Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 95.2%

Time: 21.4s

Alternatives: 13

Speedup: 32.4×

• 34.9% 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: 34.6% 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: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot k} \cdot \ell}{\frac{t}{\frac{\ell}{t_1}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\frac{t_1}{\frac{\cos k}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1e+139)
     (* 2.0 (/ (* (/ (cos k) (* k k)) l) (/ t (/ l t_1))))
     (* 2.0 (/ (* (/ l k) (/ l k)) (/ t_1 (/ (cos k) t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1e+139) {
		tmp = 2.0 * (((cos(k) / (k * k)) * l) / (t / (l / t_1)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (cos(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) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 1d+139) then
        tmp = 2.0d0 * (((cos(k) / (k * k)) * l) / (t / (l / t_1)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) / (t_1 / (cos(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 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1e+139) {
		tmp = 2.0 * (((Math.cos(k) / (k * k)) * l) / (t / (l / t_1)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (Math.cos(k) / t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1e+139:
		tmp = 2.0 * (((math.cos(k) / (k * k)) * l) / (t / (l / t_1)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (math.cos(k) / t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1e+139)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / Float64(k * k)) * l) / Float64(t / Float64(l / t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(t_1 / Float64(cos(k) / t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1e+139)
		tmp = 2.0 * (((cos(k) / (k * k)) * l) / (t / (l / t_1)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (cos(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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1e+139], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(t / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000003e139

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. associate-*l* 37.3%

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

      3. associate-/r* 37.3%

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

      4. associate-/r/ 37.8%

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

      5. *-commutative 37.8%

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

      6. times-frac 39.1%

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

      7. +-commutative 39.1%

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

      8. associate--l+ 47.4%

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

      9. metadata-eval 47.4%

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

      10. +-rgt-identity 47.4%

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

      11. times-frac 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. unpow2 73.6%

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

      2. times-frac 74.5%

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

      3. unpow2 74.5%

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

      4. associate-/l* 87.2%

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

      5. *-commutative 87.2%

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

   6. Simplified 87.2%

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

      1. associate-*r/ 90.4%

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

      2. associate-/l* 92.4%

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

   8. Applied egg-rr 92.4%

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

   if 1.00000000000000003e139 < k

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-*l* 35.7%

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

      3. associate-/r* 35.7%

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

      4. associate-/r/ 32.5%

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

      5. *-commutative 32.5%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

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

      10. +-rgt-identity 42.1%

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

      11. times-frac 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around 0 62.1%

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

      1. unpow2 62.1%

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

      2. times-frac 62.1%

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

      3. unpow2 62.1%

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

      4. associate-/l* 65.5%

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

      5. *-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in k around inf 62.1%

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

      1. *-commutative 62.1%

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

      2. *-commutative 62.1%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

      5. unpow2 62.2%

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

      6. times-frac 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in l around 0 62.1%

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

       1. associate-/r* 62.2%

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

       2. associate-*r/ 62.2%

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

       3. unpow2 62.2%

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

       4. unpow2 62.2%

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

       5. times-frac 96.7%

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

       6. unpow2 96.7%

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

       7. *-commutative 96.7%

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

       8. associate-/l* 96.7%

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

       9. associate-/l* 96.8%

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

   12. Simplified 96.8%

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

       1. unpow2 96.8%

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

   14. Applied egg-rr 96.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k \cdot k} \cdot \ell}{\frac{t}{\frac{\ell}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\frac{{\sin k}^{2}}{\frac{\cos k}{t}}}\\ \end{array} \]

Alternative 2: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow (sin k) 2.0))))
   (if (<= k 1.8e-73)
     (*
      2.0
      (/
       (*
        (/ 1.0 k)
        (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
       k))
     (if (<= k 3.8e+101)
       (* 2.0 (* (/ (cos k) (* k k)) (* l (/ l t_1))))
       (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) t_1)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t * pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.8e-73) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else if (k <= 3.8e+101) {
		tmp = 2.0 * ((cos(k) / (k * k)) * (l * (l / t_1)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / t_1));
	}
	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) :: tmp
    t_1 = t * (sin(k) ** 2.0d0)
    if (k <= 1.8d-73) then
        tmp = 2.0d0 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
    else if (k <= 3.8d+101) then
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (l * (l / t_1)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / t_1))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.8e-73) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else if (k <= 3.8e+101) {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (l * (l / t_1)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / t_1));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t * math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1.8e-73:
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)
	elif k <= 3.8e+101:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (l * (l / t_1)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t * (sin(k) ^ 2.0))
	tmp = 0.0
	if (k <= 1.8e-73)
		tmp = Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k));
	elseif (k <= 3.8e+101)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l * Float64(l / t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t_1)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t * (sin(k) ^ 2.0);
	tmp = 0.0;
	if (k <= 1.8e-73)
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	elseif (k <= 3.8e+101)
		tmp = 2.0 * ((cos(k) / (k * k)) * (l * (l / t_1)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.8e-73], N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+101], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e-73

