Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 83.3%

Time: 29.4s

Alternatives: 19

Speedup: 28.1×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

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: 54.9% 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: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;t \leq 35000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= t -1.5e+203)
     (/ 2.0 (* t_1 (* (tan k) (/ k (/ l (/ (pow t 3.0) l))))))
     (if (<= t 35000000000.0)
       (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ (pow (sin k) 2.0) (cos k)))))
       (/ 2.0 (* t_1 (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -1.5e+203) {
		tmp = 2.0 / (t_1 * (tan(k) * (k / (l / (pow(t, 3.0) / l)))));
	} else if (t <= 35000000000.0) {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (pow(sin(k), 2.0) / cos(k))));
	} else {
		tmp = 2.0 / (t_1 * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if (t <= (-1.5d+203)) then
        tmp = 2.0d0 / (t_1 * (tan(k) * (k / (l / ((t ** 3.0d0) / l)))))
    else if (t <= 35000000000.0d0) then
        tmp = 2.0d0 / ((k / l) * ((t * (k / l)) * ((sin(k) ** 2.0d0) / cos(k))))
    else
        tmp = 2.0d0 / (t_1 * (tan(k) * (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if (t <= -1.5e+203) {
		tmp = 2.0 / (t_1 * (Math.tan(k) * (k / (l / (Math.pow(t, 3.0) / l)))));
	} else if (t <= 35000000000.0) {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else {
		tmp = 2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if t <= -1.5e+203:
		tmp = 2.0 / (t_1 * (math.tan(k) * (k / (l / (math.pow(t, 3.0) / l)))))
	elif t <= 35000000000.0:
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (math.pow(math.sin(k), 2.0) / math.cos(k))))
	else:
		tmp = 2.0 / (t_1 * (math.tan(k) * (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (t <= -1.5e+203)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(k / Float64(l / Float64((t ^ 3.0) / l))))));
	elseif (t <= 35000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if (t <= -1.5e+203)
		tmp = 2.0 / (t_1 * (tan(k) * (k / (l / ((t ^ 3.0) / l)))));
	elseif (t <= 35000000000.0)
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * ((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = 2.0 / (t_1 * (tan(k) * (sin(k) * (((t ^ 1.5) / l) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+203], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 35000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e203

   1. Initial program 42.6%

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

   2. Taylor expanded in k around 0 42.6%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. associate-/l* 66.7%

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

   4. Simplified 66.7%

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

   if -1.5e203 < t < 3.5e10

   1. Initial program 49.6%

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

      1. add-sqr-sqrt 16.9%

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

      2. pow2 16.9%

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

      3. sqrt-div 16.9%

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

      4. sqrt-pow1 19.2%

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

      5. metadata-eval 19.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 9.9%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 21.5%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 21.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 71.0%

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

      1. unpow2 71.0%

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

      2. associate-*l* 72.7%

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

      3. associate-*l* 72.7%

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

      4. unpow2 72.7%

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

      5. associate-*r* 72.7%

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

      6. times-frac 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in k around inf 86.1%

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

      1. associate-*r* 86.2%

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

      2. times-frac 88.0%

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

      3. associate-/l* 87.6%

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

      4. associate-/r/ 89.6%

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

   9. Simplified 89.6%

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

   if 3.5e10 < t

   1. Initial program 64.8%

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

      1. add-sqr-sqrt 64.8%

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

      2. pow2 64.8%

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

      3. sqrt-div 64.8%

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

      4. sqrt-pow1 76.7%

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

      5. metadata-eval 76.7%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 41.9%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 84.0%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 84.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;t \leq 35000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 2: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}}\right)}^{3}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+296)
     (/
      (pow (/ (cbrt (/ 2.0 (tan k))) (/ t (cbrt (/ (* l l) (sin k))))) 3.0)
      (+ 2.0 t_1))
     (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ (pow (sin k) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296) {
		tmp = pow((cbrt((2.0 / tan(k))) / (t / cbrt(((l * l) / sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (pow(sin(k), 2.0) / cos(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) / (t / Math.cbrt(((l * l) / Math.sin(k))))), 3.0) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+296)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) / Float64(t / cbrt(Float64(Float64(l * l) / sin(k))))) ^ 3.0) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+296], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999981e295

   1. Initial program 81.1%

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

      1. associate-/r* 80.4%

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

      2. associate-*l* 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-*l* 80.4%

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

      5. *-commutative 80.4%

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

      6. sqr-neg 80.4%

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

      7. associate-/r* 80.4%

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

   3. Simplified 79.4%

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

      1. add-cube-cbrt 79.3%

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

      2. pow3 79.3%

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

      3. cbrt-div 79.2%

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

      4. cbrt-div 79.8%

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

      5. rem-cbrt-cube 89.0%

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

   5. Applied egg-rr 89.0%

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

   if 9.99999999999999981e295 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.4%

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

      1. add-sqr-sqrt 20.3%

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

      2. pow2 20.3%

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

      3. sqrt-div 20.3%

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

      4. sqrt-pow1 25.7%

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

      5. metadata-eval 25.7%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 17.0%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 33.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 33.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 56.8%

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

      1. unpow2 56.8%

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

      2. associate-*l* 59.3%

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

      3. associate-*l* 59.2%

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

      4. unpow2 59.2%

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

      5. associate-*r* 59.3%

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

      6. times-frac 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in k around inf 79.9%

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

      1. associate-*r* 80.0%

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

      2. times-frac 81.7%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq 10^{+296}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        1e+296)
     (/ 2.0 (* t_1 (* (tan k) (pow (/ t (cbrt (/ (* l l) (sin k)))) 3.0))))
     (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ (pow (sin k) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 1e+296) {
		tmp = 2.0 / (t_1 * (tan(k) * pow((t / cbrt(((l * l) / sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (pow(sin(k), 2.0) / cos(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= 1e+296) {
		tmp = 2.0 / (t_1 * (Math.tan(k) * Math.pow((t / Math.cbrt(((l * l) / Math.sin(k)))), 3.0)));
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)) <= 1e+296)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * (Float64(t / cbrt(Float64(Float64(l * l) / sin(k)))) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision]), $MachinePrecision], 1e+296], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999981e295

