Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 89.1%

Time: 20.3s

Alternatives: 17

Speedup: 28.1×

• 28.0% 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: 54.7% 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: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_3 := \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot t_3}{t_1}\right)}^{3} \cdot t_2}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t}{t_1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0))
        (t_2 (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
        (t_3 (cbrt (sin k))))
   (if (<= t -4.5e-46)
     (/ 2.0 (* (pow (/ (* t t_3) t_1) 3.0) t_2))
     (if (<= t 1.75e-18)
       (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))
       (/ 2.0 (* t_2 (pow (* (/ t t_1) (expm1 (log1p t_3))) 3.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double t_2 = tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)));
	double t_3 = cbrt(sin(k));
	double tmp;
	if (t <= -4.5e-46) {
		tmp = 2.0 / (pow(((t * t_3) / t_1), 3.0) * t_2);
	} else if (t <= 1.75e-18) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	} else {
		tmp = 2.0 / (t_2 * pow(((t / t_1) * expm1(log1p(t_3))), 3.0));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)));
	double t_3 = Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -4.5e-46) {
		tmp = 2.0 / (Math.pow(((t * t_3) / t_1), 3.0) * t_2);
	} else if (t <= 1.75e-18) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = 2.0 / (t_2 * Math.pow(((t / t_1) * Math.expm1(Math.log1p(t_3))), 3.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	t_2 = Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))
	t_3 = cbrt(sin(k))
	tmp = 0.0
	if (t <= -4.5e-46)
		tmp = Float64(2.0 / Float64((Float64(Float64(t * t_3) / t_1) ^ 3.0) * t_2));
	elseif (t <= 1.75e-18)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(Float64(t / t_1) * expm1(log1p(t_3))) ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -4.5e-46], N[(2.0 / N[(N[Power[N[(N[(t * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-18], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[(t / t$95$1), $MachinePrecision] * N[(Exp[N[Log[1 + t$95$3], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.50000000000000001e-46

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. +-commutative 66.3%

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

   3. Simplified 66.3%

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

      1. add-cube-cbrt 66.2%

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

      2. pow3 66.2%

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

      3. cbrt-prod 66.2%

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

      4. cbrt-div 66.2%

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

      5. rem-cbrt-cube 74.5%

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

      6. cbrt-prod 91.8%

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

      7. pow2 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*l/ 91.8%

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

   7. Applied egg-rr 91.8%

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

   if -4.50000000000000001e-46 < t < 1.7499999999999999e-18

   1. Initial program 48.2%

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

      1. *-commutative 48.2%

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

      2. associate-*l* 48.2%

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

      3. associate-*r* 48.2%

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

      4. +-commutative 48.2%

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

      5. associate-+r+ 48.2%

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

      6. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around inf 79.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in k around 0 78.2%

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

      1. unpow2 78.2%

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

      2. unpow2 78.2%

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

      3. times-frac 91.6%

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

   9. Simplified 91.6%

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

   if 1.7499999999999999e-18 < t

   1. Initial program 67.9%

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

      1. associate-*l* 67.9%

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

      2. +-commutative 67.9%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 67.8%

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

      3. cbrt-prod 67.8%

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

      4. cbrt-div 67.7%

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

      5. rem-cbrt-cube 75.8%

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

      6. cbrt-prod 91.3%

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

      7. pow2 91.3%

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

   5. Applied egg-rr 91.3%

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

      1. expm1-log1p-u 91.4%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Applied egg-rr 91.4%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \end{array} \]

Alternative 2: 82.6% accurate, 0.4× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= (* (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l)))) t_1) INFINITY)
     (/ 2.0 (* (* (tan k) t_1) (pow (* t (cbrt (/ (/ (sin k) l) l))) 3.0)))
     (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k)))))))

C

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

Java

k = Math.abs(k);
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 (((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / ((Math.tan(k) * t_1) * Math.pow((t * Math.cbrt(((Math.sin(k) / l) / l))), 3.0));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: k should be positive before calling this function
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[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Power[N[(t * N[Power[N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.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)) < +inf.0

   1. Initial program 85.9%

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

      1. associate-*l* 85.8%

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

      2. +-commutative 85.8%

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

   3. Simplified 85.8%

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

      1. add-cube-cbrt 85.8%

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

      2. pow3 85.8%

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

      3. cbrt-prod 85.7%

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

      4. cbrt-div 85.7%

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

      5. rem-cbrt-cube 90.1%

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

      6. cbrt-prod 93.1%

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

      7. pow2 93.1%

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

   5. Applied egg-rr 93.1%

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

      1. associate-*l/ 93.2%

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

   7. Applied egg-rr 93.2%

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

   8. Taylor expanded in t around 0 54.7%

      \[\leadsto \frac{2}{{\color{blue}{\left({\left(\frac{1 \cdot \sin k}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot t\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 54.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot {\left(\frac{1 \cdot \sin k}{{\ell}^{2}}\right)}^{0.3333333333333333}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. unpow1/3 89.1%

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

      3. *-lft-identity 89.1%

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

      4. unpow2 89.1%

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

      5. associate-/r* 91.7%

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

   10. Simplified 91.7%

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

   if +inf.0 < (*.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 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 49.4%

