\right)} - 1\right)} \]
   17. Step-by-step derivation

       1. expm1-def 69.9%

          \[\leadsto \frac{2}{k} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\tan k \cdot k} \cdot \frac{\ell}{t \cdot \sin k}\right)\right)} \]

       2. expm1-log1p 90.1%

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

       3. associate-*l/ 73.9%

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

       4. *-commutative 73.9%

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

       5. times-frac 90.2%

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

       6. associate-/r* 90.2%

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

   18. Simplified 90.2%

       \[\leadsto \frac{2}{k} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\tan k}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+149}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{\tan k}}{k}\right)\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+184}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\tan k}\right)\\ \end{array} \]

Alternative 2: 91.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := \frac{\ell}{\sin k}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{t_2}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{t_1}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))) (t_2 (/ l (sin k))))
   (if (<= k 5e-22)
     (* (/ 2.0 k) (* (/ t_2 t) (/ (/ l k) k)))
     (if (<= k 2.2e+149)
       (* (/ l (* (sin k) t)) (* (/ 2.0 k) (/ t_1 k)))
       (* t_1 (* t_2 (/ 2.0 (* k (* k t)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double t_2 = l / sin(k);
	double tmp;
	if (k <= 5e-22) {
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	} else if (k <= 2.2e+149) {
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	} else {
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this 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 = l / tan(k)
    t_2 = l / sin(k)
    if (k <= 5d-22) then
        tmp = (2.0d0 / k) * ((t_2 / t) * ((l / k) / k))
    else if (k <= 2.2d+149) then
        tmp = (l / (sin(k) * t)) * ((2.0d0 / k) * (t_1 / k))
    else
        tmp = t_1 * (t_2 * (2.0d0 / (k * (k * t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double t_2 = l / Math.sin(k);
	double tmp;
	if (k <= 5e-22) {
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	} else if (k <= 2.2e+149) {
		tmp = (l / (Math.sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	} else {
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = l / math.tan(k)
	t_2 = l / math.sin(k)
	tmp = 0
	if k <= 5e-22:
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k))
	elif k <= 2.2e+149:
		tmp = (l / (math.sin(k) * t)) * ((2.0 / k) * (t_1 / k))
	else:
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / tan(k))
	t_2 = Float64(l / sin(k))
	tmp = 0.0
	if (k <= 5e-22)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(t_2 / t) * Float64(Float64(l / k) / k)));
	elseif (k <= 2.2e+149)
		tmp = Float64(Float64(l / Float64(sin(k) * t)) * Float64(Float64(2.0 / k) * Float64(t_1 / k)));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(2.0 / Float64(k * Float64(k * t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	t_2 = l / sin(k);
	tmp = 0.0;
	if (k <= 5e-22)
		tmp = (2.0 / k) * ((t_2 / t) * ((l / k) / k));
	elseif (k <= 2.2e+149)
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * (t_1 / k));
	else
		tmp = t_1 * (t_2 * (2.0 / (k * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5e-22], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(t$95$2 / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e+149], N[(N[(l / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.99999999999999954e-22

   1. Initial program 42.2%

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

      1. associate-*l* 42.2%

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

      2. associate-*l* 42.2%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 43.7%

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

      7. +-commutative 43.7%

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

      8. associate--l+ 47.1%

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

      9. metadata-eval 47.1%

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

      10. +-rgt-identity 47.1%

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

      11. times-frac 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in t around 0 83.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.1%

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

   6. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 83.1%

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

      2. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

      3. times-frac 96.1%

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

   10. Simplified 96.1%

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

   11. Taylor expanded in k around 0 78.2%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   12. Step-by-step derivation