   1. Initial program 81.1%

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

      1. associate-/r/ 80.1%

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

      2. add-cube-cbrt 80.0%

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

      3. pow3 80.0%

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

      4. cbrt-div 80.0%

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

      5. rem-cbrt-cube 87.0%

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

   3. Applied egg-rr 87.0%

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

   if 9.99999999999999981e295 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.4%

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

      1. add-sqr-sqrt 20.3%

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

      2. pow2 20.3%

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

      3. sqrt-div 20.3%

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

      4. sqrt-pow1 25.7%

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

      5. metadata-eval 25.7%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 17.0%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 33.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 33.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 56.8%

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

      1. unpow2 56.8%

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

      2. associate-*l* 59.3%

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

      3. associate-*l* 59.2%

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

      4. unpow2 59.2%

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

      5. associate-*r* 59.3%

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

      6. times-frac 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in k around inf 79.9%

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

      1. associate-*r* 80.0%

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

      2. times-frac 81.7%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{{t}^{3} \cdot \tan k}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+296)
     (/ (* (/ l (/ (sin k) l)) (/ 2.0 (* (pow t 3.0) (tan k)))) (+ 2.0 t_1))
     (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ (pow (sin k) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296) {
		tmp = ((l / (sin(k) / l)) * (2.0 / (pow(t, 3.0) * tan(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (pow(sin(k), 2.0) / cos(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + t_1)))) <= 1d+296) then
        tmp = ((l / (sin(k) / l)) * (2.0d0 / ((t ** 3.0d0) * tan(k)))) / (2.0d0 + t_1)
    else
        tmp = 2.0d0 / ((k / l) * ((t * (k / l)) * ((sin(k) ** 2.0d0) / cos(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296) {
		tmp = ((l / (Math.sin(k) / l)) * (2.0 / (Math.pow(t, 3.0) * Math.tan(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296:
		tmp = ((l / (math.sin(k) / l)) * (2.0 / (math.pow(t, 3.0) * math.tan(k)))) / (2.0 + t_1)
	else:
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (math.pow(math.sin(k), 2.0) / math.cos(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+296)
		tmp = Float64(Float64(Float64(l / Float64(sin(k) / l)) * Float64(2.0 / Float64((t ^ 3.0) * tan(k)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+296)
		tmp = ((l / (sin(k) / l)) * (2.0 / ((t ^ 3.0) * tan(k)))) / (2.0 + t_1);
	else
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * ((sin(k) ^ 2.0) / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+296], N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[t, 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.99999999999999981e295

   1. Initial program 81.1%

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

      1. associate-/r* 80.4%

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

      2. associate-*l* 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-*l* 80.4%

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

      5. *-commutative 80.4%

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

      6. sqr-neg 80.4%

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

      7. associate-/r* 80.4%

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

   3. Simplified 79.4%

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

      1. div-inv 79.4%

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

      2. associate-/r/ 80.1%

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

   5. Applied egg-rr 80.1%

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

      1. associate-*r/ 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. *-commutative 80.1%

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

      4. associate-/l* 82.6%

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

      5. associate-/l/ 82.6%

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

   7. Simplified 82.6%

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

   if 9.99999999999999981e295 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.4%

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

      1. add-sqr-sqrt 20.3%

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

      2. pow2 20.3%

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

      3. sqrt-div 20.3%

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

      4. sqrt-pow1 25.7%

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

      5. metadata-eval 25.7%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 17.0%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 33.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 33.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 56.8%

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

      1. unpow2 56.8%

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

      2. associate-*l* 59.3%

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

      3. associate-*l* 59.2%

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

      4. unpow2 59.2%

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

      5. associate-*r* 59.3%

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

      6. times-frac 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in k around inf 79.9%

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

      1. associate-*r* 80.0%

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

      2. times-frac 81.7%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 83.2%

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

   9. Simplified 83.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+296}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 5: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;t \leq 17000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -1.5e+203)
     (/ 2.0 (* (+ 1.0 (+ 1.0 t_1)) (* (tan k) (/ k (/ l (/ (pow t 3.0) l))))))
     (if (<= t 17000000000.0)
       (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ (pow (sin k) 2.0) (cos k)))))
       (/
        (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
        (+ 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -1.5e+203) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (k / (l / (pow(t, 3.0) / l)))));
	} else if (t <= 17000000000.0) {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (pow(sin(k), 2.0) / cos(k))));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if (t <= (-1.5d+203)) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + t_1)) * (tan(k) * (k / (l / ((t ** 3.0d0) / l)))))
    else if (t <= 17000000000.0d0) then
        tmp = 2.0d0 / ((k / l) * ((t * (k / l)) * ((sin(k) ** 2.0d0) / cos(k))))
    else
        tmp = ((2.0d0 / tan(k)) / (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0))) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.5e+203) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * (k / (l / (Math.pow(t, 3.0) / l)))));
	} else if (t <= 17000000000.0) {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -1.5e+203:
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (math.tan(k) * (k / (l / (math.pow(t, 3.0) / l)))))
	elif t <= 17000000000.0:
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (math.pow(math.sin(k), 2.0) / math.cos(k))))
	else:
		tmp = ((2.0 / math.tan(k)) / (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -1.5e+203)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * Float64(k / Float64(l / Float64((t ^ 3.0) / l))))));
	elseif (t <= 17000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64((sin(k) ^ 2.0) / cos(k)))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -1.5e+203)
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (k / (l / ((t ^ 3.0) / l)))));
	elseif (t <= 17000000000.0)
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * ((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = ((2.0 / tan(k)) / (sin(k) * (((t ^ 1.5) / l) ^ 2.0))) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -1.5e+203], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 17000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e203

   1. Initial program 42.6%

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

   2. Taylor expanded in k around 0 42.6%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

      3. associate-/l* 66.7%

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

   4. Simplified 66.7%

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

   if -1.5e203 < t < 1.7e10

   1. Initial program 49.6%

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

      1. add-sqr-sqrt 16.9%

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

      2. pow2 16.9%

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

      3. sqrt-div 16.9%

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

      4. sqrt-pow1 19.2%

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

      5. metadata-eval 19.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 9.9%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 21.5%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 21.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 71.0%

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

      1. unpow2 71.0%

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

      2. associate-*l* 72.7%

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

      3. associate-*l* 72.7%

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

      4. unpow2 72.7%

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

      5. associate-*r* 72.7%

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

      6. times-frac 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in k around inf 86.1%