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

      1. associate-/l* 46.3%

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

      2. associate-/r/ 48.5%

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

      3. unpow2 48.5%

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

      4. unpow2 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in k around 0 48.5%

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

      1. unpow2 48.5%

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

      2. unpow2 48.5%

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

      3. times-frac 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 85.7%

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

Alternative 3: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
           (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))))
   (if (<= t_1 5e+266)
     t_1
     (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = 2.0 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k / t), 2.0))));
	double tmp;
	if (t_1 <= 5e+266) {
		tmp = t_1;
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / ((tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))))
    if (t_1 <= 5d+266) then
        tmp = t_1
    else
        tmp = 2.0d0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k / t), 2.0))));
	double tmp;
	if (t_1 <= 5e+266) {
		tmp = t_1;
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = 2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k / t), 2.0))))
	tmp = 0
	if t_1 <= 5e+266:
		tmp = t_1
	else:
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= 5e+266)
		tmp = t_1;
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = 2.0 / ((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k / t) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= 5e+266)
		tmp = t_1;
	else
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+266], t$95$1, N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $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))) < 4.9999999999999999e266

   1. Initial program 83.3%

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

   if 4.9999999999999999e266 < (/.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 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-*l* 26.5%

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

      3. associate-*r* 26.5%

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

      4. +-commutative 26.5%

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

      5. associate-+r+ 26.5%

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

      6. metadata-eval 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in k around inf 63.3%

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

      1. associate-/l* 61.0%

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

      2. associate-/r/ 62.5%

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

      3. unpow2 62.5%

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

      4. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in k around 0 62.5%

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

      1. unpow2 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 4: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot t_1} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_1\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l)))) t_1))
        5e+266)
     (/ 2.0 (* (* (tan k) t_1) (* (/ (sin k) l) (/ (pow t 3.0) l))))
     (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k)))))))

C

k = abs(k);
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 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+266) {
		tmp = 2.0 / ((tan(k) * t_1) * ((sin(k) / l) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))) * t_1)) <= 5d+266) then
        tmp = 2.0d0 / ((tan(k) * t_1) * ((sin(k) / l) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+266) {
		tmp = 2.0 / ((Math.tan(k) * t_1) * ((Math.sin(k) / l) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+266:
		tmp = 2.0 / ((math.tan(k) * t_1) * ((math.sin(k) / l) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
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(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * t_1)) <= 5e+266)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_1) * Float64(Float64(sin(k) / l) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * t_1)) <= 5e+266)
		tmp = 2.0 / ((tan(k) * t_1) * ((sin(k) / l) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 5e+266], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $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))) < 4.9999999999999999e266

   1. Initial program 83.3%

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

      1. associate-*l* 83.3%

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

      2. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in t around 0 83.3%

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

      1. *-commutative 83.3%

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

      2. unpow2 83.3%

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

      3. times-frac 87.1%

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

   6. Simplified 87.1%

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

   if 4.9999999999999999e266 < (/.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 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-*l* 26.5%

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

      3. associate-*r* 26.5%

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

      4. +-commutative 26.5%

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

      5. associate-+r+ 26.5%

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

      6. metadata-eval 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in k around inf 63.3%

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

      1. associate-/l* 61.0%

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

      2. associate-/r/ 62.5%

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

      3. unpow2 62.5%

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

      4. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in k around 0 62.5%

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

      1. unpow2 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+266}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 5: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-45} \lor \neg \left(t \leq 3.8 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.25e-45) (not (<= t 3.8e-18)))
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (pow (* (cbrt (sin k)) (/ t (pow (cbrt l) 2.0))) 3.0)))
   (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.25e-45) || !(t <= 3.8e-18)) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * pow((cbrt(sin(k)) * (t / pow(cbrt(l), 2.0))), 3.0));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.25e-45) || !(t <= 3.8e-18)) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))) * Math.pow((Math.cbrt(Math.sin(k)) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.25e-45) || !(t <= 3.8e-18))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))) * (Float64(cbrt(sin(k)) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -2.25e-45], N[Not[LessEqual[t, 3.8e-18]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2499999999999999e-45 or 3.7999999999999998e-18 < t

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. +-commutative 67.1%

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

   3. Simplified 67.1%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. cbrt-prod 67.0%

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

      4. cbrt-div 66.9%

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

      5. rem-cbrt-cube 75.2%

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

      6. cbrt-prod 91.6%

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

      7. pow2 91.6%

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

   5. Applied egg-rr 91.6%

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

   if -2.2499999999999999e-45 < t < 3.7999999999999998e-18

   1. Initial program 48.2%

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

      1. *-commutative 48.2%

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

      2. associate-*l* 48.2%

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

      3. associate-*r* 48.2%

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

      4. +-commutative 48.2%

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

      5. associate-+r+ 48.2%

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

      6. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around inf 79.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in k around 0 78.2%