   1. Initial program 39.3%

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

      1. associate-*l* 39.2%

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

      2. associate-*l* 39.2%

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

      3. associate-/r* 39.2%

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

      4. associate-/r/ 39.8%

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

      5. *-commutative 39.8%

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

      6. times-frac 41.4%

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

      7. +-commutative 41.4%

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

      8. associate--l+ 49.7%

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

      9. metadata-eval 49.7%

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

      10. +-rgt-identity 49.7%

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

      11. times-frac 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in t around 0 73.1%

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

      1. unpow2 73.1%

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

      2. times-frac 73.6%

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

      3. unpow2 73.6%

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

      4. associate-/l* 85.7%

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

      5. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in k around 0 68.1%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 68.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 68.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 68.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 68.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 68.1%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 68.1%

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

   9. Simplified 68.1%

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

   10. Taylor expanded in k around 0 66.1%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 66.1%

          \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. associate-/r* 66.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 66.1%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 66.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

       2. times-frac 78.2%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

       3. associate-/l* 78.2%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

   14. Applied egg-rr 78.2%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

   if 1.8e-73 < k < 3.7999999999999998e101

   1. Initial program 29.1%

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

      1. associate-*l* 29.1%

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

      2. associate-*l* 31.9%

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

      3. associate-/r* 32.0%

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

      4. associate-/r/ 31.9%

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

      5. *-commutative 31.9%

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

      6. times-frac 31.8%

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

      7. +-commutative 31.8%

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

      8. associate--l+ 42.5%

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

      9. metadata-eval 42.5%

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

      10. +-rgt-identity 42.5%

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

      11. times-frac 42.4%

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

   3. Simplified 42.4%

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

   4. Taylor expanded in t around 0 80.0%

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

      1. unpow2 80.0%

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

      2. times-frac 80.5%

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

      3. unpow2 80.5%

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

      4. associate-/l* 94.9%

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

      5. *-commutative 94.9%

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

   6. Simplified 94.9%

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

      1. associate-/r/ 95.1%

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

   8. Applied egg-rr 95.1%

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

   if 3.7999999999999998e101 < k

   1. Initial program 31.9%

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

      1. associate-*l* 31.9%

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

      2. associate-*l* 31.9%

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

      3. associate-/r* 31.9%

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

      4. associate-/r/ 29.4%

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

      5. *-commutative 29.4%

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

      6. times-frac 29.5%

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

      7. +-commutative 29.5%

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

      8. associate--l+ 36.9%

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

      9. metadata-eval 36.9%

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

      10. +-rgt-identity 36.9%

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

      11. times-frac 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around 0 61.7%

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

      1. unpow2 61.7%

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

      2. times-frac 64.3%

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

      3. unpow2 64.3%

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

      4. associate-/l* 71.5%

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

      5. *-commutative 71.5%

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

   6. Simplified 71.5%

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

   7. Taylor expanded in k around inf 61.7%

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

      1. *-commutative 61.7%

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

      2. *-commutative 61.7%

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

      3. times-frac 64.3%

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

      4. unpow2 64.3%

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

      5. unpow2 64.3%

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

      6. times-frac 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 7.1 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 7.1e-16)
   (*
    2.0
    (/
     (*
      (/ 1.0 k)
      (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
     k))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.1e-16) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	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 <= 7.1d-16) then
        tmp = 2.0d0 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    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 <= 7.1e-16) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 7.1e-16:
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 7.1e-16)
		tmp = Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.1e-16)
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 7.1e-16], N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.1e-16

   1. Initial program 38.2%

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

      1. associate-*l* 38.2%

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

      2. associate-*l* 38.7%

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

      3. associate-/r* 38.7%

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

      4. associate-/r/ 39.2%

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

      5. *-commutative 39.2%

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

      6. times-frac 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+ 48.5%

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

      9. metadata-eval 48.5%

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

      10. +-rgt-identity 48.5%

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

      11. times-frac 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. unpow2 73.6%

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

      2. times-frac 74.1%

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

      3. unpow2 74.1%

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

      4. associate-/l* 85.8%

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

      5. *-commutative 85.8%

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

   6. Simplified 85.8%

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

   7. Taylor expanded in k around 0 69.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 69.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 69.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 69.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 69.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 69.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 69.0%