       1. unpow2 78.2%

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

       2. associate-/r* 82.4%

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

   13. Simplified 82.4%

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

   if 4.99999999999999954e-22 < k < 2.2e149

   1. Initial program 28.4%

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

      1. associate-*l* 28.4%

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

      2. associate-*l* 28.4%

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

      3. associate-/r* 28.4%

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

      4. associate-/r/ 28.4%

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

      5. *-commutative 28.4%

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

      6. times-frac 31.0%

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

      7. +-commutative 31.0%

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

      8. associate--l+ 41.7%

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

      9. metadata-eval 41.7%

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

      10. +-rgt-identity 41.7%

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

      11. times-frac 41.7%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in t around 0 86.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.4%

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

   6. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 86.5%

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

      2. associate-*l* 86.5%

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

   8. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 84.1%

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

      2. *-commutative 84.1%

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

      3. frac-times 92.1%

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

      4. associate-*r* 97.2%

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

      5. associate-/l/ 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 2.2e149 < k

   1. Initial program 30.0%

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

      1. associate-*l* 30.0%

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

      2. associate-*l* 30.0%

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

      3. associate-/r* 30.0%

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

      4. associate-/r/ 30.0%

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

      5. *-commutative 30.0%

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

      6. times-frac 27.8%

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

      7. +-commutative 27.8%

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

      8. associate--l+ 45.3%

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

      9. metadata-eval 45.3%

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

      10. +-rgt-identity 45.3%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 60.4%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 60.4%

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

   6. Simplified 60.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 60.4%

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

      2. associate-*l* 70.2%

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

   8. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ 70.1%

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

      2. associate-*r* 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{\tan k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 3: 93.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot \tan k} \cdot \frac{\ell}{\sin k \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ 2.0 k) (* (/ (/ l (sin k)) t) (/ (/ l k) k)))
   (* (/ 2.0 k) (* (/ l (* k (tan k))) (/ l (* (sin k) t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
	} else {
		tmp = (2.0 / k) * ((l / (k * tan(k))) * (l / (sin(k) * t)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this 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) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 / k) * (((l / sin(k)) / t) * ((l / k) / k))
    else
        tmp = (2.0d0 / k) * ((l / (k * tan(k))) * (l / (sin(k) * t)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / k) * (((l / Math.sin(k)) / t) * ((l / k) / k));
	} else {
		tmp = (2.0 / k) * ((l / (k * Math.tan(k))) * (l / (Math.sin(k) * t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 / k) * (((l / math.sin(k)) / t) * ((l / k) / k))
	else:
		tmp = (2.0 / k) * ((l / (k * math.tan(k))) * (l / (math.sin(k) * t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / sin(k)) / t) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(k * tan(k))) * Float64(l / Float64(sin(k) * t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
	else
		tmp = (2.0 / k) * ((l / (k * tan(k))) * (l / (sin(k) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. associate-*l* 28.9%

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

      3. associate-/r* 28.9%

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

      4. associate-/r/ 28.8%

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

      5. *-commutative 28.8%

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

      6. times-frac 28.8%

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

      7. +-commutative 28.8%

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

      8. associate--l+ 41.0%

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

      9. metadata-eval 41.0%

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

      10. +-rgt-identity 41.0%

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

      11. times-frac 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in t around 0 83.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.5%

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

   6. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 83.5%

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

      2. associate-*l* 89.4%

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

   8. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 90.8%

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

      2. *-commutative 90.8%

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

      3. times-frac 95.5%

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

   10. Simplified 95.5%

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

   11. Taylor expanded in k around 0 88.4%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   12. Step-by-step derivation

       1. unpow2 88.4%

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

       2. associate-/r* 95.5%

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

   13. Simplified 95.5%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.6%

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

      1. associate-*l* 41.6%

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

      2. associate-*l* 41.6%

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

      3. associate-/r* 41.5%

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

      4. associate-/r/ 41.5%

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

      5. *-commutative 41.5%

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

      6. times-frac 43.1%

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

      7. +-commutative 43.1%

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

      8. associate--l+ 47.8%

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

      9. metadata-eval 47.8%

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

      10. +-rgt-identity 47.8%

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

      11. times-frac 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in t around 0 78.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 78.8%