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

      1. unpow2 78.2%

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

      2. unpow2 78.2%

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

      3. times-frac 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-45} \lor \neg \left(t \leq 3.8 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 6: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_3 := \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot t_3}{t_1}\right)}^{3} \cdot t_2}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(t_3 \cdot \frac{t}{t_1}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0))
        (t_2 (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
        (t_3 (cbrt (sin k))))
   (if (<= t -1.05e-43)
     (/ 2.0 (* (pow (/ (* t t_3) t_1) 3.0) t_2))
     (if (<= t 3e-17)
       (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))
       (/ 2.0 (* t_2 (pow (* t_3 (/ t t_1)) 3.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double t_2 = tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)));
	double t_3 = cbrt(sin(k));
	double tmp;
	if (t <= -1.05e-43) {
		tmp = 2.0 / (pow(((t * t_3) / t_1), 3.0) * t_2);
	} else if (t <= 3e-17) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	} else {
		tmp = 2.0 / (t_2 * pow((t_3 * (t / t_1)), 3.0));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)));
	double t_3 = Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -1.05e-43) {
		tmp = 2.0 / (Math.pow(((t * t_3) / t_1), 3.0) * t_2);
	} else if (t <= 3e-17) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = 2.0 / (t_2 * Math.pow((t_3 * (t / t_1)), 3.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	t_2 = Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))
	t_3 = cbrt(sin(k))
	tmp = 0.0
	if (t <= -1.05e-43)
		tmp = Float64(2.0 / Float64((Float64(Float64(t * t_3) / t_1) ^ 3.0) * t_2));
	elseif (t <= 3e-17)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(t_3 * Float64(t / t_1)) ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -1.05e-43], N[(2.0 / N[(N[Power[N[(N[(t * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-17], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(t$95$3 * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e-43

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. +-commutative 66.3%

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

   3. Simplified 66.3%

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

      1. add-cube-cbrt 66.2%

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

      2. pow3 66.2%

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

      3. cbrt-prod 66.2%

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

      4. cbrt-div 66.2%

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

      5. rem-cbrt-cube 74.5%

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

      6. cbrt-prod 91.8%

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

      7. pow2 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*l/ 91.8%

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

   7. Applied egg-rr 91.8%

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

   if -1.05e-43 < t < 3.00000000000000006e-17

   1. Initial program 48.2%

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

      1. *-commutative 48.2%

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

      2. associate-*l* 48.2%

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

      3. associate-*r* 48.2%

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

      4. +-commutative 48.2%

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

      5. associate-+r+ 48.2%

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

      6. metadata-eval 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around inf 79.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in k around 0 78.2%

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

      1. unpow2 78.2%

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

      2. unpow2 78.2%

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

      3. times-frac 91.6%

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

   9. Simplified 91.6%

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

   if 3.00000000000000006e-17 < t

   1. Initial program 67.9%

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

      1. associate-*l* 67.9%

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

      2. +-commutative 67.9%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.8%

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

      2. pow3 67.8%

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

      3. cbrt-prod 67.8%

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

      4. cbrt-div 67.7%

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

      5. rem-cbrt-cube 75.8%

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

      6. cbrt-prod 91.3%

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

      7. pow2 91.3%

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

   5. Applied egg-rr 91.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 7: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-45} \lor \neg \left(t \leq 2.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.55e-45) (not (<= t 2.2e+44)))
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.55e-45) || !(t <= 2.2e+44)) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.55e-45) || !(t <= 2.2e+44)) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.55e-45) || !(t <= 2.2e+44))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -2.55e-45], N[Not[LessEqual[t, 2.2e+44]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5499999999999999e-45 or 2.19999999999999996e44 < t

   1. Initial program 67.3%

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

      1. associate-*l* 67.3%

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

      2. +-commutative 67.3%

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

   3. Simplified 67.3%

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

      1. add-cube-cbrt 67.3%

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

      2. pow3 67.3%

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

      3. cbrt-div 67.2%

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

      4. rem-cbrt-cube 75.4%

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

      5. cbrt-prod 90.9%

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

      6. pow2 90.9%

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

   5. Applied egg-rr 90.9%

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

   if -2.5499999999999999e-45 < t < 2.19999999999999996e44

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 49.4%

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

      3. associate-*r* 49.4%

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

      4. +-commutative 49.4%

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

      5. associate-+r+ 49.4%

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

      6. metadata-eval 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in k around inf 78.4%