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

   9. Simplified 69.0%

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

   10. Taylor expanded in k around 0 67.1%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 67.1%

          \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. associate-/r* 67.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 67.1%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 67.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

       2. times-frac 78.8%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

       3. associate-/l* 78.8%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

   14. Applied egg-rr 78.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

   if 7.1e-16 < k

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-/r* 32.1%

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

      4. associate-/r/ 30.4%

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

      5. *-commutative 30.4%

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

      6. times-frac 30.5%

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

      7. +-commutative 30.5%

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

      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 41.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in t around 0 67.6%

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

      1. unpow2 67.6%

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

      2. times-frac 69.5%

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

      3. unpow2 69.5%

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

      4. associate-/l* 80.6%

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

      5. *-commutative 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in k around inf 67.6%

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

      1. *-commutative 67.6%

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

      2. *-commutative 67.6%

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

      3. times-frac 69.6%

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

      4. unpow2 69.6%

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

      5. unpow2 69.6%

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

      6. times-frac 93.3%

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

   9. Simplified 93.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.1 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \ell}{\left(k \cdot k\right) \cdot \frac{t}{\frac{\ell}{t_1}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 8.4e+102)
     (* 2.0 (/ (* (cos k) l) (* (* k k) (/ t (/ l t_1)))))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 8.4e+102) {
		tmp = 2.0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	}
	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) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 8.4d+102) then
        tmp = 2.0d0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 8.4e+102) {
		tmp = 2.0 * ((Math.cos(k) * l) / ((k * k) * (t / (l / t_1))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 8.4e+102:
		tmp = 2.0 * ((math.cos(k) * l) / ((k * k) * (t / (l / t_1))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 8.4e+102)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * l) / Float64(Float64(k * k) * Float64(t / Float64(l / t_1)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 8.4e+102)
		tmp = 2.0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 8.4e+102], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.40000000000000006e102

   1. Initial program 37.5%

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

      1. associate-*l* 37.5%

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

      2. associate-*l* 37.9%

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

      3. associate-/r* 37.9%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. times-frac 39.8%

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

      7. +-commutative 39.8%

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

      8. associate--l+ 48.4%

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

      9. metadata-eval 48.4%

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

      10. +-rgt-identity 48.4%

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

      11. times-frac 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in t around 0 73.8%

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

      1. unpow2 73.8%

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

      2. times-frac 74.3%

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

      3. unpow2 74.3%

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

      4. associate-/l* 87.1%

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

      5. *-commutative 87.1%

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

   6. Simplified 87.1%

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

      1. frac-times 89.6%

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

      2. associate-/l* 90.5%

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

   8. Applied egg-rr 90.5%

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

   if 8.40000000000000006e102 < k

   1. Initial program 32.7%

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

      1. associate-*l* 32.7%

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

      2. associate-*l* 32.7%

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

      3. associate-/r* 32.7%

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

      4. associate-/r/ 30.2%

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

      5. *-commutative 30.2%

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

      6. times-frac 30.2%

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

      7. +-commutative 30.2%

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

      8. associate--l+ 37.8%

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

      9. metadata-eval 37.8%

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

      10. +-rgt-identity 37.8%

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

      11. times-frac 37.8%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in t around 0 63.2%

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

      1. unpow2 63.2%

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

      2. times-frac 65.8%

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

      3. unpow2 65.8%

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

      4. associate-/l* 70.8%

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

      5. *-commutative 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in k around inf 63.2%

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

      1. *-commutative 63.2%

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

      2. *-commutative 63.2%

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

      3. times-frac 65.8%

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

      4. unpow2 65.8%

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

      5. unpow2 65.8%

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

      6. times-frac 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.4 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \ell}{\left(k \cdot k\right) \cdot \frac{t}{\frac{\ell}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.04 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \ell}{\left(k \cdot k\right) \cdot \frac{t}{\frac{\ell}{t_1}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\frac{t_1}{\frac{\cos k}{t}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.04e+139)
     (* 2.0 (/ (* (cos k) l) (* (* k k) (/ t (/ l t_1)))))
     (* 2.0 (/ (* (/ l k) (/ l k)) (/ t_1 (/ (cos k) t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.04e+139) {
		tmp = 2.0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (cos(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) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 1.04d+139) then
        tmp = 2.0d0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) / (t_1 / (cos(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 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.04e+139) {
		tmp = 2.0 * ((Math.cos(k) * l) / ((k * k) * (t / (l / t_1))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (Math.cos(k) / t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1.04e+139:
		tmp = 2.0 * ((math.cos(k) * l) / ((k * k) * (t / (l / t_1))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (math.cos(k) / t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.04e+139)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * l) / Float64(Float64(k * k) * Float64(t / Float64(l / t_1)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(t_1 / Float64(cos(k) / t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.04e+139)
		tmp = 2.0 * ((cos(k) * l) / ((k * k) * (t / (l / t_1))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) / (t_1 / (cos(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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.04e+139], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.04e139