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

      1. associate-*r* 86.2%

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

      2. times-frac 88.0%

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

      3. associate-/l* 87.6%

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

      4. associate-/r/ 89.6%

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

   9. Simplified 89.6%

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

   if 1.7e10 < t

   1. Initial program 64.8%

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

      1. associate-/r* 64.8%

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

      2. associate-*l* 56.9%

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

      3. sqr-neg 56.9%

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

      4. associate-*l* 64.8%

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

      5. *-commutative 64.8%

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

      6. sqr-neg 64.8%

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

      7. associate-/r* 64.8%

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

   3. Simplified 64.6%

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

      1. associate-/r/ 64.8%

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

      2. add-sqr-sqrt 64.8%

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

      3. pow2 64.8%

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

      4. sqrt-div 64.8%

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

      5. sqrt-pow1 76.7%

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

      6. metadata-eval 76.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 41.8%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 84.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 84.0%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;t \leq 17000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 6: 69.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\sin k}^{2}}{\cos k}\\ \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{t \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (pow (sin k) 2.0) (cos k))))
   (if (<= k 3.05e-133)
     (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
     (if (<= k 9e-23)
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
         (* (tan k) (/ k (/ l (/ (pow t 3.0) l))))))
       (if (<= k 2.1e+218)
         (/ 2.0 (/ (* k (* t_1 (/ (* t k) l))) l))
         (/ 2.0 (* (* t (/ k l)) (* (/ k l) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0) / cos(k);
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 9e-23) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (k / (l / (pow(t, 3.0) / l)))));
	} else if (k <= 2.1e+218) {
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sin(k) ** 2.0d0) / cos(k)
    if (k <= 3.05d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 9d-23) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (k / (l / ((t ** 3.0d0) / l)))))
    else if (k <= 2.1d+218) then
        tmp = 2.0d0 / ((k * (t_1 * ((t * k) / l))) / l)
    else
        tmp = 2.0d0 / ((t * (k / l)) * ((k / l) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0) / Math.cos(k);
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 9e-23) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (k / (l / (Math.pow(t, 3.0) / l)))));
	} else if (k <= 2.1e+218) {
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0) / math.cos(k)
	tmp = 0
	if k <= 3.05e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 9e-23:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (k / (l / (math.pow(t, 3.0) / l)))))
	elif k <= 2.1e+218:
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l)
	else:
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64((sin(k) ^ 2.0) / cos(k))
	tmp = 0.0
	if (k <= 3.05e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 9e-23)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(k / Float64(l / Float64((t ^ 3.0) / l))))));
	elseif (k <= 2.1e+218)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_1 * Float64(Float64(t * k) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(Float64(k / l) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (sin(k) ^ 2.0) / cos(k);
	tmp = 0.0;
	if (k <= 3.05e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 9e-23)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (k / (l / ((t ^ 3.0) / l)))));
	elseif (k <= 2.1e+218)
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	else
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.05e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9e-23], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e+218], N[(2.0 / N[(N[(k * N[(t$95$1 * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.0500000000000002e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.0500000000000002e-133 < k < 8.9999999999999995e-23

   1. Initial program 68.7%

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

   2. Taylor expanded in k around 0 68.7%

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

      1. associate-/l* 68.8%

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

      2. unpow2 68.8%

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

      3. associate-/l* 74.4%

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

   4. Simplified 74.4%

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

   if 8.9999999999999995e-23 < k < 2.0999999999999999e218

   1. Initial program 52.7%

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

      1. add-sqr-sqrt 22.9%

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

      2. pow2 22.9%

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

      3. sqrt-div 22.9%

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

      4. sqrt-pow1 27.4%

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

      5. metadata-eval 27.4%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 9.1%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 29.6%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 29.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 80.3%

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

      1. unpow2 80.3%

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

      2. associate-*l* 82.4%

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

      3. associate-*l* 82.4%

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

      4. unpow2 82.4%

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

      5. associate-*r* 82.5%

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

      6. times-frac 93.2%

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

   6. Simplified 93.2%

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

      1. associate-*l/ 93.1%

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

      2. times-frac 93.1%

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

   8. Applied egg-rr 93.1%

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

   if 2.0999999999999999e218 < k

   1. Initial program 46.2%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. sqrt-div 38.5%

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

      4. sqrt-pow1 38.6%

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

      5. metadata-eval 38.6%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 30.8%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 46.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 46.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 62.3%

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

      1. unpow2 62.3%

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

      2. associate-*l* 62.3%

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

      3. associate-*l* 62.3%

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

      4. unpow2 62.3%

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

      5. associate-*r* 62.3%

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

      6. times-frac 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in k around inf 62.3%

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

      1. associate-/r* 62.3%

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

      2. unpow2 62.3%

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

      3. associate-*r* 62.3%

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

      4. associate-*r* 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. times-frac 77.7%

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

      8. associate-*l/ 77.9%

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

      9. associate-*l/ 77.9%

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

      10. associate-*r/ 77.9%

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

      11. associate-*l* 78.1%

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

      12. associate-/l* 99.3%

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

      13. associate-/r/ 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 7: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 3.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+163}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t_1}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 3.6e-133)
     (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
     (if (<= k 4.2e-50)
       (/ (* l (/ l (pow t 3.0))) (* k k))
       (if (<= k 3.8e+163)
         (* 2.0 (/ (* (* l (cos k)) (/ l (* k (* t k)))) t_1))
         (/ 2.0 (* (* t (/ k l)) (* (/ k l) (/ t_1 (cos k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 3.6e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 4.2e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else if (k <= 3.8e+163) {
		tmp = 2.0 * (((l * cos(k)) * (l / (k * (t * k)))) / t_1);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * (t_1 / cos(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 3.6d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 4.2d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else if (k <= 3.8d+163) then
        tmp = 2.0d0 * (((l * cos(k)) * (l / (k * (t * k)))) / t_1)
    else
        tmp = 2.0d0 / ((t * (k / l)) * ((k / l) * (t_1 / cos(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 3.6e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 4.2e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else if (k <= 3.8e+163) {
		tmp = 2.0 * (((l * Math.cos(k)) * (l / (k * (t * k)))) / t_1);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * (t_1 / Math.cos(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 3.6e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 4.2e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	elif k <= 3.8e+163:
		tmp = 2.0 * (((l * math.cos(k)) * (l / (k * (t * k)))) / t_1)
	else:
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * (t_1 / math.cos(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 3.6e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 4.2e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	elseif (k <= 3.8e+163)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * cos(k)) * Float64(l / Float64(k * Float64(t * k)))) / t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(Float64(k / l) * Float64(t_1 / cos(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 3.6e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 4.2e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	elseif (k <= 3.8e+163)
		tmp = 2.0 * (((l * cos(k)) * (l / (k * (t * k)))) / t_1);
	else
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * (t_1 / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 3.6e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.2e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+163], N[(2.0 * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.6000000000000004e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.6000000000000004e-133 < k < 4.2000000000000002e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 4.2000000000000002e-50 < k < 3.80000000000000008e163