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

      1. associate-/l* 76.5%

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

      2. associate-/r/ 77.1%

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

      3. unpow2 77.1%

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

      4. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in k around 0 77.1%

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

      1. unpow2 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 90.2%

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

   9. Simplified 90.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{-45} \lor \neg \left(t \leq 2.2 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 8: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq -135:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{t_1}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\frac{t_1}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)) (t_2 (* (sin k) (tan k))))
   (if (<= t -135.0)
     (/ 2.0 (* (pow (* (cbrt (sin k)) (/ t t_1)) 3.0) (* 2.0 k)))
     (if (<= t 5.6e+42)
       (/ 2.0 (* (* t (* (/ k l) (/ k l))) t_2))
       (* (/ 2.0 (* t_2 (+ 2.0 (pow (/ k t) 2.0)))) (pow (/ t_1 t) 3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double t_2 = sin(k) * tan(k);
	double tmp;
	if (t <= -135.0) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t / t_1)), 3.0) * (2.0 * k));
	} else if (t <= 5.6e+42) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * t_2);
	} else {
		tmp = (2.0 / (t_2 * (2.0 + pow((k / t), 2.0)))) * pow((t_1 / t), 3.0);
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= -135.0) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t / t_1)), 3.0) * (2.0 * k));
	} else if (t <= 5.6e+42) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * t_2);
	} else {
		tmp = (2.0 / (t_2 * (2.0 + Math.pow((k / t), 2.0)))) * Math.pow((t_1 / t), 3.0);
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	t_2 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= -135.0)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t / t_1)) ^ 3.0) * Float64(2.0 * k)));
	elseif (t <= 5.6e+42)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * t_2));
	else
		tmp = Float64(Float64(2.0 / Float64(t_2 * Float64(2.0 + (Float64(k / t) ^ 2.0)))) * (Float64(t_1 / t) ^ 3.0));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -135.0], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+42], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$2 * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -135

   1. Initial program 65.4%

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

      1. associate-*l* 65.4%

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

      2. +-commutative 65.4%

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

   3. Simplified 65.4%

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

      1. add-cube-cbrt 65.3%

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

      2. pow3 65.3%

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

      3. cbrt-prod 65.3%

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

      4. cbrt-div 65.3%

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

      5. rem-cbrt-cube 74.5%

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

      6. cbrt-prod 91.1%

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

      7. pow2 91.1%

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

   5. Applied egg-rr 91.1%

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

   6. Taylor expanded in k around 0 84.6%

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

   if -135 < t < 5.5999999999999999e42

   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. *-commutative 50.5%

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

      2. associate-*l* 50.4%

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

      3. associate-*r* 50.4%

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

      4. +-commutative 50.4%

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

      5. associate-+r+ 50.4%

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

      6. metadata-eval 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in k around inf 77.9%

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

      1. associate-/l* 76.1%

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

      2. associate-/r/ 76.7%

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

      3. unpow2 76.7%

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

      4. unpow2 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in k around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

      3. times-frac 89.4%

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

   9. Simplified 89.4%

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

   if 5.5999999999999999e42 < t

   1. Initial program 68.6%

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

      1. associate-*l* 68.6%

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

      2. associate-/l/ 68.6%

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

      3. *-commutative 68.6%

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

      4. associate-*r/ 68.6%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 66.2%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.8%

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

      2. pow3 66.7%

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

      3. associate-*l/ 66.0%

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

      4. cbrt-div 65.9%

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

      5. cbrt-prod 66.8%

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

      6. pow2 66.8%

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

      7. rem-cbrt-cube 87.2%

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

   5. Applied egg-rr 87.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -135:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \end{array} \]

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -14.5 \lor \neg \left(t \leq 2.9 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -14.5) (not (<= t 2.9e+98)))
   (/ 2.0 (* (pow (* (cbrt (sin k)) (/ t (pow (cbrt l) 2.0))) 3.0) (* 2.0 k)))
   (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -14.5) || !(t <= 2.9e+98)) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -14.5) || !(t <= 2.9e+98)) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -14.5) || !(t <= 2.9e+98))
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -14.5], N[Not[LessEqual[t, 2.9e+98]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -14.5 or 2.9000000000000001e98 < t

   1. Initial program 61.5%

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

      1. associate-*l* 61.5%

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

      2. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. cbrt-prod 61.5%

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

      4. cbrt-div 61.4%

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

      5. rem-cbrt-cube 72.6%

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

      6. cbrt-prod 91.0%

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

      7. pow2 91.0%

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

   5. Applied egg-rr 91.0%

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

   6. Taylor expanded in k around 0 81.2%

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

   if -14.5 < t < 2.9000000000000001e98

   1. Initial program 55.0%

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

      1. *-commutative 55.0%

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

      2. associate-*l* 54.9%

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

      3. associate-*r* 54.9%

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

      4. +-commutative 54.9%

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

      5. associate-+r+ 54.9%

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

      6. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in k around inf 77.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in k around 0 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

      3. times-frac 89.3%

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

   9. Simplified 89.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -14.5 \lor \neg \left(t \leq 2.9 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 10: 77.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -430000000 \lor \neg \left(t \leq 7 \cdot 10^{+98}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -430000000.0) (not (<= t 7e+98)))
   (* (* l l) (/ 2.0 (* (* 2.0 k) (pow (* t (cbrt (sin k))) 3.0))))
   (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -430000000.0) || !(t <= 7e+98)) {
		tmp = (l * l) * (2.0 / ((2.0 * k) * pow((t * cbrt(sin(k))), 3.0)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -430000000.0) || !(t <= 7e+98)) {
		tmp = (l * l) * (2.0 / ((2.0 * k) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -430000000.0) || !(t <= 7e+98))
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t * cbrt(sin(k))) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -430000000.0], N[Not[LessEqual[t, 7e+98]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3e8 or 7e98 < t