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. associate-*l* 37.3%

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

      3. associate-/r* 37.3%

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

      4. associate-/r/ 37.8%

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

      5. *-commutative 37.8%

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

      6. times-frac 39.1%

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

      7. +-commutative 39.1%

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

      8. associate--l+ 47.4%

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

      9. metadata-eval 47.4%

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

      10. +-rgt-identity 47.4%

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

      11. times-frac 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. unpow2 73.6%

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

      2. times-frac 74.5%

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

      3. unpow2 74.5%

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

      4. associate-/l* 87.2%

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

      5. *-commutative 87.2%

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

   6. Simplified 87.2%

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

      1. frac-times 90.0%

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

      2. associate-/l* 90.9%

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

   8. Applied egg-rr 90.9%

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

   if 1.04e139 < k

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-*l* 35.7%

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

      3. associate-/r* 35.7%

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

      4. associate-/r/ 32.5%

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

      5. *-commutative 32.5%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

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

      10. +-rgt-identity 42.1%

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

      11. times-frac 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around 0 62.1%

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

      1. unpow2 62.1%

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

      2. times-frac 62.1%

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

      3. unpow2 62.1%

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

      4. associate-/l* 65.5%

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

      5. *-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in k around inf 62.1%

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

      1. *-commutative 62.1%

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

      2. *-commutative 62.1%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

      5. unpow2 62.2%

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

      6. times-frac 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in l around 0 62.1%

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

       1. associate-/r* 62.2%

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

       2. associate-*r/ 62.2%

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

       3. unpow2 62.2%

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

       4. unpow2 62.2%

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

       5. times-frac 96.7%

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

       6. unpow2 96.7%

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

       7. *-commutative 96.7%

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

       8. associate-/l* 96.7%

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

       9. associate-/l* 96.8%

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

   12. Simplified 96.8%

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

       1. unpow2 96.8%

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

   14. Applied egg-rr 96.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.04 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \ell}{\left(k \cdot k\right) \cdot \frac{t}{\frac{\ell}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\frac{{\sin k}^{2}}{\frac{\cos k}{t}}}\\ \end{array} \]

Alternative 6: 83.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.8%

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

   2. associate-*l* 37.1%

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

   3. associate-/r* 37.1%

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

   4. associate-/r/ 37.1%

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

   5. *-commutative 37.1%

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

   6. times-frac 38.3%

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

   7. +-commutative 38.3%

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

   8. associate--l+ 46.8%

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

   9. metadata-eval 46.8%

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

   10. +-rgt-identity 46.8%

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

   11. times-frac 51.6%

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

3. Simplified 51.6%

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

4. Taylor expanded in t around 0 79.5%

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

   1. unpow2 79.5%

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

   2. associate-*l* 82.8%

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

6. Simplified 82.8%

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

7. Final simplification 82.8%

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

Alternative 7: 76.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0019:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0019)
   (*
    2.0
    (/
     (*
      (/ 1.0 k)
      (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
     k))
   (* 2.0 (* 0.3333333333333333 (/ (* (cos k) (pow (/ l k) 2.0)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0019) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) * pow((l / k), 2.0)) / 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 (k <= 0.0019d0) then
        tmp = 2.0d0 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * ((cos(k) * ((l / k) ** 2.0d0)) / 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 (k <= 0.0019) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((Math.cos(k) * Math.pow((l / k), 2.0)) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 0.0019:
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)
	else:
		tmp = 2.0 * (0.3333333333333333 * ((math.cos(k) * math.pow((l / k), 2.0)) / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0019)
		tmp = Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(cos(k) * (Float64(l / k) ^ 2.0)) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.0019)
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	else
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) * ((l / k) ^ 2.0)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.0019], N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0019

   1. Initial program 38.3%

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

      1. associate-*l* 38.3%

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

      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* 38.8%

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

      4. associate-/r/ 39.3%

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

      5. *-commutative 39.3%

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

      6. times-frac 40.8%

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

      7. +-commutative 40.8%

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

      8. associate--l+ 48.5%

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

      9. metadata-eval 48.5%

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

      10. +-rgt-identity 48.5%

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

      11. times-frac 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in t around 0 73.9%

