Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 86.5%

Time: 33.0s

Alternatives: 23

Speedup: 3.8×

• 39.7% 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: 35.5% 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: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := \frac{\frac{\frac{\sqrt{2}}{t\_1}}{k}}{t\_2}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(t\_3 \cdot {\left(t \cdot \left(t\_1 \cdot {\left(\sqrt[3]{t\_2}\right)}^{3}\right)\right)}^{-2}\right)\right) \cdot t\_4\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(t \cdot {\left(\sqrt[3]{t\_3 \cdot {\left(t\_2 \cdot \left(t \cdot t\_1\right)\right)}^{-2}}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (/ (/ (/ (sqrt 2.0) t_1) k) t_2)))
   (if (<= (* l l) 0.0)
     (* (* t (* t_3 (pow (* t (* t_1 (pow (cbrt t_2) 3.0))) -2.0))) t_4)
     (if (<= (* l l) 1e+225)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (* t_4 (* t (pow (cbrt (* t_3 (pow (* t_2 (* t t_1)) -2.0))) 3.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = ((sqrt(2.0) / t_1) / k) / t_2;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * (t_3 * pow((t * (t_1 * pow(cbrt(t_2), 3.0))), -2.0))) * t_4;
	} else if ((l * l) <= 1e+225) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else {
		tmp = t_4 * (t * pow(cbrt((t_3 * pow((t_2 * (t * t_1)), -2.0))), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = ((Math.sqrt(2.0) / t_1) / k) / t_2;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * (t_3 * Math.pow((t * (t_1 * Math.pow(Math.cbrt(t_2), 3.0))), -2.0))) * t_4;
	} else if ((l * l) <= 1e+225) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else {
		tmp = t_4 * (t * Math.pow(Math.cbrt((t_3 * Math.pow((t_2 * (t * t_1)), -2.0))), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(Float64(Float64(sqrt(2.0) / t_1) / k) / t_2)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t * Float64(t_3 * (Float64(t * Float64(t_1 * (cbrt(t_2) ^ 3.0))) ^ -2.0))) * t_4);
	elseif (Float64(l * l) <= 1e+225)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	else
		tmp = Float64(t_4 * Float64(t * (cbrt(Float64(t_3 * (Float64(t_2 * Float64(t * t_1)) ^ -2.0))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / k), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t * N[(t$95$3 * N[Power[N[(t * N[(t$95$1 * N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+225], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(t * N[Power[N[Power[N[(t$95$3 * N[Power[N[(t$95$2 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 19.0%