   1. Initial program 63.4%

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

      1. associate-/r* 63.5%

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

      2. associate-*l* 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-*l* 63.5%

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

      5. *-commutative 63.5%

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

      6. sqr-neg 63.5%

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

      7. associate-*l/ 63.5%

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

      8. associate-*r/ 63.6%

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

      9. associate-/r/ 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around inf 81.2%

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

      1. unpow2 81.2%

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

      2. associate-*l* 81.2%

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

      3. associate-*r* 81.2%

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

      4. times-frac 91.4%

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

      5. unpow2 91.4%

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

   6. Simplified 91.4%

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

      1. associate-*r/ 91.5%

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

      2. associate-*l* 93.6%

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

   8. Applied egg-rr 93.6%

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

   if 3.80000000000000008e163 < k

   1. Initial program 38.9%

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

      1. add-sqr-sqrt 27.8%

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

      2. pow2 27.8%

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

      3. sqrt-div 27.8%

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

      4. sqrt-pow1 27.9%

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

      5. metadata-eval 27.9%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 22.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 33.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 33.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 67.2%

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

      1. unpow2 67.2%

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

      2. associate-*l* 67.2%

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

      3. associate-*l* 67.2%

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

      4. unpow2 67.2%

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

      5. associate-*r* 67.2%

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

      6. times-frac 84.0%

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

   6. Simplified 84.0%

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

   7. Taylor expanded in k around inf 67.2%

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

      1. associate-/r* 67.2%

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

      2. unpow2 67.2%

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

      3. associate-*r* 67.2%

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

      4. associate-*r* 67.2%

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

      5. *-commutative 67.2%

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

      6. unpow2 67.2%

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

      7. times-frac 83.8%

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

      8. associate-*l/ 84.0%

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

      9. associate-*l/ 84.0%

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

      10. associate-*r/ 84.0%

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

      11. associate-*l* 84.1%

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

      12. associate-/l* 99.4%

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

      13. associate-/r/ 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+163}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 8: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 4.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t_1}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{t_1}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 4.8e-133)
     (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
     (if (<= k 3.5e-50)
       (/ (* l (/ l (pow t 3.0))) (* k k))
       (if (<= k 7e+152)
         (/ 2.0 (* (/ (* t (* k k)) l) (/ t_1 (* l (cos k)))))
         (/ 2.0 (* (/ k l) (* (* t (/ k l)) (/ t_1 (cos k))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 4.8e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 3.5e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else if (k <= 7e+152) {
		tmp = 2.0 / (((t * (k * k)) / l) * (t_1 / (l * cos(k))));
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (t_1 / cos(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 4.8d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 3.5d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else if (k <= 7d+152) then
        tmp = 2.0d0 / (((t * (k * k)) / l) * (t_1 / (l * cos(k))))
    else
        tmp = 2.0d0 / ((k / l) * ((t * (k / l)) * (t_1 / cos(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 4.8e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 3.5e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else if (k <= 7e+152) {
		tmp = 2.0 / (((t * (k * k)) / l) * (t_1 / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (t_1 / Math.cos(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 4.8e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 3.5e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	elif k <= 7e+152:
		tmp = 2.0 / (((t * (k * k)) / l) * (t_1 / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (t_1 / math.cos(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 4.8e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 3.5e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	elseif (k <= 7e+152)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k * k)) / l) * Float64(t_1 / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t * Float64(k / l)) * Float64(t_1 / cos(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 4.8e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 3.5e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	elseif (k <= 7e+152)
		tmp = 2.0 / (((t * (k * k)) / l) * (t_1 / (l * cos(k))));
	else
		tmp = 2.0 / ((k / l) * ((t * (k / l)) * (t_1 / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 4.8e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7e+152], N[(2.0 / N[(N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 4.8e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 4.8e-133 < k < 3.49999999999999997e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 3.49999999999999997e-50 < k < 6.99999999999999963e152

   1. Initial program 63.2%

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

   2. Taylor expanded in t around 0 82.2%

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

      1. associate-*r* 82.2%

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

      2. unpow2 82.2%

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

      3. associate-*l* 82.2%

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

      4. times-frac 92.9%

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

      5. unpow2 92.9%

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

   4. Simplified 92.9%

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

   if 6.99999999999999963e152 < k

   1. Initial program 42.9%

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

      1. add-sqr-sqrt 28.6%

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

      2. pow2 28.6%

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

      3. sqrt-div 28.6%

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

      4. sqrt-pow1 28.7%

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

      5. metadata-eval 28.7%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 19.0%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 33.3%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 33.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 67.3%

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

      1. unpow2 67.3%

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

      2. associate-*l* 71.8%

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

      3. associate-*l* 71.9%

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

      4. unpow2 71.9%

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

      5. associate-*r* 71.9%

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

      6. times-frac 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in k around inf 86.1%