   1. Initial program 62.2%

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

      1. associate-/l/ 62.2%

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

      2. associate-*l/ 62.2%

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

      3. associate-*l/ 61.2%

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

      4. associate-/r/ 61.2%

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

      5. *-commutative 61.2%

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

      6. associate-/l/ 61.2%

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

      7. associate-*r* 61.2%

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

      8. *-commutative 61.2%

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

      9. associate-*r* 61.2%

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

      10. *-commutative 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

   5. Applied egg-rr 75.0%

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

      1. cube-prod 71.8%

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

      2. rem-cube-cbrt 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in k around 0 71.9%

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

      1. *-commutative 71.9%

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

   10. Simplified 71.9%

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

   if -4.3e8 < t < 7e98

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.6%

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

      3. associate-*r* 54.6%

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

      4. +-commutative 54.6%

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

      5. associate-+r+ 54.6%

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

      6. metadata-eval 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in k around inf 77.2%

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

      1. associate-/l* 76.2%

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

      2. associate-/r/ 76.7%

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

      3. unpow2 76.7%

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

      4. unpow2 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in k around 0 76.7%

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

      1. unpow2 76.7%

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

      2. unpow2 76.7%

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

      3. times-frac 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -430000000 \lor \neg \left(t \leq 7 \cdot 10^{+98}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 11: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+147}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left(k \cdot {t}^{3}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -4e+147)
   (* (* l l) (/ 2.0 (* (tan k) (* 2.0 (* k (pow t 3.0))))))
   (if (<= t 1.25e+99)
     (/ 2.0 (* (* t (* (/ k l) (/ k l))) (* (sin k) (tan k))))
     (* (pow (/ (cbrt l) t) 3.0) (/ l (* k k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+147) {
		tmp = (l * l) * (2.0 / (tan(k) * (2.0 * (k * pow(t, 3.0)))));
	} else if (t <= 1.25e+99) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (sin(k) * tan(k)));
	} else {
		tmp = pow((cbrt(l) / t), 3.0) * (l / (k * k));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+147) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (2.0 * (k * Math.pow(t, 3.0)))));
	} else if (t <= 1.25e+99) {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * (l / (k * k));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -4e+147)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(2.0 * Float64(k * (t ^ 3.0))))));
	elseif (t <= 1.25e+99)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(l / Float64(k * k)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -4e+147], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+99], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.9999999999999999e147

   1. Initial program 74.7%

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

      1. associate-/l/ 74.7%

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

      2. associate-*l/ 74.7%

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

      3. associate-*l/ 74.7%

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

      4. associate-/r/ 74.7%

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

      5. *-commutative 74.7%

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

      6. associate-/l/ 74.7%

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

      7. associate-*r* 74.7%

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

      8. *-commutative 74.7%

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

      9. associate-*r* 74.7%

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

      10. *-commutative 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in k around 0 74.7%

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

   if -3.9999999999999999e147 < t < 1.25000000000000002e99

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-*l* 53.8%

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

      3. associate-*r* 53.8%

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

      4. +-commutative 53.8%

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

      5. associate-+r+ 53.8%

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

      6. metadata-eval 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in k around inf 74.7%

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

      1. associate-/l* 73.8%

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

      2. associate-/r/ 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

   6. Simplified 74.2%

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

   7. Taylor expanded in k around 0 74.2%

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

      1. unpow2 74.2%

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

      2. unpow2 74.2%

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

      3. times-frac 84.7%

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

   9. Simplified 84.7%

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

   if 1.25000000000000002e99 < t

   1. Initial program 55.2%

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

      1. associate-/l/ 55.2%

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

      2. associate-*l/ 55.2%

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

      3. associate-*l/ 55.2%

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

      4. associate-/r/ 55.3%

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

      5. *-commutative 55.3%

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

      6. associate-/l/ 55.3%

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

      7. associate-*r* 55.3%

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

      8. *-commutative 55.3%

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

      9. associate-*r* 55.3%

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

      10. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in k around 0 51.7%

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

      1. unpow2 51.7%

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

      2. *-commutative 51.7%

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

      3. times-frac 52.7%

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

      4. unpow2 52.7%

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

   6. Simplified 52.7%

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

      1. add-cube-cbrt 52.7%

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

      2. pow2 52.7%

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

      3. cbrt-div 52.7%

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

      4. rem-cbrt-cube 52.7%

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

      5. cbrt-div 52.7%

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

      6. rem-cbrt-cube 63.3%

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

   8. Applied egg-rr 63.3%

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

      1. pow-plus 63.3%

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

      2. metadata-eval 63.3%

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

   10. Simplified 63.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+147}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left(k \cdot {t}^{3}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 12: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.52 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.52e-6)
   (* (pow (/ (cbrt l) t) 3.0) (/ l (* k k)))
   (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.52e-6) {
		tmp = pow((cbrt(l) / t), 3.0) * (l / (k * k));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.52e-6) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * (l / (k * k));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.52e-6)
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.52e-6], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.52000000000000006e-6

   1. Initial program 60.3%

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

      1. associate-/l/ 60.3%

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

      2. associate-*l/ 60.3%

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

      3. associate-*l/ 59.3%

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

      4. associate-/r/ 58.8%

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

      5. *-commutative 58.8%

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

      6. associate-/l/ 58.8%

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

      7. associate-*r* 59.3%

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

      8. *-commutative 59.3%

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

      9. associate-*r* 59.3%

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

      10. *-commutative 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. unpow2 57.3%