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

      1. *-commutative 19.0%

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

      2. associate-/r* 19.0%

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

   3. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 24.0%

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

      2. add-cube-cbrt 24.0%

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

      3. times-frac 24.0%

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

   6. Applied egg-rr 81.4%

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

      1. associate-/r/ 81.4%

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

      2. associate-/r* 81.4%

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

      3. associate-/r/ 81.4%

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

   8. Simplified 81.4%

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

      1. associate-/l* 81.4%

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

      2. div-inv 81.4%

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

      3. pow-flip 81.4%

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

      4. metadata-eval 81.4%

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

   10. Applied egg-rr 81.4%

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

       1. associate-*l/ 81.4%

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

       2. associate-/r* 81.4%

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

       3. *-inverses 81.4%

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

       4. associate-*r/ 81.4%

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

   12. Simplified 81.4%

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

       1. div-inv 81.4%

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

       2. pow-flip 81.4%

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

       3. div-inv 81.4%

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

       4. pow-flip 81.4%

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

       5. metadata-eval 81.4%

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

       6. metadata-eval 81.4%

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

   14. Applied egg-rr 81.4%

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

       1. *-commutative 81.4%

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

       2. associate-*l* 87.5%

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

       3. associate-*l* 90.7%

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

   16. Simplified 90.7%

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

       1. add-cube-cbrt 90.7%

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

       2. pow3 90.7%

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

   18. Applied egg-rr 90.7%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/r* 46.1%

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

   3. Simplified 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.4%

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

      2. associate-/l/ 52.3%

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

      3. div-inv 52.3%

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

      4. *-commutative 52.3%

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

      5. associate-*l/ 53.0%

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

      6. associate-/l* 53.0%

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

      7. pow2 53.0%

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

   6. Applied egg-rr 53.0%

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

      1. associate-*r/ 53.0%

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

      2. metadata-eval 53.0%

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

      3. *-commutative 53.0%

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

      4. *-commutative 53.0%

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

      5. associate-/r/ 52.6%

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

      6. associate-/l* 52.6%

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

      7. associate-*l* 52.6%

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

   8. Simplified 52.6%

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

   9. Taylor expanded in k around inf 88.8%

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

       1. times-frac 93.3%

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

   11. Simplified 93.3%

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

   if 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-/r* 36.6%

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

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 36.7%

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

      2. add-cube-cbrt 36.7%

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

      3. times-frac 36.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/r/ 79.1%

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

      2. associate-/r* 79.1%

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

      3. associate-/r/ 79.1%

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

   8. Simplified 79.1%

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

      1. associate-/l* 79.1%

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

      2. div-inv 79.1%

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

      3. pow-flip 79.1%

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

      4. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. associate-*l/ 79.1%

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

       2. associate-/r* 79.1%

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

       3. *-inverses 79.1%

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

       4. associate-*r/ 79.1%

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

   12. Simplified 79.1%

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

       1. div-inv 79.1%

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

       2. pow-flip 79.1%

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

       3. div-inv 79.1%

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

       4. pow-flip 79.1%

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

       5. metadata-eval 79.1%

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

       6. metadata-eval 79.1%

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

   14. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

       2. associate-*l* 86.2%

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

       3. associate-*l* 86.3%

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

   16. Simplified 86.3%

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

       1. add-cube-cbrt 86.2%

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

       2. pow3 86.2%

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

       3. associate-*r* 86.3%

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

   18. Applied egg-rr 86.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t \cdot {\left(\sqrt[3]{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}}\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := \frac{\frac{\frac{\sqrt{2}}{t\_1}}{k}}{t\_2}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(t\_3 \cdot {\left(t \cdot \left(t\_1 \cdot {\left(\sqrt[3]{t\_2}\right)}^{3}\right)\right)}^{-2}\right)\right) \cdot t\_4\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(t \cdot \left(t\_3 \cdot {\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (/ (/ (/ (sqrt 2.0) t_1) k) t_2)))
   (if (<= (* l l) 0.0)
     (* (* t (* t_3 (pow (* t (* t_1 (pow (cbrt t_2) 3.0))) -2.0))) t_4)
     (if (<= (* l l) 1e+225)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (* t_4 (* t (* t_3 (pow (* t (* t_1 t_2)) -2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = ((sqrt(2.0) / t_1) / k) / t_2;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * (t_3 * pow((t * (t_1 * pow(cbrt(t_2), 3.0))), -2.0))) * t_4;
	} else if ((l * l) <= 1e+225) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else {
		tmp = t_4 * (t * (t_3 * pow((t * (t_1 * t_2)), -2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = ((Math.sqrt(2.0) / t_1) / k) / t_2;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * (t_3 * Math.pow((t * (t_1 * Math.pow(Math.cbrt(t_2), 3.0))), -2.0))) * t_4;
	} else if ((l * l) <= 1e+225) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else {
		tmp = t_4 * (t * (t_3 * Math.pow((t * (t_1 * t_2)), -2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(Float64(Float64(sqrt(2.0) / t_1) / k) / t_2)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t * Float64(t_3 * (Float64(t * Float64(t_1 * (cbrt(t_2) ^ 3.0))) ^ -2.0))) * t_4);
	elseif (Float64(l * l) <= 1e+225)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	else
		tmp = Float64(t_4 * Float64(t * Float64(t_3 * (Float64(t * Float64(t_1 * t_2)) ^ -2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / k), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t * N[(t$95$3 * N[Power[N[(t * N[(t$95$1 * N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+225], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(t * N[(t$95$3 * N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 19.0%