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

      1. unpow2 73.9%

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

      2. times-frac 74.3%

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

      3. unpow2 74.3%

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

      4. associate-/l* 86.0%

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

      5. *-commutative 86.0%

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

   6. Simplified 86.0%

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

   7. Taylor expanded in k around 0 69.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 69.3%

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

   9. Simplified 69.3%

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

   10. Taylor expanded in k around 0 67.4%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 67.4%

          \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. associate-/r* 67.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 67.4%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 67.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

       2. times-frac 79.0%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

       3. associate-/l* 79.0%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

   14. Applied egg-rr 79.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

   if 0.0019 < k

   1. Initial program 31.5%

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

      1. associate-*l* 31.5%

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

      2. associate-*l* 31.5%

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

      3. associate-/r* 31.5%

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

      4. associate-/r/ 29.7%

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

      5. *-commutative 29.7%

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

      6. times-frac 29.8%

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

      7. +-commutative 29.8%

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

      8. associate--l+ 40.8%

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

      9. metadata-eval 40.8%

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

      10. +-rgt-identity 40.8%

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

      11. times-frac 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in t around 0 66.5%

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

      1. unpow2 66.5%

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

      2. times-frac 68.5%

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

      3. unpow2 68.5%

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

      4. associate-/l* 80.0%

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

      5. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in k around 0 54.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in k around inf 53.8%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 53.8%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot t}\right) \]

       2. times-frac 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right) \]

       3. unpow2 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

       4. unpow2 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right) \]

       5. times-frac 59.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]

       6. unpow2 59.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

   12. Simplified 59.6%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 59.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}\right) \]

   14. Applied egg-rr 59.6%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0019:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]

Alternative 8: 76.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0019:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0019)
   (*
    2.0
    (/
     (*
      (/ 1.0 k)
      (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
     k))
   (* 2.0 (* 0.3333333333333333 (* (* (/ l k) (/ l k)) (/ (cos k) t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0019) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(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 (k <= 0.0019d0) then
        tmp = 2.0d0 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * (((l / k) * (l / k)) * (cos(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 (k <= 0.0019) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (Math.cos(k) / t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 0.0019:
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)
	else:
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (math.cos(k) / t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0019)
		tmp = Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.0019)
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	else
		tmp = 2.0 * (0.3333333333333333 * (((l / k) * (l / k)) * (cos(k) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.0019], N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0019

   1. Initial program 38.3%

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

      1. associate-*l* 38.3%

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

      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* 38.8%

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

      4. associate-/r/ 39.3%

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

      5. *-commutative 39.3%

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

      6. times-frac 40.8%

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

      7. +-commutative 40.8%

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

      8. associate--l+ 48.5%

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

      9. metadata-eval 48.5%

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

      10. +-rgt-identity 48.5%

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

      11. times-frac 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in t around 0 73.9%

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

      1. unpow2 73.9%

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

      2. times-frac 74.3%

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

      3. unpow2 74.3%

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

      4. associate-/l* 86.0%

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

      5. *-commutative 86.0%

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

   6. Simplified 86.0%

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

   7. Taylor expanded in k around 0 69.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 69.3%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 69.3%

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

   9. Simplified 69.3%

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

   10. Taylor expanded in k around 0 67.4%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 67.4%

          \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. associate-/r* 67.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 67.4%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 67.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

       2. times-frac 79.0%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

       3. associate-/l* 79.0%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

   14. Applied egg-rr 79.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

   if 0.0019 < k

   1. Initial program 31.5%

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

      1. associate-*l* 31.5%

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

      2. associate-*l* 31.5%

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

      3. associate-/r* 31.5%

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

      4. associate-/r/ 29.7%

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

      5. *-commutative 29.7%

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

      6. times-frac 29.8%

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

      7. +-commutative 29.8%

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

      8. associate--l+ 40.8%

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

      9. metadata-eval 40.8%

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

      10. +-rgt-identity 40.8%

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

      11. times-frac 40.7%

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

   3. Simplified 40.7%

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

   4. Taylor expanded in t around 0 66.5%

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

      1. unpow2 66.5%

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

      2. times-frac 68.5%

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

      3. unpow2 68.5%

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

      4. associate-/l* 80.0%

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

      5. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in k around 0 54.0%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 54.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in k around inf 53.8%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 53.8%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot t}\right) \]

       2. times-frac 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right) \]

       3. unpow2 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

       4. unpow2 54.5%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right) \]

       5. times-frac 59.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]