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

      2. *-commutative 57.3%

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

      3. times-frac 60.4%

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

      4. unpow2 60.4%

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

   6. Simplified 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow2 60.4%

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

      3. cbrt-div 60.3%

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

      4. rem-cbrt-cube 60.3%

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

      5. cbrt-div 60.3%

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

      6. rem-cbrt-cube 64.9%

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

   8. Applied egg-rr 64.9%

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

      1. pow-plus 64.9%

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

      2. metadata-eval 64.9%

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

   10. Simplified 64.9%

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

   if 1.52000000000000006e-6 < k

   1. Initial program 47.4%

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

      1. associate-/l/ 47.4%

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

      2. associate-*l/ 47.4%

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

      3. associate-*l/ 47.4%

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

      4. associate-/r/ 46.6%

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

      5. *-commutative 46.6%

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

      6. associate-/l/ 45.9%

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

      7. associate-*r* 45.9%

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

      8. *-commutative 45.9%

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

      9. associate-*r* 45.9%

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

      10. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in k around inf 66.6%

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

      1. unpow2 66.6%

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

      2. *-commutative 66.6%

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

   6. Simplified 66.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.52 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \]

Alternative 13: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq 205000000000:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(t_1 \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))))
   (if (<= k 205000000000.0)
     (* (pow (/ (cbrt l) t) 3.0) t_1)
     (* 2.0 (* (/ (cos k) (* k k)) (* t_1 (/ l t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (k <= 205000000000.0) {
		tmp = pow((cbrt(l) / t), 3.0) * t_1;
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (t_1 * (l / t)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (k <= 205000000000.0) {
		tmp = Math.pow((Math.cbrt(l) / t), 3.0) * t_1;
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (t_1 * (l / t)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (k <= 205000000000.0)
		tmp = Float64((Float64(cbrt(l) / t) ^ 3.0) * t_1);
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(t_1 * Float64(l / t))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 205000000000.0], N[(N[Power[N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.05e11

   1. Initial program 60.4%

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

      1. associate-/l/ 60.4%

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

      2. associate-*l/ 60.4%

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

      3. associate-*l/ 59.4%

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

      4. associate-/r/ 59.0%

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

      5. *-commutative 59.0%

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

      6. associate-/l/ 59.0%

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

      7. associate-*r* 59.4%

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

      8. *-commutative 59.4%

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

      9. associate-*r* 59.4%

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

      10. *-commutative 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in k around 0 58.0%

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

      1. unpow2 58.0%

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

      2. *-commutative 58.0%

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

      3. times-frac 61.0%

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

      4. unpow2 61.0%

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

   6. Simplified 61.0%

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

      1. add-cube-cbrt 61.0%

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

      2. pow2 61.0%

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

      3. cbrt-div 60.9%

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

      4. rem-cbrt-cube 60.9%

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

      5. cbrt-div 60.9%

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

      6. rem-cbrt-cube 65.4%

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

   8. Applied egg-rr 65.4%

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

      1. pow-plus 65.4%

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

      2. metadata-eval 65.4%

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

   10. Simplified 65.4%

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

   if 2.05e11 < k

   1. Initial program 46.4%

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

      1. associate-/l/ 46.4%

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

      2. associate-*l/ 46.4%

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

      3. associate-*l/ 46.4%

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

      4. associate-/r/ 45.6%

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

      5. *-commutative 45.6%

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

      6. associate-/l/ 44.8%

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

      7. associate-*r* 44.8%

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

      8. *-commutative 44.8%

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

      9. associate-*r* 44.8%

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

      10. *-commutative 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in k around inf 66.5%