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

      1. *-commutative 19.0%

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

      2. associate-/r* 19.0%

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

   3. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 24.0%

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

      2. add-cube-cbrt 24.0%

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

      3. times-frac 24.0%

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

   6. Applied egg-rr 81.4%

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

      1. associate-/r/ 81.4%

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

      2. associate-/r* 81.4%

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

      3. associate-/r/ 81.4%

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

   8. Simplified 81.4%

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

      1. associate-/l* 81.4%

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

      2. div-inv 81.4%

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

      3. pow-flip 81.4%

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

      4. metadata-eval 81.4%

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

   10. Applied egg-rr 81.4%

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

       1. associate-*l/ 81.4%

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

       2. associate-/r* 81.4%

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

       3. *-inverses 81.4%

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

       4. associate-*r/ 81.4%

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

   12. Simplified 81.4%

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

       1. div-inv 81.4%

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

       2. pow-flip 81.4%

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

       3. div-inv 81.4%

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

       4. pow-flip 81.4%

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

       5. metadata-eval 81.4%

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

       6. metadata-eval 81.4%

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

   14. Applied egg-rr 81.4%

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

       1. *-commutative 81.4%

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

       2. associate-*l* 87.5%

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

       3. associate-*l* 90.7%

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

   16. Simplified 90.7%

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

       1. add-cube-cbrt 90.7%

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

       2. pow3 90.7%

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

   18. Applied egg-rr 90.7%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/r* 46.1%

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

   3. Simplified 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.4%

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

      2. associate-/l/ 52.3%

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

      3. div-inv 52.3%

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

      4. *-commutative 52.3%

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

      5. associate-*l/ 53.0%

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

      6. associate-/l* 53.0%

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

      7. pow2 53.0%

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

   6. Applied egg-rr 53.0%

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

      1. associate-*r/ 53.0%

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

      2. metadata-eval 53.0%

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

      3. *-commutative 53.0%

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

      4. *-commutative 53.0%

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

      5. associate-/r/ 52.6%

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

      6. associate-/l* 52.6%

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

      7. associate-*l* 52.6%

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

   8. Simplified 52.6%

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

   9. Taylor expanded in k around inf 88.8%

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

       1. times-frac 93.3%

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

   11. Simplified 93.3%

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

   if 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-/r* 36.6%

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

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 36.7%

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

      2. add-cube-cbrt 36.7%

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

      3. times-frac 36.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/r/ 79.1%

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

      2. associate-/r* 79.1%

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

      3. associate-/r/ 79.1%

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

   8. Simplified 79.1%

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

      1. associate-/l* 79.1%

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

      2. div-inv 79.1%

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

      3. pow-flip 79.1%

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

      4. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. associate-*l/ 79.1%

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

       2. associate-/r* 79.1%

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

       3. *-inverses 79.1%

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

       4. associate-*r/ 79.1%

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

   12. Simplified 79.1%

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

       1. div-inv 79.1%

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

       2. pow-flip 79.1%

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

       3. div-inv 79.1%

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

       4. pow-flip 79.1%

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

       5. metadata-eval 79.1%

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

       6. metadata-eval 79.1%

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

   14. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

       2. associate-*l* 86.2%

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

       3. associate-*l* 86.3%

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

   16. Simplified 86.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+225}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+225}\right):\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{t\_1}}{k}}{t\_2} \cdot \left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0)) (t_2 (cbrt (* (sin k) (tan k)))))
   (if (or (<= (* l l) 0.0) (not (<= (* l l) 1e+225)))
     (*
      (/ (/ (/ (sqrt 2.0) t_1) k) t_2)
      (* t (* (/ (sqrt 2.0) k) (pow (* t (* t_1 t_2)) -2.0))))
     (/
      2.0
      (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+225)) {
		tmp = (((sqrt(2.0) / t_1) / k) / t_2) * (t * ((sqrt(2.0) / k) * pow((t * (t_1 * t_2)), -2.0)));
	} else {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+225)) {
		tmp = (((Math.sqrt(2.0) / t_1) / k) / t_2) * (t * ((Math.sqrt(2.0) / k) * Math.pow((t * (t_1 * t_2)), -2.0)));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if ((Float64(l * l) <= 0.0) || !(Float64(l * l) <= 1e+225))
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / t_1) / k) / t_2) * Float64(t * Float64(Float64(sqrt(2.0) / k) * (Float64(t * Float64(t_1 * t_2)) ^ -2.0))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(l * l), $MachinePrecision], 1e+225]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / k), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0 or 9.99999999999999928e224 < (*.f64 l l)