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

   6. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 78.8%

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

      2. associate-*l* 83.0%

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

   8. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 88.2%

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

      2. *-commutative 88.2%

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

      3. times-frac 94.3%

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

   10. Simplified 94.3%

       \[\leadsto \color{blue}{\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)} \]
   11. Step-by-step derivation

       1. frac-times 88.2%

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

   12. Applied egg-rr 88.2%

       \[\leadsto \frac{2}{k} \cdot \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}}{k \cdot t}} \]
   13. Step-by-step derivation

       1. times-frac 94.3%

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

       2. associate-/l/ 94.3%

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

       3. associate-/r* 94.3%

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

       4. *-commutative 94.3%

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

   14. Simplified 94.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot \tan k} \cdot \frac{\ell}{\sin k \cdot t}\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{\tan k}}{k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.8e-21)
   (* (/ 2.0 k) (* (/ (/ l (sin k)) t) (/ (/ l k) k)))
   (* (/ l (* (sin k) t)) (* (/ 2.0 k) (/ (/ l (tan k)) k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-21) {
		tmp = (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
	} else {
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * ((l / tan(k)) / k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this 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) :: tmp
    if (k <= 3.8d-21) then
        tmp = (2.0d0 / k) * (((l / sin(k)) / t) * ((l / k) / k))
    else
        tmp = (l / (sin(k) * t)) * ((2.0d0 / k) * ((l / tan(k)) / k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-21) {
		tmp = (2.0 / k) * (((l / Math.sin(k)) / t) * ((l / k) / k));
	} else {
		tmp = (l / (Math.sin(k) * t)) * ((2.0 / k) * ((l / Math.tan(k)) / k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.8e-21:
		tmp = (2.0 / k) * (((l / math.sin(k)) / t) * ((l / k) / k))
	else:
		tmp = (l / (math.sin(k) * t)) * ((2.0 / k) * ((l / math.tan(k)) / k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.8e-21)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / sin(k)) / t) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(Float64(l / Float64(sin(k) * t)) * Float64(Float64(2.0 / k) * Float64(Float64(l / tan(k)) / k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.8e-21)
		tmp = (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
	else
		tmp = (l / (sin(k) * t)) * ((2.0 / k) * ((l / tan(k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.8e-21], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7999999999999998e-21

   1. Initial program 42.2%

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

      1. associate-*l* 42.2%

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

      2. associate-*l* 42.2%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 43.7%

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

      7. +-commutative 43.7%

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

      8. associate--l+ 47.1%

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

      9. metadata-eval 47.1%

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

      10. +-rgt-identity 47.1%

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

      11. times-frac 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in t around 0 83.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 83.1%

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

   6. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 83.1%

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

      2. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

      3. times-frac 96.1%

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

   10. Simplified 96.1%

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

   11. Taylor expanded in k around 0 78.2%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   12. Step-by-step derivation

       1. unpow2 78.2%

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

       2. associate-/r* 82.4%

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

   13. Simplified 82.4%

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

   if 3.7999999999999998e-21 < k

   1. Initial program 29.2%

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

      1. associate-*l* 29.2%

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

      2. associate-*l* 29.2%

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

      3. associate-/r* 29.2%

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

      4. associate-/r/ 29.2%

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

      5. *-commutative 29.2%

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

      6. times-frac 29.3%

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

      7. +-commutative 29.3%

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

      8. associate--l+ 43.6%

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

      9. metadata-eval 43.6%

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

      10. +-rgt-identity 43.6%

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

      11. times-frac 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in t around 0 72.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 72.7%

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

   6. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 72.7%

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

      2. associate-*l* 77.9%

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

   8. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 83.4%

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

      2. *-commutative 83.4%

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

      3. frac-times 91.1%

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

      4. associate-*r* 84.0%

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

      5. associate-/l/ 84.0%

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

   10. Applied egg-rr 84.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot t} \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{\tan k}}{k}\right)\\ \end{array} \]