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

      1. associate-*r* 86.3%

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

      2. times-frac 86.1%

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

      3. associate-/l* 95.1%

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

      4. associate-/r/ 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 9: 70.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\sin k}^{2}}{\cos k}\\ \mathbf{if}\;k \leq 5.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{t \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (pow (sin k) 2.0) (cos k))))
   (if (<= k 5.2e-133)
     (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
     (if (<= k 6.2e-50)
       (/ (* l (/ l (pow t 3.0))) (* k k))
       (if (<= k 2.1e+218)
         (/ 2.0 (/ (* k (* t_1 (/ (* t k) l))) l))
         (/ 2.0 (* (* t (/ k l)) (* (/ k l) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0) / cos(k);
	double tmp;
	if (k <= 5.2e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 6.2e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else if (k <= 2.1e+218) {
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sin(k) ** 2.0d0) / cos(k)
    if (k <= 5.2d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 6.2d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else if (k <= 2.1d+218) then
        tmp = 2.0d0 / ((k * (t_1 * ((t * k) / l))) / l)
    else
        tmp = 2.0d0 / ((t * (k / l)) * ((k / l) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0) / Math.cos(k);
	double tmp;
	if (k <= 5.2e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 6.2e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else if (k <= 2.1e+218) {
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	} else {
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0) / math.cos(k)
	tmp = 0
	if k <= 5.2e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 6.2e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	elif k <= 2.1e+218:
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l)
	else:
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64((sin(k) ^ 2.0) / cos(k))
	tmp = 0.0
	if (k <= 5.2e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 6.2e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	elseif (k <= 2.1e+218)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_1 * Float64(Float64(t * k) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(Float64(k / l) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (sin(k) ^ 2.0) / cos(k);
	tmp = 0.0;
	if (k <= 5.2e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 6.2e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	elseif (k <= 2.1e+218)
		tmp = 2.0 / ((k * (t_1 * ((t * k) / l))) / l);
	else
		tmp = 2.0 / ((t * (k / l)) * ((k / l) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.2e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.2e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e+218], N[(2.0 / N[(N[(k * N[(t$95$1 * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 5.1999999999999999e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 5.1999999999999999e-133 < k < 6.2000000000000004e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 6.2000000000000004e-50 < k < 2.0999999999999999e218

   1. Initial program 59.2%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-div 21.7%

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

      4. sqrt-pow1 25.6%

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

      5. metadata-eval 25.6%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 9.8%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 27.5%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 27.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 81.1%

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

      1. unpow2 81.1%

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

      2. associate-*l* 83.0%

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

      3. associate-*l* 83.0%

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

      4. unpow2 83.0%

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

      5. associate-*r* 83.0%

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

      6. times-frac 92.3%

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

   6. Simplified 92.3%

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

      1. associate-*l/ 92.2%

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

      2. times-frac 92.2%

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

   8. Applied egg-rr 92.2%

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

   if 2.0999999999999999e218 < k

   1. Initial program 46.2%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. sqrt-div 38.5%

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

      4. sqrt-pow1 38.6%

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

      5. metadata-eval 38.6%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. sqrt-prod 30.8%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. add-sqr-sqrt 46.2%

         \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied egg-rr 46.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around 0 62.3%

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

      1. unpow2 62.3%

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

      2. associate-*l* 62.3%

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

      3. associate-*l* 62.3%

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

      4. unpow2 62.3%

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

      5. associate-*r* 62.3%

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

      6. times-frac 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in k around inf 62.3%

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

      1. associate-/r* 62.3%

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

      2. unpow2 62.3%

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

      3. associate-*r* 62.3%

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

      4. associate-*r* 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. times-frac 77.7%

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

      8. associate-*l/ 77.9%

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

      9. associate-*l/ 77.9%

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

      10. associate-*r/ 77.9%

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

      11. associate-*l* 78.1%

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

      12. associate-/l* 99.3%

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

      13. associate-/r/ 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 10: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2} \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.05e-133)
   (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
   (if (<= k 2.1e-50)
     (/ (* l (/ l (pow t 3.0))) (* k k))
     (* 2.0 (* (/ l t) (/ (* l (cos k)) (* (pow (sin k) 2.0) (* k k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 2.1e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / t) * ((l * cos(k)) / (pow(sin(k), 2.0) * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.05d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 2.1d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * ((l / t) * ((l * cos(k)) / ((sin(k) ** 2.0d0) * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 2.1e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / t) * ((l * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.05e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 2.1e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * ((l / t) * ((l * math.cos(k)) / (math.pow(math.sin(k), 2.0) * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.05e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 2.1e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(Float64(l * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.05e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 2.1e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * ((l / t) * ((l * cos(k)) / ((sin(k) ^ 2.0) * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.05e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.0500000000000002e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.0500000000000002e-133 < k < 2.1000000000000001e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 2.1000000000000001e-50 < k

   1. Initial program 56.5%

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

      1. associate-/r* 56.6%

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

      2. associate-*l* 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-*l* 56.6%

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

      5. *-commutative 56.6%

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

      6. sqr-neg 56.6%

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

      7. associate-/r* 56.6%

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

   3. Simplified 56.6%

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

      1. associate-/r/ 56.6%

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

   5. Applied egg-rr 56.6%

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

   6. Taylor expanded in k around inf 77.3%

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

      1. unpow2 77.3%

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

      2. associate-*r* 77.3%

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

      3. associate-*r* 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-*l* 77.3%

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

      6. times-frac 80.4%

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

      7. unpow2 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2} \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 11: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 6.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.7e-133)
   (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
   (if (<= k 6.9e-50)
     (/ (* l (/ l (pow t 3.0))) (* k k))
     (* 2.0 (* (/ l (* k (* t k))) (/ (* l (cos k)) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.7e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 6.9e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / (k * (t * k))) * ((l * cos(k)) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.7d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 6.9d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * ((l / (k * (t * k))) * ((l * cos(k)) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.7e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 6.9e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / (k * (t * k))) * ((l * Math.cos(k)) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.7e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 6.9e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * ((l / (k * (t * k))) * ((l * math.cos(k)) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.7e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 6.9e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(t * k))) * Float64(Float64(l * cos(k)) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.7e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 6.9e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * ((l / (k * (t * k))) * ((l * cos(k)) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.7e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.9e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.70000000000000003e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 4.70000000000000003e-133 < k < 6.9000000000000001e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 6.9000000000000001e-50 < k

   1. Initial program 56.5%

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

      1. associate-/r* 56.6%

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

      2. associate-*l* 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-*l* 56.6%

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

      5. *-commutative 56.6%

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

      6. sqr-neg 56.6%

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

      7. associate-*l/ 56.6%

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

      8. associate-*r/ 56.6%

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

      9. associate-/r/ 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in k around inf 77.3%

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

      1. unpow2 77.3%

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

      2. associate-*l* 77.3%

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

      3. associate-*r* 77.3%

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

      4. times-frac 84.9%

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

      5. unpow2 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in l around 0 84.9%