   1. Initial program 29.5%

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

      1. *-commutative 29.5%

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

      2. associate-/r* 29.5%

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

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 31.5%

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

      2. add-cube-cbrt 31.5%

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

      3. times-frac 31.5%

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

   6. Applied egg-rr 80.0%

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

      1. associate-/r/ 80.0%

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

      2. associate-/r* 80.0%

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

      3. associate-/r/ 80.0%

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

   8. Simplified 80.0%

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

      1. associate-/l* 80.0%

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

      2. div-inv 80.0%

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

      3. pow-flip 80.0%

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

      4. metadata-eval 80.0%

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

   10. Applied egg-rr 80.0%

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

       1. associate-*l/ 80.0%

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

       2. associate-/r* 80.0%

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

       3. *-inverses 80.0%

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

       4. associate-*r/ 80.0%

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

   12. Simplified 80.0%

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

       1. div-inv 80.0%

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

       2. pow-flip 80.0%

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

       3. div-inv 80.0%

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

       4. pow-flip 80.0%

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

       5. metadata-eval 80.0%

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

       6. metadata-eval 80.0%

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

   14. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

       2. associate-*l* 86.8%

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

       3. associate-*l* 88.1%

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

   16. Simplified 88.1%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999928e224

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

      2. associate-/r* 46.1%

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

   3. Simplified 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.4%

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

      2. associate-/l/ 52.3%

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

      3. div-inv 52.3%

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

      4. *-commutative 52.3%

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

      5. associate-*l/ 53.0%

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

      6. associate-/l* 53.0%

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

      7. pow2 53.0%

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

   6. Applied egg-rr 53.0%

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

      1. associate-*r/ 53.0%

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

      2. metadata-eval 53.0%

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

      3. *-commutative 53.0%

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

      4. *-commutative 53.0%

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

      5. associate-/r/ 52.6%

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

      6. associate-/l* 52.6%

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

      7. associate-*l* 52.6%

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

   8. Simplified 52.6%

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

   9. Taylor expanded in k around inf 88.8%

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

       1. times-frac 93.3%

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

   11. Simplified 93.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+225}\right):\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+268}\right):\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{t\_1}}{k}}{t\_2} \cdot \left(t \cdot \left(\sqrt{2} \cdot \frac{{\left(t\_2 \cdot \left(t \cdot t\_1\right)\right)}^{-2}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0)) (t_2 (cbrt (* (sin k) (tan k)))))
   (if (or (<= (* l l) 0.0) (not (<= (* l l) 1e+268)))
     (*
      (/ (/ (/ (sqrt 2.0) t_1) k) t_2)
      (* t (* (sqrt 2.0) (/ (pow (* t_2 (* t t_1)) -2.0) k))))
     (/
      2.0
      (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+268)) {
		tmp = (((sqrt(2.0) / t_1) / k) / t_2) * (t * (sqrt(2.0) * (pow((t_2 * (t * t_1)), -2.0) / k)));
	} else {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 1e+268)) {
		tmp = (((Math.sqrt(2.0) / t_1) / k) / t_2) * (t * (Math.sqrt(2.0) * (Math.pow((t_2 * (t * t_1)), -2.0) / k)));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if ((Float64(l * l) <= 0.0) || !(Float64(l * l) <= 1e+268))
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) / t_1) / k) / t_2) * Float64(t * Float64(sqrt(2.0) * Float64((Float64(t_2 * Float64(t * t_1)) ^ -2.0) / k))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(l * l), $MachinePrecision], 1e+268]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$1), $MachinePrecision] / k), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[(t$95$2 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0 or 9.9999999999999997e267 < (*.f64 l l)

   1. Initial program 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/r* 28.2%

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

   3. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.3%

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

      3. times-frac 30.3%

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

   6. Applied egg-rr 80.4%

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

      1. associate-/r/ 80.4%

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

      2. associate-/r* 80.4%

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

      3. associate-/r/ 80.4%

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

   8. Simplified 80.4%

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

      1. associate-/l* 80.4%

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

      2. div-inv 80.4%

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

      3. pow-flip 80.4%

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

      4. metadata-eval 80.4%

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

   10. Applied egg-rr 80.4%

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

       1. associate-*l/ 80.4%

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

       2. associate-/r* 80.4%

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

       3. *-inverses 80.4%

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

       4. associate-*r/ 80.4%

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

   12. Simplified 80.4%

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

       1. div-inv 80.4%

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

       2. pow-flip 80.4%

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

       3. div-inv 80.4%

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

       4. pow-flip 80.4%

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

       5. metadata-eval 80.4%

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

       6. metadata-eval 80.4%

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

   14. Applied egg-rr 80.4%

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

       1. *-commutative 80.4%

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

       2. associate-*l* 87.4%

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

       3. associate-*l* 88.8%

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

   16. Simplified 88.8%

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

       1. associate-*l/ 88.8%

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

       2. associate-*r* 87.4%

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

   18. Applied egg-rr 87.4%

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

       1. associate-/l* 87.4%

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

   20. Simplified 87.4%

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

   if 0.0 < (*.f64 l l) < 9.9999999999999997e267

   1. Initial program 46.8%

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

      1. *-commutative 46.8%

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

      2. associate-/r* 46.8%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.7%

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

      2. associate-/l/ 52.7%

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

      3. div-inv 52.7%

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

      4. *-commutative 52.7%

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

      5. associate-*l/ 53.2%

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

      6. associate-/l* 53.3%

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

      7. pow2 53.3%

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

   6. Applied egg-rr 53.3%

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

      1. associate-*r/ 53.3%

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

      2. metadata-eval 53.3%

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

      3. *-commutative 53.3%

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

      4. *-commutative 53.3%

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

      5. associate-/r/ 52.9%

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

      6. associate-/l* 52.9%

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

      7. associate-*l* 52.9%

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

   8. Simplified 52.9%

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

   9. Taylor expanded in k around inf 86.8%

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

       1. times-frac 91.9%

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

   11. Simplified 91.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 10^{+268}\right):\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{k}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t \cdot \left(\sqrt{2} \cdot \frac{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(t \cdot \frac{\sqrt{2}}{k}\right)}^{2}}{{t\_1}^{2}}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (cbrt (* (sin k) (tan k))) (/ t (pow (cbrt l) 2.0)))))
   (if (<= k 1.6e-165)
     (/
      (/ 2.0 (pow (/ k t) 2.0))
      (log (pow (exp (* (pow t 3.0) (pow l -2.0))) (pow k 2.0))))
     (if (<= k 4.1e+152)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (/ (/ (pow (* t (/ (sqrt 2.0) k)) 2.0) (pow t_1 2.0)) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k))) * (t / pow(cbrt(l), 2.0));
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / pow((k / t), 2.0)) / log(pow(exp((pow(t, 3.0) * pow(l, -2.0))), pow(k, 2.0)));
	} else if (k <= 4.1e+152) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else {
		tmp = (pow((t * (sqrt(2.0) / k)), 2.0) / pow(t_1, 2.0)) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k))) * (t / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / Math.log(Math.pow(Math.exp((Math.pow(t, 3.0) * Math.pow(l, -2.0))), Math.pow(k, 2.0)));
	} else if (k <= 4.1e+152) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else {
		tmp = (Math.pow((t * (Math.sqrt(2.0) / k)), 2.0) / Math.pow(t_1, 2.0)) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (k <= 1.6e-165)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / log((exp(Float64((t ^ 3.0) * (l ^ -2.0))) ^ (k ^ 2.0))));
	elseif (k <= 4.1e+152)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	else
		tmp = Float64(Float64((Float64(t * Float64(sqrt(2.0) / k)) ^ 2.0) / (t_1 ^ 2.0)) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.6e-165], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Exp[N[(N[Power[t, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[k, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+152], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000006e-165