Alternative 5: 94.8% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 k) (* (/ (/ l k) (tan k)) (/ (/ l (sin k)) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 / k) * (((l / k) / tan(k)) * ((l / sin(k)) / t));
}

Fortran

NOTE: k should be positive before calling this function
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 / k) * (((l / k) / tan(k)) * ((l / sin(k)) / t))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (2.0 / k) * (((l / k) / Math.tan(k)) * ((l / Math.sin(k)) / t));
}

Python

k = abs(k)
def code(t, l, k):
	return (2.0 / k) * (((l / k) / math.tan(k)) * ((l / math.sin(k)) / t))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / k) / tan(k)) * Float64(Float64(l / sin(k)) / t)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (2.0 / k) * (((l / k) / tan(k)) * ((l / sin(k)) / t));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate-*l* 38.3%

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

   3. associate-/r* 38.2%

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

   4. associate-/r/ 38.2%

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

   5. *-commutative 38.2%

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

   6. times-frac 39.4%

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

   7. +-commutative 39.4%

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

   8. associate--l+ 46.0%

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

   9. metadata-eval 46.0%

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

   10. +-rgt-identity 46.0%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 80.0%

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

6. Simplified 80.0%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.7%

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

8. Applied egg-rr 84.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
9. Step-by-step derivation

   1. times-frac 88.8%

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

   2. *-commutative 88.8%

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

   3. times-frac 94.6%

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

10. Simplified 94.6%

    \[\leadsto \color{blue}{\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\tan k}}{k} \cdot \frac{\frac{\ell}{\sin k}}{t}\right)} \]
11. Step-by-step derivation

    1. div-inv 94.5%

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

12. Applied egg-rr 94.5%

    \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{1}{k}\right)} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
13. Step-by-step derivation

    1. associate-*l/ 94.5%

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

    2. associate-*r/ 94.6%

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

    3. *-rgt-identity 94.6%

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

14. Simplified 94.6%

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

15. Final simplification 94.6%

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

Alternative 6: 74.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\frac{k}{\frac{\ell}{k}}}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e-155)
   (/ (/ (* 2.0 l) (/ k (/ l k))) (* k (* k t)))
   (* (/ 2.0 k) (* (/ (/ l (sin k)) t) (/ l (* k k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = ((2.0 * l) / (k / (l / k))) / (k * (k * t));
	} else {
		tmp = (2.0 / k) * (((l / sin(k)) / t) * (l / (k * k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this 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) :: tmp
    if (k <= 2.2d-155) then
        tmp = ((2.0d0 * l) / (k / (l / k))) / (k * (k * t))
    else
        tmp = (2.0d0 / k) * (((l / sin(k)) / t) * (l / (k * k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-155) {
		tmp = ((2.0 * l) / (k / (l / k))) / (k * (k * t));
	} else {
		tmp = (2.0 / k) * (((l / Math.sin(k)) / t) * (l / (k * k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.2e-155:
		tmp = ((2.0 * l) / (k / (l / k))) / (k * (k * t))
	else:
		tmp = (2.0 / k) * (((l / math.sin(k)) / t) * (l / (k * k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e-155)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k / Float64(l / k))) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / sin(k)) / t) * Float64(l / Float64(k * k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e-155)
		tmp = ((2.0 * l) / (k / (l / k))) / (k * (k * t));
	else
		tmp = (2.0 / k) * (((l / sin(k)) / t) * (l / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.2e-155], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999999e-155

   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. Step-by-step derivation

      1. associate-*l* 38.8%

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

      2. associate-*l* 38.8%

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

      3. associate-/r* 38.6%

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

      4. associate-/r/ 38.6%

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

      5. *-commutative 38.6%

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

      6. times-frac 40.7%

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

      7. +-commutative 40.7%

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

      8. associate--l+ 44.7%

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

      9. metadata-eval 44.7%

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

      10. +-rgt-identity 44.7%

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

      11. times-frac 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in t around 0 82.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 82.3%