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

      1. times-frac 63.3%

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

      2. unpow2 63.3%

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

      3. unpow2 63.3%

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

      4. *-commutative 63.3%

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

      5. times-frac 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

      2. times-frac 64.1%

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

      3. unpow2 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 205000000000:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell}}{t}\right)}^{3} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 14: 60.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 1.08 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 220000000000:\\ \;\;\;\;\frac{2}{2 \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\sin k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) (/ k l))))
   (if (<= k 1.08e-135)
     (/ 2.0 (* t_1 (* k (* t k))))
     (if (<= k 220000000000.0)
       (* (/ 2.0 (* 2.0 (* k k))) (* l (/ l (pow t 3.0))))
       (/ 2.0 (* t_1 (* (sin k) (* t k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (k / l) * (k / l);
	double tmp;
	if (k <= 1.08e-135) {
		tmp = 2.0 / (t_1 * (k * (t * k)));
	} else if (k <= 220000000000.0) {
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / pow(t, 3.0)));
	} else {
		tmp = 2.0 / (t_1 * (sin(k) * (t * k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * (k / l)
    if (k <= 1.08d-135) then
        tmp = 2.0d0 / (t_1 * (k * (t * k)))
    else if (k <= 220000000000.0d0) then
        tmp = (2.0d0 / (2.0d0 * (k * k))) * (l * (l / (t ** 3.0d0)))
    else
        tmp = 2.0d0 / (t_1 * (sin(k) * (t * k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (k / l) * (k / l);
	double tmp;
	if (k <= 1.08e-135) {
		tmp = 2.0 / (t_1 * (k * (t * k)));
	} else if (k <= 220000000000.0) {
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / (t_1 * (Math.sin(k) * (t * k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (k / l) * (k / l)
	tmp = 0
	if k <= 1.08e-135:
		tmp = 2.0 / (t_1 * (k * (t * k)))
	elif k <= 220000000000.0:
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / math.pow(t, 3.0)))
	else:
		tmp = 2.0 / (t_1 * (math.sin(k) * (t * k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(k / l) * Float64(k / l))
	tmp = 0.0
	if (k <= 1.08e-135)
		tmp = Float64(2.0 / Float64(t_1 * Float64(k * Float64(t * k))));
	elseif (k <= 220000000000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 * Float64(k * k))) * Float64(l * Float64(l / (t ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(sin(k) * Float64(t * k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (k / l) * (k / l);
	tmp = 0.0;
	if (k <= 1.08e-135)
		tmp = 2.0 / (t_1 * (k * (t * k)));
	elseif (k <= 220000000000.0)
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / (t ^ 3.0)));
	else
		tmp = 2.0 / (t_1 * (sin(k) * (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.08e-135], N[(2.0 / N[(t$95$1 * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 220000000000.0], N[(N[(2.0 / N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0799999999999999e-135

   1. Initial program 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-*l* 55.4%

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

      3. associate-*r* 55.4%

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

      4. +-commutative 55.4%

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

      5. associate-+r+ 55.4%

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

      6. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in k around inf 69.7%

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

      1. associate-/l* 68.1%

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

      2. associate-/r/ 67.6%

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

      3. unpow2 67.6%

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

      4. unpow2 67.6%

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

   6. Simplified 67.6%

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

      1. pow1 67.6%

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

      2. associate-*l* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. unpow1 67.1%

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

      2. times-frac 72.2%

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

      3. *-commutative 72.2%

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

      4. associate-*r* 72.2%

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

   10. Simplified 72.2%

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

   11. Taylor expanded in k around 0 63.8%

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

       1. unpow2 63.8%

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

       2. associate-*l* 63.8%

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

   13. Simplified 63.8%

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

   if 1.0799999999999999e-135 < k < 2.2e11

   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-*l* 61.9%

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

      2. associate-/l/ 62.0%

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

      3. *-commutative 62.0%

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

      4. associate-*r/ 62.0%

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

      5. associate-/l* 62.0%

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

      6. associate-/r/ 62.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in k around 0 84.9%

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

      1. unpow2 84.9%

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

   6. Simplified 84.9%

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

   if 2.2e11 < k

   1. Initial program 46.4%

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

      1. *-commutative 46.4%

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

      2. associate-*l* 46.4%

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

      3. associate-*r* 46.3%

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

      4. +-commutative 46.3%

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

      5. associate-+r+ 46.3%

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

      6. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in k around inf 66.6%

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

      1. associate-/l* 63.3%

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

      2. associate-/r/ 66.6%

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

      3. unpow2 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. pow1 66.6%

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

      2. associate-*l* 66.6%

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

   8. Applied egg-rr 66.6%

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

      1. unpow1 66.6%

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

      2. times-frac 86.1%

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

      3. *-commutative 86.1%

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

      4. associate-*r* 86.2%

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

   10. Simplified 86.2%

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

   11. Taylor expanded in k around 0 62.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 220000000000:\\ \;\;\;\;\frac{2}{2 \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\sin k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]

Alternative 15: 61.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 1.08 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 8800000000000:\\ \;\;\;\;\frac{2}{2 \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot t_1\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) (/ k l))))
   (if (<= k 1.08e-135)
     (/ 2.0 (* t_1 (* k (* t k))))
     (if (<= k 8800000000000.0)
       (* (/ 2.0 (* 2.0 (* k k))) (* l (/ l (pow t 3.0))))
       (/ 2.0 (* (* t t_1) (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (k / l) * (k / l);
	double tmp;
	if (k <= 1.08e-135) {
		tmp = 2.0 / (t_1 * (k * (t * k)));
	} else if (k <= 8800000000000.0) {
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((t * t_1) * (k * k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * (k / l)
    if (k <= 1.08d-135) then
        tmp = 2.0d0 / (t_1 * (k * (t * k)))
    else if (k <= 8800000000000.0d0) then
        tmp = (2.0d0 / (2.0d0 * (k * k))) * (l * (l / (t ** 3.0d0)))
    else
        tmp = 2.0d0 / ((t * t_1) * (k * k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (k / l) * (k / l);
	double tmp;
	if (k <= 1.08e-135) {
		tmp = 2.0 / (t_1 * (k * (t * k)));
	} else if (k <= 8800000000000.0) {
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((t * t_1) * (k * k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (k / l) * (k / l)
	tmp = 0
	if k <= 1.08e-135:
		tmp = 2.0 / (t_1 * (k * (t * k)))
	elif k <= 8800000000000.0:
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / math.pow(t, 3.0)))
	else:
		tmp = 2.0 / ((t * t_1) * (k * k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(k / l) * Float64(k / l))
	tmp = 0.0
	if (k <= 1.08e-135)
		tmp = Float64(2.0 / Float64(t_1 * Float64(k * Float64(t * k))));
	elseif (k <= 8800000000000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 * Float64(k * k))) * Float64(l * Float64(l / (t ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * t_1) * Float64(k * k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (k / l) * (k / l);
	tmp = 0.0;
	if (k <= 1.08e-135)
		tmp = 2.0 / (t_1 * (k * (t * k)));
	elseif (k <= 8800000000000.0)
		tmp = (2.0 / (2.0 * (k * k))) * (l * (l / (t ^ 3.0)));
	else
		tmp = 2.0 / ((t * t_1) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.08e-135], N[(2.0 / N[(t$95$1 * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8800000000000.0], N[(N[(2.0 / N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * t$95$1), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0799999999999999e-135