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 38.6%

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

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 23.7%

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

      2. exp-prod 30.5%

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

      3. div-inv 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      4. pow2 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot \frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      5. pow-flip 30.5%

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

      6. metadata-eval 30.5%

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

   6. Applied egg-rr 30.5%

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

   7. Taylor expanded in k around 0 30.5%

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

   if 1.60000000000000006e-165 < k < 4.0999999999999998e152

   1. Initial program 23.5%

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

      1. *-commutative 23.5%

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

      2. associate-/r* 23.5%

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

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 32.0%

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

      2. associate-/l/ 32.0%

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

      3. div-inv 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-*l/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. pow2 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. associate-*r/ 32.0%

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

      2. metadata-eval 32.0%

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

      3. *-commutative 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-/r/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. associate-*l* 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in k around inf 79.0%

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

       1. times-frac 83.8%

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

   11. Simplified 83.8%

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

   if 4.0999999999999998e152 < k

   1. Initial program 46.7%

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

      1. *-commutative 46.7%

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

      2. associate-/r* 46.7%

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

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

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

      2. add-cube-cbrt 50.0%

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

      3. times-frac 50.0%

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

   6. Applied egg-rr 83.0%

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

      1. associate-*r/ 82.5%

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

      2. associate-*l/ 76.8%

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

      3. unpow2 76.8%

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

      4. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(t \cdot \frac{\sqrt{2}}{k}\right)}^{2}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_1 \cdot t\_2\right)}^{2}} \cdot \frac{\frac{{\left(\frac{k}{t}\right)}^{-2}}{t\_2}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k)))) (t_2 (/ t (pow (cbrt l) 2.0))))
   (if (<= k 1.6e-165)
     (/
      (/ 2.0 (pow (/ k t) 2.0))
      (log (pow (exp (* (pow t 3.0) (pow l -2.0))) (pow k 2.0))))
     (if (<= k 2.6e+150)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (* (/ 2.0 (pow (* t_1 t_2) 2.0)) (/ (/ (pow (/ k t) -2.0) t_2) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = t / pow(cbrt(l), 2.0);
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / pow((k / t), 2.0)) / log(pow(exp((pow(t, 3.0) * pow(l, -2.0))), pow(k, 2.0)));
	} else if (k <= 2.6e+150) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else {
		tmp = (2.0 / pow((t_1 * t_2), 2.0)) * ((pow((k / t), -2.0) / t_2) / t_1);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = t / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / Math.log(Math.pow(Math.exp((Math.pow(t, 3.0) * Math.pow(l, -2.0))), Math.pow(k, 2.0)));
	} else if (k <= 2.6e+150) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else {
		tmp = (2.0 / Math.pow((t_1 * t_2), 2.0)) * ((Math.pow((k / t), -2.0) / t_2) / t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(t / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (k <= 1.6e-165)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / log((exp(Float64((t ^ 3.0) * (l ^ -2.0))) ^ (k ^ 2.0))));
	elseif (k <= 2.6e+150)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	else
		tmp = Float64(Float64(2.0 / (Float64(t_1 * t_2) ^ 2.0)) * Float64(Float64((Float64(k / t) ^ -2.0) / t_2) / t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.6e-165], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Exp[N[(N[Power[t, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[k, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.6e+150], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(t$95$1 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000006e-165

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 38.6%

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

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 23.7%

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

      2. exp-prod 30.5%

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

      3. div-inv 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      4. pow2 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot \frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      5. pow-flip 30.5%

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

      6. metadata-eval 30.5%

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

   6. Applied egg-rr 30.5%

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

   7. Taylor expanded in k around 0 30.5%

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

   if 1.60000000000000006e-165 < k < 2.60000000000000006e150

   1. Initial program 23.5%

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

      1. *-commutative 23.5%

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

      2. associate-/r* 23.5%

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

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 32.0%

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

      2. associate-/l/ 32.0%

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

      3. div-inv 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-*l/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. pow2 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. associate-*r/ 32.0%

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

      2. metadata-eval 32.0%

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

      3. *-commutative 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-/r/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. associate-*l* 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in k around inf 79.0%

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

       1. times-frac 83.8%

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

   11. Simplified 83.8%

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

   if 2.60000000000000006e150 < k

   1. Initial program 46.7%

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

      1. *-commutative 46.7%

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

      2. associate-/r* 46.7%

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

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 50.0%

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

      2. div-inv 50.0%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 50.0%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 76.8%