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

   6. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 82.3%

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

      2. associate-*l* 87.6%

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

   8. Applied egg-rr 87.6%

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

   9. Taylor expanded in k around 0 62.6%

      \[\leadsto \frac{2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot \left(k \cdot t\right)} \]
   10. Step-by-step derivation

       1. unpow2 62.6%

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

       2. unpow2 62.6%

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

   11. Simplified 62.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}}}{k \cdot \left(k \cdot t\right)} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 62.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{\ell \cdot \ell}{k \cdot k}\right)\right)}}{k \cdot \left(k \cdot t\right)} \]

       2. expm1-udef 60.4%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{\ell \cdot \ell}{k \cdot k}\right)} - 1}}{k \cdot \left(k \cdot t\right)} \]

       3. associate-/l* 68.2%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\frac{\ell}{\frac{k \cdot k}{\ell}}}\right)} - 1}{k \cdot \left(k \cdot t\right)} \]

   13. Applied egg-rr 68.2%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \frac{\ell}{\frac{k \cdot k}{\ell}}\right)} - 1}}{k \cdot \left(k \cdot t\right)} \]
   14. Step-by-step derivation

       1. expm1-def 72.8%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{\ell}{\frac{k \cdot k}{\ell}}\right)\right)}}{k \cdot \left(k \cdot t\right)} \]

       2. expm1-log1p 73.0%

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

       3. associate-*r/ 73.0%

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

       4. associate-/l* 75.6%

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

   15. Simplified 75.6%

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

   if 2.1999999999999999e-155 < k

   1. Initial program 37.7%

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

      1. associate-*l* 37.7%

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

      2. associate-*l* 37.7%

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

      3. associate-/r* 37.7%

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

      4. associate-/r/ 37.7%

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

      5. *-commutative 37.7%

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

      6. times-frac 37.7%

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

      7. +-commutative 37.7%

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

      8. associate--l+ 47.9%

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

      9. metadata-eval 47.9%

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

      10. +-rgt-identity 47.9%

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

      11. times-frac 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in t around 0 76.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 76.9%

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

   6. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   7. Step-by-step derivation

      1. associate-*l/ 76.9%

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

      2. associate-*l* 80.6%

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

   8. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
   9. Step-by-step derivation

      1. times-frac 85.4%

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

      2. *-commutative 85.4%

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

      3. times-frac 92.8%

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

   10. Simplified 92.8%

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

   11. Taylor expanded in k around 0 73.0%

       \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
   12. Step-by-step derivation

       1. unpow2 73.0%

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

   13. Simplified 73.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\frac{k}{\frac{\ell}{k}}}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \]

Alternative 7: 74.5% accurate, 3.7× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 k) (* (/ (/ l (sin k)) t) (/ (/ l k) k))))

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
}

Fortran

NOTE: k should be positive before calling this function
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 / k) * (((l / sin(k)) / t) * ((l / k) / k))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (2.0 / k) * (((l / Math.sin(k)) / t) * ((l / k) / k));
}

Python

k = abs(k)
def code(t, l, k):
	return (2.0 / k) * (((l / math.sin(k)) / t) * ((l / k) / k))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / sin(k)) / t) * Float64(Float64(l / k) / k)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (2.0 / k) * (((l / sin(k)) / t) * ((l / k) / k));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate-*l* 38.3%

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

   3. associate-/r* 38.2%

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

   4. associate-/r/ 38.2%

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

   5. *-commutative 38.2%

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

   6. times-frac 39.4%

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

   7. +-commutative 39.4%

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

   8. associate--l+ 46.0%

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

   9. metadata-eval 46.0%

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

   10. +-rgt-identity 46.0%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 80.0%

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

6. Simplified 80.0%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.7%

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

8. Applied egg-rr 84.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
9. Step-by-step derivation

   1. times-frac 88.8%

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

   2. *-commutative 88.8%

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

   3. times-frac 94.6%

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

10. Simplified 94.6%

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

11. Taylor expanded in k around 0 73.8%

    \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
12. Step-by-step derivation