       6. unpow2 59.6%

          \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

   12. Simplified 59.6%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 93.1%

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

   14. Applied egg-rr 59.6%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0019:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 14.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{1}{k}}{k} + -0.5\right) \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.75e+174)
   (*
    2.0
    (/
     (*
      (/ 1.0 k)
      (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
     k))
   (*
    2.0
    (*
     (+ (/ (/ 1.0 k) k) -0.5)
     (+ (/ (* l l) (* k (* k t))) (/ (* (* l l) 0.3333333333333333) t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.75e+174) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * ((((1.0 / k) / k) + -0.5) * (((l * l) / (k * (k * t))) + (((l * l) * 0.3333333333333333) / 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 <= 1.75d+174) then
        tmp = 2.0d0 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
    else
        tmp = 2.0d0 * ((((1.0d0 / k) / k) + (-0.5d0)) * (((l * l) / (k * (k * t))) + (((l * l) * 0.3333333333333333d0) / 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 <= 1.75e+174) {
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	} else {
		tmp = 2.0 * ((((1.0 / k) / k) + -0.5) * (((l * l) / (k * (k * t))) + (((l * l) * 0.3333333333333333) / t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if l <= 1.75e+174:
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)
	else:
		tmp = 2.0 * ((((1.0 / k) / k) + -0.5) * (((l * l) / (k * (k * t))) + (((l * l) * 0.3333333333333333) / t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.75e+174)
		tmp = Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(1.0 / k) / k) + -0.5) * Float64(Float64(Float64(l * l) / Float64(k * Float64(k * t))) + Float64(Float64(Float64(l * l) * 0.3333333333333333) / t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.75e+174)
		tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
	else
		tmp = 2.0 * ((((1.0 / k) / k) + -0.5) * (((l * l) / (k * (k * t))) + (((l * l) * 0.3333333333333333) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.75e+174], N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(1.0 / k), $MachinePrecision] / k), $MachinePrecision] + -0.5), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7500000000000001e174

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. associate-*l* 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* 38.3%

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

      4. associate-/r/ 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 39.6%

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

      7. +-commutative 39.6%

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

      8. associate--l+ 49.0%

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

      9. metadata-eval 49.0%

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

      10. +-rgt-identity 49.0%

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

      11. times-frac 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in t around 0 74.5%

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

      1. unpow2 74.5%

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

      2. times-frac 75.4%

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

      3. unpow2 75.4%

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

      4. associate-/l* 87.1%

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

      5. *-commutative 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in k around 0 67.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 67.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 67.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 67.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 67.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 67.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 67.4%

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

   9. Simplified 67.4%

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

   10. Taylor expanded in k around 0 65.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 65.0%

          \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. associate-/r* 65.0%

          \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 65.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   13. Step-by-step derivation

       1. associate-*l/ 65.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

       2. times-frac 75.1%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

       3. associate-/l* 75.1%

          \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

   14. Applied egg-rr 75.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

   if 1.7500000000000001e174 < l

   1. Initial program 26.9%

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

      1. associate-*l* 26.9%

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

      2. associate-*l* 26.9%

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

      3. associate-/r* 26.9%

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

      4. associate-/r/ 26.9%

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

      5. *-commutative 26.9%

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

      6. times-frac 26.9%

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

      7. +-commutative 26.9%

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

      8. associate--l+ 26.9%

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

      9. metadata-eval 26.9%

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

      10. +-rgt-identity 26.9%

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

      11. times-frac 26.9%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in t around 0 51.4%

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

      1. unpow2 51.4%

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

      2. times-frac 51.7%

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

      3. unpow2 51.7%

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

      4. associate-/l* 62.7%

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

      5. *-commutative 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in k around 0 51.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 51.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 51.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*l* 51.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. associate-*r/ 51.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

      5. *-commutative 51.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

      6. unpow2 51.4%

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

   9. Simplified 51.4%

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

   10. Taylor expanded in k around 0 58.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
   11. Step-by-step derivation

       1. sub-neg 58.2%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       2. unpow2 58.2%

          \[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       3. associate-/r* 58.2%

          \[\leadsto 2 \cdot \left(\left(\color{blue}{\frac{\frac{1}{k}}{k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

       4. metadata-eval 58.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\frac{1}{k}}{k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

   12. Simplified 58.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\frac{1}{k}}{k} + -0.5\right)} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+174}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\frac{1}{k}}{k} + -0.5\right) \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right)\\ \end{array} \]

Alternative 10: 71.9% accurate, 16.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (/
   (*
    (/ 1.0 k)
    (+ (* (/ l k) (/ l (* k t))) (/ (* l l) (/ t 0.3333333333333333))))
   k)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / 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 * (((1.0d0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333d0)))) / k)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k)

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(Float64(l * l) / Float64(t / 0.3333333333333333)))) / k))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (((1.0 / k) * (((l / k) * (l / (k * t))) + ((l * l) / (t / 0.3333333333333333)))) / k);
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[(t / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.8%