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

      1. associate-/r* 76.8%

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

   8. Simplified 76.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{t\_1}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t\_1 \cdot \frac{\tan k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= k 1.6e-165)
     (/ (/ 2.0 t_1) (log (pow (exp (* (pow t 3.0) (pow l -2.0))) (pow k 2.0))))
     (if (<= k 1.7e+154)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (if (<= k 1.08e+204)
         (/
          2.0
          (* (sin k) (* t_1 (/ (tan k) (pow (/ (pow (cbrt l) 2.0) t) 3.0)))))
         (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / t_1) / log(pow(exp((pow(t, 3.0) * pow(l, -2.0))), pow(k, 2.0)));
	} else if (k <= 1.7e+154) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else if (k <= 1.08e+204) {
		tmp = 2.0 / (sin(k) * (t_1 * (tan(k) / pow((pow(cbrt(l), 2.0) / t), 3.0))));
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (k <= 1.6e-165) {
		tmp = (2.0 / t_1) / Math.log(Math.pow(Math.exp((Math.pow(t, 3.0) * Math.pow(l, -2.0))), Math.pow(k, 2.0)));
	} else if (k <= 1.7e+154) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else if (k <= 1.08e+204) {
		tmp = 2.0 / (Math.sin(k) * (t_1 * (Math.tan(k) / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0))));
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (k <= 1.6e-165)
		tmp = Float64(Float64(2.0 / t_1) / log((exp(Float64((t ^ 3.0) * (l ^ -2.0))) ^ (k ^ 2.0))));
	elseif (k <= 1.7e+154)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	elseif (k <= 1.08e+204)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(t_1 * Float64(tan(k) / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0)))));
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.6e-165], N[(N[(2.0 / t$95$1), $MachinePrecision] / N[Log[N[Power[N[Exp[N[(N[Power[t, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[k, 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e+154], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.08e+204], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.60000000000000006e-165

   1. Initial program 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/r* 38.6%

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

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 23.7%

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

      2. exp-prod 30.5%

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

      3. div-inv 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      4. pow2 30.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot \frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      5. pow-flip 30.5%

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

      6. metadata-eval 30.5%

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

   6. Applied egg-rr 30.5%

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

   7. Taylor expanded in k around 0 30.5%

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

   if 1.60000000000000006e-165 < k < 1.69999999999999987e154

   1. Initial program 23.5%

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

      1. *-commutative 23.5%

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

      2. associate-/r* 23.5%

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

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 32.0%

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

      2. associate-/l/ 32.0%

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

      3. div-inv 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-*l/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. pow2 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. associate-*r/ 32.0%

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

      2. metadata-eval 32.0%

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

      3. *-commutative 32.0%

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

      4. *-commutative 32.0%

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

      5. associate-/r/ 32.0%

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

      6. associate-/l* 32.0%

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

      7. associate-*l* 32.0%

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

   8. Simplified 32.0%

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

   9. Taylor expanded in k around inf 79.0%

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

       1. times-frac 83.8%

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

   11. Simplified 83.8%

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

   if 1.69999999999999987e154 < k < 1.08e204

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/r* 20.0%

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

   3. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 40.0%

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

      2. associate-/l/ 40.0%

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

      3. div-inv 40.0%

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

      4. *-commutative 40.0%

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

      5. associate-*l/ 40.0%

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

      6. associate-/l* 40.0%

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

      7. pow2 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. associate-*r/ 40.0%

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

      2. metadata-eval 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. associate-/r/ 40.0%

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

      6. associate-/l* 40.0%

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

      7. associate-*l* 40.0%

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

   8. Simplified 40.0%

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

      1. pow2 40.0%

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

      2. add-cube-cbrt 40.0%

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

      3. pow2 40.0%

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

      4. cbrt-div 40.0%

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

      5. cbrt-prod 40.0%

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

      6. unpow2 40.0%

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

      7. unpow3 40.0%

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

      8. add-cbrt-cube 40.0%

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

      9. cbrt-div 40.0%

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

      10. cbrt-prod 60.7%

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

      11. unpow2 60.7%

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

      12. unpow3 60.7%

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

      13. add-cbrt-cube 79.0%

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

   10. Applied egg-rr 79.0%

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

       1. unpow2 79.0%

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

       2. unpow3 79.0%

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

   12. Simplified 79.0%

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

   if 1.08e204 < k

   1. Initial program 52.0%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in k around 0 71.9%

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

      1. add-log-exp 71.9%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 71.9%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 76.4%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 76.4%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. *-commutative 76.4%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}\right) \]