    1. unpow2 73.8%

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

    2. associate-/r* 76.8%

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

13. Simplified 76.8%

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

14. Final simplification 76.8%

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

Alternative 8: 71.5% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* l (/ (/ l k) k)) (* k (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l * ((l / k) / k)) / (k * (k * t)));
}

Fortran

NOTE: k should be positive before calling this function
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 * ((l * ((l / k) / k)) / (k * (k * t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l * ((l / k) / k)) / (k * (k * t)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l * ((l / k) / k)) / (k * (k * t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(Float64(l / k) / k)) / Float64(k * Float64(k * t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l * ((l / k) / k)) / (k * (k * t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate-*l* 38.3%

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

   3. associate-/r* 38.2%

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

   4. associate-/r/ 38.2%

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

   5. *-commutative 38.2%

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

   6. times-frac 39.4%

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

   7. +-commutative 39.4%

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

   8. associate--l+ 46.0%

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

   9. metadata-eval 46.0%

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

   10. +-rgt-identity 46.0%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 80.0%

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

6. Simplified 80.0%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.7%

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

8. Applied egg-rr 84.7%

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

9. Taylor expanded in k around 0 63.8%

   \[\leadsto \frac{2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot \left(k \cdot t\right)} \]
10. Step-by-step derivation

    1. unpow2 63.8%

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

    2. unpow2 63.8%

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

11. Simplified 63.8%

    \[\leadsto \frac{2 \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot k}}}{k \cdot \left(k \cdot t\right)} \]
12. Step-by-step derivation

    1. div-inv 63.8%

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

    2. associate-/l* 70.9%

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

13. Applied egg-rr 70.9%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{\frac{k \cdot k}{\ell}}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}} \]
14. Step-by-step derivation

    1. associate-*l* 70.9%

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

    2. associate-*r/ 71.0%

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

    3. *-rgt-identity 71.0%

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

    4. unpow2 71.0%

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

    5. associate-/r/ 71.0%

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

    6. unpow2 71.0%

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

    7. associate-/r* 72.5%

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

15. Simplified 72.5%

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

16. Final simplification 72.5%

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

Alternative 9: 73.7% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 k) (* (/ (/ l k) k) (/ l (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 / k) * (((l / k) / k) * (l / (k * t)));
}

Fortran

NOTE: k should be positive before calling this function
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 / k) * (((l / k) / k) * (l / (k * t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (2.0 / k) * (((l / k) / k) * (l / (k * t)));
}

Python

k = abs(k)
def code(t, l, k):
	return (2.0 / k) * (((l / k) / k) * (l / (k * t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / k) / k) * Float64(l / Float64(k * t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (2.0 / k) * (((l / k) / k) * (l / (k * t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate-*l* 38.3%

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

   3. associate-/r* 38.2%

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

   4. associate-/r/ 38.2%

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

   5. *-commutative 38.2%

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

   6. times-frac 39.4%

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

   7. +-commutative 39.4%

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

   8. associate--l+ 46.0%

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

   9. metadata-eval 46.0%

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

   10. +-rgt-identity 46.0%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 80.0%

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

6. Simplified 80.0%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.7%

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

8. Applied egg-rr 84.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
9. Step-by-step derivation

   1. times-frac 88.8%

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

   2. *-commutative 88.8%

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

   3. times-frac 94.6%

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

10. Simplified 94.6%

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

11. Taylor expanded in k around 0 73.8%

    \[\leadsto \frac{2}{k} \cdot \left(\color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\frac{\ell}{\sin k}}{t}\right) \]
12. Step-by-step derivation

    1. unpow2 73.8%

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

    2. associate-/r* 76.8%

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

13. Simplified 76.8%

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

14. Taylor expanded in k around 0 74.0%

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

15. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2023229 
(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))))