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

   2. associate-*l* 37.1%

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

   3. associate-/r* 37.1%

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

   4. associate-/r/ 37.1%

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

   5. *-commutative 37.1%

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

   6. times-frac 38.3%

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

   7. +-commutative 38.3%

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

   8. associate--l+ 46.8%

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

   9. metadata-eval 46.8%

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

   10. +-rgt-identity 46.8%

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

   11. times-frac 51.6%

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

3. Simplified 51.6%

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

4. Taylor expanded in t around 0 72.2%

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

   1. unpow2 72.2%

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

   2. times-frac 73.0%

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

   3. unpow2 73.0%

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

   4. associate-/l* 84.6%

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

   5. *-commutative 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 65.8%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   2. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   3. associate-*l* 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   4. associate-*r/ 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

   5. *-commutative 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

   6. unpow2 65.8%

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

9. Simplified 65.8%

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

10. Taylor expanded in k around 0 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 63.2%

       \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

    2. associate-/r* 63.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

12. Simplified 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
13. Step-by-step derivation

    1. associate-*l/ 63.3%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k}} \]

    2. times-frac 72.3%

       \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)}{k} \]

    3. associate-/l* 72.3%

       \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}}\right)}{k} \]

14. Applied egg-rr 72.3%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k}} \]

15. Final simplification 72.3%

    \[\leadsto 2 \cdot \frac{\frac{1}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + \frac{\ell \cdot \ell}{\frac{t}{0.3333333333333333}}\right)}{k} \]

Alternative 11: 63.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{k \cdot k}{\ell \cdot \ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (/ (+ (/ 0.3333333333333333 t) (/ 1.0 (* k (* k t)))) (/ (* k k) (* l l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (((0.3333333333333333 / t) + (1.0 / (k * (k * t)))) / ((k * k) / (l * l)));
}

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 * (((0.3333333333333333d0 / t) + (1.0d0 / (k * (k * t)))) / ((k * k) / (l * l)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (((0.3333333333333333 / t) + (1.0 / (k * (k * t)))) / ((k * k) / (l * l)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (((0.3333333333333333 / t) + (1.0 / (k * (k * t)))) / ((k * k) / (l * l)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(0.3333333333333333 / t) + Float64(1.0 / Float64(k * Float64(k * t)))) / Float64(Float64(k * k) / Float64(l * l))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (((0.3333333333333333 / t) + (1.0 / (k * (k * t)))) / ((k * k) / (l * l)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.8%

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

   2. associate-*l* 37.1%

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

   3. associate-/r* 37.1%

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

   4. associate-/r/ 37.1%

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

   5. *-commutative 37.1%

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

   6. times-frac 38.3%

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

   7. +-commutative 38.3%

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

   8. associate--l+ 46.8%

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

   9. metadata-eval 46.8%

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

   10. +-rgt-identity 46.8%

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

   11. times-frac 51.6%

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

3. Simplified 51.6%

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

4. Taylor expanded in t around 0 72.2%

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

   1. unpow2 72.2%

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

   2. times-frac 73.0%

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

   3. unpow2 73.0%

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

   4. associate-/l* 84.6%

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

   5. *-commutative 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 65.8%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   2. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   3. associate-*l* 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   4. associate-*r/ 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

   5. *-commutative 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

   6. unpow2 65.8%

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

9. Simplified 65.8%

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

10. Taylor expanded in k around 0 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 63.2%

       \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

    2. associate-/r* 63.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

12. Simplified 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

13. Taylor expanded in l around 0 63.3%

    \[\leadsto 2 \cdot \color{blue}{\frac{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}{{k}^{2}}} \]
14. Step-by-step derivation

    1. associate-/l* 63.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

    2. associate-*r/ 63.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    3. metadata-eval 63.4%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{0.3333333333333333}}{t} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    4. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    5. associate-*r* 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    6. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

    7. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

15. Simplified 63.4%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{0.3333333333333333}{t} + \frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{k \cdot k}{\ell \cdot \ell}}} \]

16. Final simplification 63.4%

    \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{k \cdot k}{\ell \cdot \ell}} \]

Alternative 12: 72.2% accurate, 20.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\frac{k}{\ell} \cdot \frac{k}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (/ (+ (/ 0.3333333333333333 t) (/ (/ 1.0 k) (* k t))) (* (/ k l) (/ k l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (((0.3333333333333333 / t) + ((1.0 / k) / (k * t))) / ((k / l) * (k / l)));
}