   7. Applied egg-rr 76.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left({k}^{2}\right)}\right)}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \frac{\tan k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \frac{\tan k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.2e-166)
   (pow
    (* l (/ (sqrt 2.0) (* (sqrt (* (sin k) (tan k))) (* (/ k t) (pow t 1.5)))))
    2.0)
   (if (<= k 1.35e+154)
     (/
      2.0
      (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
     (if (<= k 1.4e+204)
       (/
        2.0
        (*
         (sin k)
         (* (pow (/ k t) 2.0) (/ (tan k) (pow (/ (pow (cbrt l) 2.0) t) 3.0)))))
       (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e-166) {
		tmp = pow((l * (sqrt(2.0) / (sqrt((sin(k) * tan(k))) * ((k / t) * pow(t, 1.5))))), 2.0);
	} else if (k <= 1.35e+154) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else if (k <= 1.4e+204) {
		tmp = 2.0 / (sin(k) * (pow((k / t), 2.0) * (tan(k) / pow((pow(cbrt(l), 2.0) / t), 3.0))));
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e-166) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sqrt((Math.sin(k) * Math.tan(k))) * ((k / t) * Math.pow(t, 1.5))))), 2.0);
	} else if (k <= 1.35e+154) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else if (k <= 1.4e+204) {
		tmp = 2.0 / (Math.sin(k) * (Math.pow((k / t), 2.0) * (Math.tan(k) / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0))));
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.2e-166)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sqrt(Float64(sin(k) * tan(k))) * Float64(Float64(k / t) * (t ^ 1.5))))) ^ 2.0;
	elseif (k <= 1.35e+154)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	elseif (k <= 1.4e+204)
		tmp = Float64(2.0 / Float64(sin(k) * Float64((Float64(k / t) ^ 2.0) * Float64(tan(k) / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0)))));
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.2e-166], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.35e+154], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+204], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.20000000000000001e-166

   1. Initial program 38.8%

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

   3. Simplified 40.3%

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

      1. add-sqr-sqrt 25.7%

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

      2. pow2 25.7%

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

   6. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l* 28.2%

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

   8. Simplified 28.2%

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

   if 3.20000000000000001e-166 < k < 1.35000000000000003e154

   1. Initial program 23.1%

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

      1. *-commutative 23.1%

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

      2. associate-/r* 23.2%

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

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 31.5%

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

      2. associate-/l/ 31.5%

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

      3. div-inv 31.5%

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

      4. *-commutative 31.5%

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

      5. associate-*l/ 31.4%

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

      6. associate-/l* 31.4%

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

      7. pow2 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-*r/ 31.4%

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

      2. metadata-eval 31.4%

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

      3. *-commutative 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-/r/ 31.5%

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

      6. associate-/l* 31.5%

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

      7. associate-*l* 31.5%

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

   8. Simplified 31.5%

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

   9. Taylor expanded in k around inf 77.9%

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

       1. times-frac 82.6%

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

   11. Simplified 82.6%

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

   if 1.35000000000000003e154 < k < 1.40000000000000012e204

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/r* 20.0%

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

   3. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 40.0%

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

      2. associate-/l/ 40.0%

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

      3. div-inv 40.0%

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

      4. *-commutative 40.0%

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

      5. associate-*l/ 40.0%

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

      6. associate-/l* 40.0%

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

      7. pow2 40.0%

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

   6. Applied egg-rr 40.0%

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

      1. associate-*r/ 40.0%

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

      2. metadata-eval 40.0%

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

      3. *-commutative 40.0%

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

      4. *-commutative 40.0%

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

      5. associate-/r/ 40.0%

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

      6. associate-/l* 40.0%

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

      7. associate-*l* 40.0%

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

   8. Simplified 40.0%

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

      1. pow2 40.0%

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

      2. add-cube-cbrt 40.0%

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

      3. pow2 40.0%

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

      4. cbrt-div 40.0%

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

      5. cbrt-prod 40.0%

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

      6. unpow2 40.0%

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

      7. unpow3 40.0%

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

      8. add-cbrt-cube 40.0%

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

      9. cbrt-div 40.0%

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

      10. cbrt-prod 60.7%

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

      11. unpow2 60.7%

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

      12. unpow3 60.7%

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

      13. add-cbrt-cube 79.0%

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

   10. Applied egg-rr 79.0%

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

       1. unpow2 79.0%

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

       2. unpow3 79.0%

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

   12. Simplified 79.0%

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

   if 1.40000000000000012e204 < k

   1. Initial program 52.0%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in k around 0 71.9%

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

      1. add-log-exp 71.9%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 71.9%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 76.4%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 76.4%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. *-commutative 76.4%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}\right) \]