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

      1. unpow2 84.9%

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

      2. associate-*r* 86.4%

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

   9. Simplified 86.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 6.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 12: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.05e-133)
   (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
   (if (<= k 5.8e-50)
     (/ (* l (/ l (pow t 3.0))) (* k k))
     (* 2.0 (/ (* (* l (cos k)) (/ l (* k (* t k)))) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 5.8e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (((l * cos(k)) * (l / (k * (t * k)))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.05d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 5.8d-50) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * (((l * cos(k)) * (l / (k * (t * k)))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 5.8e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * (((l * Math.cos(k)) * (l / (k * (t * k)))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.05e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 5.8e-50:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * (((l * math.cos(k)) * (l / (k * (t * k)))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.05e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 5.8e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * cos(k)) * Float64(l / Float64(k * Float64(t * k)))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.05e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 5.8e-50)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * (((l * cos(k)) * (l / (k * (t * k)))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.05e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.0500000000000002e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.0500000000000002e-133 < k < 5.80000000000000016e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 5.80000000000000016e-50 < k

   1. Initial program 56.5%

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

      1. associate-/r* 56.6%

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

      2. associate-*l* 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-*l* 56.6%

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

      5. *-commutative 56.6%

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

      6. sqr-neg 56.6%

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

      7. associate-*l/ 56.6%

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

      8. associate-*r/ 56.6%

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

      9. associate-/r/ 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in k around inf 77.3%

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

      1. unpow2 77.3%

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

      2. associate-*l* 77.3%

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

      3. associate-*r* 77.3%

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

      4. times-frac 84.9%

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

      5. unpow2 84.9%

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

   6. Simplified 84.9%

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

      1. associate-*r/ 84.9%

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

      2. associate-*l* 86.5%

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

   8. Applied egg-rr 86.5%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 13: 61.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k \cdot k}{\ell}\\ t_2 := t \cdot t_1\\ \mathbf{if}\;k \leq 3.25 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \sqrt[3]{t_2 \cdot \left(t_2 \cdot t_2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* k k) l)) (t_2 (* t t_1)))
   (if (<= k 3.25e-133)
     (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
     (if (<= k 2.8e-50)
       (/ (* l (/ l (pow t 3.0))) (* k k))
       (/ 2.0 (* t_1 (cbrt (* t_2 (* t_2 t_2)))))))))

C

double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double t_2 = t * t_1;
	double tmp;
	if (k <= 3.25e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 2.8e-50) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 / (t_1 * cbrt((t_2 * (t_2 * t_2))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double t_2 = t * t_1;
	double tmp;
	if (k <= 3.25e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 2.8e-50) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 / (t_1 * Math.cbrt((t_2 * (t_2 * t_2))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k * k) / l)
	t_2 = Float64(t * t_1)
	tmp = 0.0
	if (k <= 3.25e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 2.8e-50)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 / Float64(t_1 * cbrt(Float64(t_2 * Float64(t_2 * t_2)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t * t$95$1), $MachinePrecision]}, If[LessEqual[k, 3.25e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e-50], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[Power[N[(t$95$2 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.2500000000000001e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.2500000000000001e-133 < k < 2.7999999999999998e-50

   1. Initial program 50.5%

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

      1. associate-/r* 50.5%

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

      2. associate-*l* 50.3%

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

      3. sqr-neg 50.3%

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

      4. associate-*l* 50.5%

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

      5. *-commutative 50.5%

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

      6. sqr-neg 50.5%

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

      7. associate-*l/ 50.5%

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

      8. associate-*r/ 50.5%

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

      9. associate-/r/ 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around 0 58.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 65.6%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 65.6%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.7%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 2.7999999999999998e-50 < k

   1. Initial program 56.5%

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

   2. Taylor expanded in t around 0 77.3%

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

      1. associate-*r* 77.3%

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

      2. unpow2 77.3%

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

      3. associate-*l* 77.3%

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

      4. times-frac 84.8%

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

      5. unpow2 84.8%

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

   4. Simplified 84.8%

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

   5. Taylor expanded in k around 0 65.8%

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

      1. unpow2 65.8%

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

   7. Simplified 65.8%

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

      1. add-cbrt-cube 67.1%

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

      2. associate-/l* 67.1%

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

      3. associate-/l* 67.1%

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

      4. associate-/l* 67.1%

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

   9. Applied egg-rr 67.1%

      \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \cdot \frac{k \cdot k}{\ell}} \]
   10. Step-by-step derivation

       1. associate-*l* 67.1%

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

       2. unpow2 67.1%

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

       3. associate-/r/ 67.1%

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

       4. unpow2 67.1%

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

       5. unpow2 67.1%

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

       6. associate-/r/ 67.1%

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

       7. unpow2 67.1%

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

       8. unpow2 67.1%

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

       9. associate-/r/ 67.1%

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

       10. unpow2 67.1%

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

   11. Simplified 67.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.25 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \sqrt[3]{\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}}\\ \end{array} \]

Alternative 14: 61.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.65e-127)
   (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
   (if (<= k 4.7e+95)
     (* (/ l (pow t 3.0)) (/ l (* k k)))
     (* 2.0 (* (/ (/ (/ l k) t) k) (* l -0.16666666666666666))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.65e-127) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 4.7e+95) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.65d-127) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 4.7d+95) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((((l / k) / t) / k) * (l * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.65e-127) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 4.7e+95) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.65e-127:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 4.7e+95:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.65e-127)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 4.7e+95)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / t) / k) * Float64(l * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.65e-127)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 4.7e+95)
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.65e-127], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.7e+95], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] * N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-127

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 59.7%

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

      1. associate-*r* 59.7%

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

      2. unpow2 59.7%

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

      3. associate-*l* 59.7%

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

      4. times-frac 68.5%

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

      5. unpow2 68.5%

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

   4. Simplified 68.5%

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

   5. Taylor expanded in k around 0 59.7%

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

      1. unpow2 59.7%

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

   7. Simplified 59.7%

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

      1. div-inv 59.7%

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

      2. *-commutative 59.7%

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

      3. associate-/l* 59.6%

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

      4. associate-/l* 61.3%

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

   9. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.3%

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

       2. metadata-eval 61.3%

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

       3. associate-/r* 61.3%

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

       4. associate-/r/ 61.3%

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

       5. associate-/l* 63.5%

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

   11. Simplified 63.5%

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

   if 1.6499999999999999e-127 < k < 4.69999999999999972e95

   1. Initial program 62.5%

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

      1. associate-/r* 62.5%

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

      2. associate-*l* 62.5%

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

      3. sqr-neg 62.5%

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

      4. associate-*l* 62.5%

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

      5. *-commutative 62.5%

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

      6. sqr-neg 62.5%

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

      7. associate-*l/ 62.5%

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

      8. associate-*r/ 62.6%

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

      9. associate-/r/ 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 65.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 65.5%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 68.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   if 4.69999999999999972e95 < k