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 * (((0.3333333333333333d0 / t) + ((1.0d0 / k) / (k * t))) / ((k / l) * (k / l)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (((0.3333333333333333 / t) + ((1.0 / k) / (k * t))) / ((k / l) * (k / l)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (((0.3333333333333333 / t) + ((1.0 / k) / (k * t))) / ((k / l) * (k / l)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(0.3333333333333333 / t) + Float64(Float64(1.0 / k) / Float64(k * t))) / Float64(Float64(k / l) * Float64(k / l))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (((0.3333333333333333 / t) + ((1.0 / k) / (k * t))) / ((k / l) * (k / l)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(N[(1.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.8%

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

   2. associate-*l* 37.1%

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

   3. associate-/r* 37.1%

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

   4. associate-/r/ 37.1%

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

   5. *-commutative 37.1%

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

   6. times-frac 38.3%

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

   7. +-commutative 38.3%

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

   8. associate--l+ 46.8%

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

   9. metadata-eval 46.8%

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

   10. +-rgt-identity 46.8%

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

   11. times-frac 51.6%

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

3. Simplified 51.6%

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

4. Taylor expanded in t around 0 72.2%

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

   1. unpow2 72.2%

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

   2. times-frac 73.0%

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

   3. unpow2 73.0%

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

   4. associate-/l* 84.6%

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

   5. *-commutative 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 65.8%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   2. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   3. associate-*l* 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   4. associate-*r/ 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

   5. *-commutative 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

   6. unpow2 65.8%

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

9. Simplified 65.8%

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

10. Taylor expanded in k around 0 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 63.2%

       \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

    2. associate-/r* 63.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

12. Simplified 63.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}\right)\right) \]

13. Taylor expanded in l around 0 63.3%

    \[\leadsto 2 \cdot \color{blue}{\frac{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) \cdot {\ell}^{2}}{{k}^{2}}} \]
14. Step-by-step derivation

    1. associate-/l* 63.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

    2. associate-*r/ 63.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    3. metadata-eval 63.4%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{0.3333333333333333}}{t} + \frac{1}{{k}^{2} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    4. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    5. associate-*r* 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{1}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    6. associate-/r* 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \color{blue}{\frac{\frac{1}{k}}{k \cdot t}}}{\frac{{k}^{2}}{{\ell}^{2}}} \]

    7. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

    8. unpow2 63.4%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

    9. times-frac 71.7%

       \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell}}} \]

15. Simplified 71.7%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\frac{k}{\ell} \cdot \frac{k}{\ell}}} \]

16. Final simplification 71.7%

    \[\leadsto 2 \cdot \frac{\frac{0.3333333333333333}{t} + \frac{\frac{1}{k}}{k \cdot t}}{\frac{k}{\ell} \cdot \frac{k}{\ell}} \]

Alternative 13: 56.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (* 0.3333333333333333 (/ (* l l) (* k (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (0.3333333333333333 * ((l * l) / (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 * (0.3333333333333333d0 * ((l * l) / (k * (k * t))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(l * l) / Float64(k * Float64(k * t)))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.8%

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

   2. associate-*l* 37.1%

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

   3. associate-/r* 37.1%

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

   4. associate-/r/ 37.1%

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

   5. *-commutative 37.1%

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

   6. times-frac 38.3%

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

   7. +-commutative 38.3%

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

   8. associate--l+ 46.8%

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

   9. metadata-eval 46.8%

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

   10. +-rgt-identity 46.8%

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

   11. times-frac 51.6%

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

3. Simplified 51.6%

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

4. Taylor expanded in t around 0 72.2%

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

   1. unpow2 72.2%

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

   2. times-frac 73.0%

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

   3. unpow2 73.0%

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

   4. associate-/l* 84.6%

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

   5. *-commutative 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 65.8%

   \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
8. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   2. unpow2 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   3. associate-*l* 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

   4. associate-*r/ 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{0.3333333333333333 \cdot {\ell}^{2}}{t}}\right)\right) \]

   5. *-commutative 65.8%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{{\ell}^{2} \cdot 0.3333333333333333}}{t}\right)\right) \]

   6. unpow2 65.8%

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

9. Simplified 65.8%

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

10. Taylor expanded in k around inf 63.3%

    \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. *-commutative 63.3%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot t}\right) \]

    2. times-frac 62.4%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right) \]

    3. unpow2 62.4%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

    4. unpow2 62.4%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right) \]

    5. times-frac 63.2%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right) \]

    6. unpow2 63.2%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{t}\right)\right) \]

12. Simplified 63.2%

    \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)} \]

13. Taylor expanded in k around 0 60.8%

    \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right) \]
14. Step-by-step derivation

    1. unpow2 60.8%

       \[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

    2. unpow2 60.8%

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

    3. associate-*r* 58.7%

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

15. Simplified 58.7%

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

16. Final simplification 58.7%

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

Reproduce

herbie shell --seed 2023255 
(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))))