   1. Initial program 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-*l* 55.4%

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

      3. associate-*r* 55.4%

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

      4. +-commutative 55.4%

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

      5. associate-+r+ 55.4%

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

      6. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in k around inf 69.7%

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

      1. associate-/l* 68.1%

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

      2. associate-/r/ 67.6%

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

      3. unpow2 67.6%

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

      4. unpow2 67.6%

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

   6. Simplified 67.6%

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

      1. pow1 67.6%

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

      2. associate-*l* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. unpow1 67.1%

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

      2. times-frac 72.2%

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

      3. *-commutative 72.2%

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

      4. associate-*r* 72.2%

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

   10. Simplified 72.2%

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

   11. Taylor expanded in k around 0 63.8%

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

       1. unpow2 63.8%

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

       2. associate-*l* 63.8%

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

   13. Simplified 63.8%

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

   if 1.0799999999999999e-135 < k < 8.8e12

   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-*l* 61.9%

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

      2. associate-/l/ 62.0%

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

      3. *-commutative 62.0%

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

      4. associate-*r/ 62.0%

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

      5. associate-/l* 62.0%

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

      6. associate-/r/ 62.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in k around 0 84.9%

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

      1. unpow2 84.9%

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

   6. Simplified 84.9%

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

   if 8.8e12 < k

   1. Initial program 46.4%

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

      1. *-commutative 46.4%

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

      2. associate-*l* 46.4%

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

      3. associate-*r* 46.3%

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

      4. +-commutative 46.3%

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

      5. associate-+r+ 46.3%

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

      6. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in k around inf 66.6%

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

      1. associate-/l* 63.3%

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

      2. associate-/r/ 66.6%

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

      3. unpow2 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in k around 0 66.6%

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

      1. unpow2 66.6%

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

      2. unpow2 66.6%

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

      3. times-frac 86.1%

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

   9. Simplified 86.1%

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

   10. Taylor expanded in k around 0 54.8%

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

       1. unpow2 54.8%

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

   12. Simplified 54.8%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 8800000000000:\\ \;\;\;\;\frac{2}{2 \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 16: 61.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((cos(k) / (k * k)) * ((l / (k * k)) * (l / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. associate-/l/ 57.3%

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

   2. associate-*l/ 57.3%

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

   3. associate-*l/ 56.6%

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

   4. associate-/r/ 56.0%

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

   5. *-commutative 56.0%

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

   6. associate-/l/ 55.9%

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

   7. associate-*r* 56.2%

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

   8. *-commutative 56.2%

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

   9. associate-*r* 56.2%

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

   10. *-commutative 56.2%

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

3. Simplified 56.2%

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

4. Taylor expanded in k around inf 66.3%

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

   1. times-frac 66.4%

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

   2. unpow2 66.4%

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

   3. unpow2 66.4%

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

   4. *-commutative 66.4%

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

   5. times-frac 70.4%

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

6. Simplified 70.4%

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

7. Taylor expanded in k around 0 63.2%

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

   1. unpow2 63.2%

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

   2. times-frac 67.1%

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

   3. unpow2 67.1%

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

9. Simplified 67.1%

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

10. Final simplification 67.1%

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

Alternative 17: 58.2% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((t * ((k / l) * (k / l))) * (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. *-commutative 57.3%

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

   2. associate-*l* 54.0%

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

   3. associate-*r* 54.0%

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

   4. +-commutative 54.0%

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

   5. associate-+r+ 54.0%

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

   6. metadata-eval 54.0%

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

3. Simplified 54.0%

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

4. Taylor expanded in k around inf 68.2%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 66.8%

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

   3. unpow2 66.8%

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

   4. unpow2 66.8%

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

6. Simplified 66.8%

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

7. Taylor expanded in k around 0 66.8%

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

   1. unpow2 66.8%

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

   2. unpow2 66.8%

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

   3. times-frac 75.3%

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

9. Simplified 75.3%

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

10. Taylor expanded in k around 0 63.1%

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

    1. unpow2 63.1%

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

12. Simplified 63.1%

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

13. Final simplification 63.1%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))