   1. Initial program 49.0%

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

      1. associate-/r* 49.0%

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

      2. associate-*l* 49.0%

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

      3. sqr-neg 49.0%

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

      4. associate-*l* 49.0%

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

      5. *-commutative 49.0%

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

      6. sqr-neg 49.0%

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

      7. associate-*l/ 49.0%

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

      8. associate-*r/ 49.0%

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

      9. associate-/r/ 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around inf 78.8%

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

      1. unpow2 78.8%

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

      2. associate-*l* 78.7%

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

      3. associate-*r* 78.7%

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

      4. times-frac 81.7%

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

      5. unpow2 81.7%

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

   6. Simplified 81.7%

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

   7. Taylor expanded in k around 0 60.1%

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

      1. cancel-sign-sub-inv 60.1%

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

      2. fma-def 60.1%

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

      3. unpow2 60.1%

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

      4. metadata-eval 60.1%

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

   9. Simplified 60.1%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. unpow2 59.4%

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

       2. associate-*r* 59.4%

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

       3. associate-/l* 60.0%

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

       4. associate-/r/ 60.0%

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

       5. associate-/r* 60.7%

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

       6. *-commutative 60.7%

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

       7. associate-/r* 60.7%

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

       8. distribute-rgt-out 60.7%

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

       9. metadata-eval 60.7%

          \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 60.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 15: 61.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.05e-133)
   (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k)))
   (if (<= k 1.5e+94)
     (/ (* l (/ l (pow t 3.0))) (* k k))
     (* 2.0 (* (/ (/ (/ l k) t) k) (* l -0.16666666666666666))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 1.5e+94) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.05d-133) then
        tmp = (2.0d0 / (k * (k / l))) / (k / ((l / t) / k))
    else if (k <= 1.5d+94) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * ((((l / k) / t) / k) * (l * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.05e-133) {
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	} else if (k <= 1.5e+94) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.05e-133:
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k))
	elif k <= 1.5e+94:
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.05e-133)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)));
	elseif (k <= 1.5e+94)
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / t) / k) * Float64(l * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.05e-133)
		tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
	elseif (k <= 1.5e+94)
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.05e-133], N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+94], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] * N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.0500000000000002e-133

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 60.3%

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

      4. times-frac 69.2%

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

      5. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Simplified 60.3%

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

      1. div-inv 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-/l* 60.3%

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

      4. associate-/l* 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

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

       3. associate-/r* 61.5%

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

       4. associate-/r/ 61.5%

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

       5. associate-/l* 63.7%

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

   11. Simplified 63.7%

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

   if 3.0500000000000002e-133 < k < 1.5e94

   1. Initial program 61.9%

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

      1. associate-/r* 61.9%

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

      2. associate-*l* 61.9%

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

      3. sqr-neg 61.9%

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

      4. associate-*l* 61.9%

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

      5. *-commutative 61.9%

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

      6. sqr-neg 61.9%

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

      7. associate-*l/ 61.9%

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

      8. associate-*r/ 62.0%

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

      9. associate-/r/ 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in k around 0 64.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 64.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 67.2%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 67.2%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*l/ 67.5%

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   8. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}} \]

   if 1.5e94 < k

   1. Initial program 49.0%

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

      1. associate-/r* 49.0%

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

      2. associate-*l* 49.0%

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

      3. sqr-neg 49.0%

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

      4. associate-*l* 49.0%

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

      5. *-commutative 49.0%

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

      6. sqr-neg 49.0%

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

      7. associate-*l/ 49.0%

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

      8. associate-*r/ 49.0%

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

      9. associate-/r/ 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around inf 78.8%

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

      1. unpow2 78.8%

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

      2. associate-*l* 78.7%

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

      3. associate-*r* 78.7%

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

      4. times-frac 81.7%

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

      5. unpow2 81.7%

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

   6. Simplified 81.7%

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

   7. Taylor expanded in k around 0 60.1%

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

      1. cancel-sign-sub-inv 60.1%

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

      2. fma-def 60.1%

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

      3. unpow2 60.1%

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

      4. metadata-eval 60.1%

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

   9. Simplified 60.1%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. unpow2 59.4%

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

       2. associate-*r* 59.4%

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

       3. associate-/l* 60.0%

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

       4. associate-/r/ 60.0%

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

       5. associate-/r* 60.7%

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

       6. *-commutative 60.7%

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

       7. associate-/r* 60.7%

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

       8. distribute-rgt-out 60.7%

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

       9. metadata-eval 60.7%

          \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot \color{blue}{-0.16666666666666666}\right)\right) \]

   12. Simplified 60.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 16: 56.7% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (* k k)) (/ l (* t (* k k))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / (k * k)) * (l / (t * (k * k))));
}

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 * ((l / (k * k)) * (l / (t * (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / (k * k)) * (l / (t * (k * k))));
}

Python

def code(t, l, k):
	return 2.0 * ((l / (k * k)) * (l / (t * (k * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(k * k)) * Float64(l / Float64(t * Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k * k)) * (l / (t * (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 52.7%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 52.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 52.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 62.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. unpow2 62.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*l* 62.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   3. associate-*r* 62.8%

      \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   4. times-frac 71.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

   5. unpow2 71.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right) \]

6. Simplified 71.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 60.2%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right) \]
8. Step-by-step derivation

   1. unpow2 60.2%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \]

9. Simplified 60.2%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

10. Final simplification 60.2%

    \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \]

Alternative 17: 60.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ 2.0 (* k (/ k l))) (/ k (/ (/ l t) k))))

C

double code(double t, double l, double k) {
	return (2.0 / (k * (k / l))) / (k / ((l / t) / k));
}

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 / (k * (k / l))) / (k / ((l / t) / k))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * (k / l))) / (k / ((l / t) / k));
}

Python

def code(t, l, k):
	return (2.0 / (k * (k / l))) / (k / ((l / t) / k))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * Float64(k / l))) / Float64(k / Float64(Float64(l / t) / k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k * (k / l))) / (k / ((l / t) / k));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 63.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 63.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. unpow2 63.0%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

   3. associate-*l* 63.0%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   4. times-frac 71.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. unpow2 71.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

4. Simplified 71.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

5. Taylor expanded in k around 0 60.2%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
6. Step-by-step derivation

   1. unpow2 60.2%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

7. Simplified 60.2%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]
8. Step-by-step derivation

   1. div-inv 60.2%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{k \cdot k}{\ell}}} \]

   2. *-commutative 60.2%

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}}} \]

   3. associate-/l* 60.1%

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}} \]

   4. associate-/l* 60.6%

      \[\leadsto 2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}}} \]

9. Applied egg-rr 60.6%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 60.6%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}} \]

    2. metadata-eval 60.6%

       \[\leadsto \frac{\color{blue}{2}}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \]

    3. associate-/r* 60.6%

       \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{\ell}{k}}}}{\frac{k \cdot k}{\frac{\ell}{t}}}} \]

    4. associate-/r/ 60.6%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{\ell} \cdot k}}}{\frac{k \cdot k}{\frac{\ell}{t}}} \]

    5. associate-/l* 62.2%

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell} \cdot k}}{\color{blue}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}} \]

11. Simplified 62.2%

    \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell} \cdot k}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}}} \]

12. Final simplification 62.2%

    \[\leadsto \frac{\frac{2}{k \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\frac{\ell}{t}}{k}}} \]

Alternative 18: 32.6% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot -0.16666666666666666}{t}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (* k k)) (/ (* l -0.16666666666666666) t))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / (k * k)) * ((l * -0.16666666666666666) / t));
}

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 * ((l / (k * k)) * ((l * (-0.16666666666666666d0)) / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / (k * k)) * ((l * -0.16666666666666666) / t));
}

Python

def code(t, l, k):
	return 2.0 * ((l / (k * k)) * ((l * -0.16666666666666666) / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(k * k)) * Float64(Float64(l * -0.16666666666666666) / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k * k)) * ((l * -0.16666666666666666) / t));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * -0.16666666666666666), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 52.7%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 52.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 52.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 62.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. unpow2 62.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*l* 62.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   3. associate-*r* 62.8%

      \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   4. times-frac 71.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

   5. unpow2 71.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right) \]

6. Simplified 71.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 59.0%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell\right)}\right) \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) + \left(--0.3333333333333333\right) \cdot \ell\right)}\right) \]

   2. fma-def 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{{k}^{2}}\right)} + \left(--0.3333333333333333\right) \cdot \ell\right)\right) \]

   3. unpow2 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{\color{blue}{k \cdot k}}\right) + \left(--0.3333333333333333\right) \cdot \ell\right)\right) \]

   4. metadata-eval 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{k \cdot k}\right) + \color{blue}{0.3333333333333333} \cdot \ell\right)\right) \]

9. Simplified 59.0%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{k \cdot k}\right) + 0.3333333333333333 \cdot \ell\right)}\right) \]

10. Taylor expanded in k around inf 31.4%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. times-frac 31.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell}{t}\right)} \]

    2. unpow2 31.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell}{t}\right) \]

    3. distribute-rgt-out 31.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \left(-0.5 + 0.3333333333333333\right)}}{t}\right) \]

    4. metadata-eval 31.6%

       \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \color{blue}{-0.16666666666666666}}{t}\right) \]

12. Simplified 31.6%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot -0.16666666666666666}{t}\right)} \]

13. Final simplification 31.6%

    \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot -0.16666666666666666}{t}\right) \]

Alternative 19: 33.9% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (/ (/ l k) t) k) (* l -0.16666666666666666))))

C

double code(double t, double l, double k) {
	return 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
}

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 * ((((l / k) / t) / k) * (l * (-0.16666666666666666d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
}

Python

def code(t, l, k):
	return 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / t) / k) * Float64(l * -0.16666666666666666)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((((l / k) / t) / k) * (l * -0.16666666666666666));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] * N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. associate-*l* 50.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. sqr-neg 50.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 52.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 52.7%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 52.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 52.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 62.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. unpow2 62.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   2. associate-*l* 62.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   3. associate-*r* 62.8%

      \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   4. times-frac 71.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

   5. unpow2 71.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right) \]

6. Simplified 71.0%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 59.0%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell\right)}\right) \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) + \left(--0.3333333333333333\right) \cdot \ell\right)}\right) \]

   2. fma-def 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{{k}^{2}}\right)} + \left(--0.3333333333333333\right) \cdot \ell\right)\right) \]

   3. unpow2 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{\color{blue}{k \cdot k}}\right) + \left(--0.3333333333333333\right) \cdot \ell\right)\right) \]

   4. metadata-eval 59.0%

      \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{k \cdot k}\right) + \color{blue}{0.3333333333333333} \cdot \ell\right)\right) \]

9. Simplified 59.0%

   \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \ell, \frac{\ell}{k \cdot k}\right) + 0.3333333333333333 \cdot \ell\right)}\right) \]

10. Taylor expanded in k around inf 31.4%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 31.4%

       \[\leadsto 2 \cdot \frac{\ell \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    2. associate-*r* 32.6%

       \[\leadsto 2 \cdot \frac{\ell \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    3. associate-/l* 34.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell}}} \]

    4. associate-/r/ 34.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)\right)} \]

    5. associate-/r* 34.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{k \cdot t}} \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)\right) \]

    6. *-commutative 34.0%

       \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{k}}{\color{blue}{t \cdot k}} \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)\right) \]

    7. associate-/r* 32.5%

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\frac{\ell}{k}}{t}}{k}} \cdot \left(-0.5 \cdot \ell + 0.3333333333333333 \cdot \ell\right)\right) \]

    8. distribute-rgt-out 32.5%

       \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \color{blue}{\left(\ell \cdot \left(-0.5 + 0.3333333333333333\right)\right)}\right) \]

    9. metadata-eval 32.5%

       \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot \color{blue}{-0.16666666666666666}\right)\right) \]

12. Simplified 32.5%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]

13. Final simplification 32.5%

    \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right) \]

Reproduce

herbie shell --seed 2023279 
(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))))