   7. Applied egg-rr 76.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-166}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+204}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \frac{\tan k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\sin k \cdot \tan k}\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{t\_1 \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+201}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k}}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{t}{t\_1}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sqrt (* (sin k) (tan k)))))
   (if (<= k 1.85e-166)
     (pow (* l (/ (sqrt 2.0) (* t_1 (* (/ k t) (pow t 1.5))))) 2.0)
     (if (<= k 1.3e+150)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (if (<= k 3.5e+201)
         (pow (* (/ (/ (sqrt 2.0) k) (/ (pow t 1.5) l)) (/ t t_1)) 2.0)
         (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = sqrt((sin(k) * tan(k)));
	double tmp;
	if (k <= 1.85e-166) {
		tmp = pow((l * (sqrt(2.0) / (t_1 * ((k / t) * pow(t, 1.5))))), 2.0);
	} else if (k <= 1.3e+150) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else if (k <= 3.5e+201) {
		tmp = pow((((sqrt(2.0) / k) / (pow(t, 1.5) / l)) * (t / t_1)), 2.0);
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((sin(k) * tan(k)))
    if (k <= 1.85d-166) then
        tmp = (l * (sqrt(2.0d0) / (t_1 * ((k / t) * (t ** 1.5d0))))) ** 2.0d0
    else if (k <= 1.3d+150) then
        tmp = 2.0d0 / (sin(k) * (((k ** 2.0d0) / (l ** 2.0d0)) * ((t * sin(k)) / cos(k))))
    else if (k <= 3.5d+201) then
        tmp = (((sqrt(2.0d0) / k) / ((t ** 1.5d0) / l)) * (t / t_1)) ** 2.0d0
    else
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (k <= 1.85e-166) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (t_1 * ((k / t) * Math.pow(t, 1.5))))), 2.0);
	} else if (k <= 1.3e+150) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else if (k <= 3.5e+201) {
		tmp = Math.pow((((Math.sqrt(2.0) / k) / (Math.pow(t, 1.5) / l)) * (t / t_1)), 2.0);
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sqrt((math.sin(k) * math.tan(k)))
	tmp = 0
	if k <= 1.85e-166:
		tmp = math.pow((l * (math.sqrt(2.0) / (t_1 * ((k / t) * math.pow(t, 1.5))))), 2.0)
	elif k <= 1.3e+150:
		tmp = 2.0 / (math.sin(k) * ((math.pow(k, 2.0) / math.pow(l, 2.0)) * ((t * math.sin(k)) / math.cos(k))))
	elif k <= 3.5e+201:
		tmp = math.pow((((math.sqrt(2.0) / k) / (math.pow(t, 1.5) / l)) * (t / t_1)), 2.0)
	else:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sqrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (k <= 1.85e-166)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(t_1 * Float64(Float64(k / t) * (t ^ 1.5))))) ^ 2.0;
	elseif (k <= 1.3e+150)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	elseif (k <= 3.5e+201)
		tmp = Float64(Float64(Float64(sqrt(2.0) / k) / Float64((t ^ 1.5) / l)) * Float64(t / t_1)) ^ 2.0;
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sqrt((sin(k) * tan(k)));
	tmp = 0.0;
	if (k <= 1.85e-166)
		tmp = (l * (sqrt(2.0) / (t_1 * ((k / t) * (t ^ 1.5))))) ^ 2.0;
	elseif (k <= 1.3e+150)
		tmp = 2.0 / (sin(k) * (((k ^ 2.0) / (l ^ 2.0)) * ((t * sin(k)) / cos(k))));
	elseif (k <= 3.5e+201)
		tmp = (((sqrt(2.0) / k) / ((t ^ 1.5) / l)) * (t / t_1)) ^ 2.0;
	else
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 1.85e-166], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * N[(N[(k / t), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.3e+150], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5e+201], N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] / N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.8500000000000001e-166

   1. Initial program 38.8%

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

   3. Simplified 40.3%

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

      1. add-sqr-sqrt 25.7%

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

      2. pow2 25.7%

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

   6. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l* 28.2%

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

   8. Simplified 28.2%

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

   if 1.8500000000000001e-166 < k < 1.30000000000000003e150

   1. Initial program 23.1%

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

      1. *-commutative 23.1%

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

      2. associate-/r* 23.2%

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

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 31.5%

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

      2. associate-/l/ 31.5%

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

      3. div-inv 31.5%

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

      4. *-commutative 31.5%

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

      5. associate-*l/ 31.4%

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

      6. associate-/l* 31.4%

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

      7. pow2 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-*r/ 31.4%

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

      2. metadata-eval 31.4%

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

      3. *-commutative 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-/r/ 31.5%

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

      6. associate-/l* 31.5%

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

      7. associate-*l* 31.5%

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

   8. Simplified 31.5%

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

   9. Taylor expanded in k around inf 77.9%

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

       1. times-frac 82.6%

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

   11. Simplified 82.6%

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

   if 1.30000000000000003e150 < k < 3.5000000000000002e201

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-/r* 25.0%

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

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 50.0%

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

   6. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 25.0%

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

      2. associate-/r/ 25.0%

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

      3. times-frac 25.0%

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

   8. Simplified 25.0%

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

   if 3.5000000000000002e201 < k

   1. Initial program 50.0%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in k around 0 69.2%

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

      1. add-log-exp 69.2%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 69.2%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 73.6%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 73.6%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. *-commutative 73.6%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}\right) \]

   7. Applied egg-rr 73.6%

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

Alternative 10: 47.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.9 \cdot 10^{-166}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{t\_1} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{t\_1 \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.9e-166)
     (pow (* l (/ (sqrt 2.0) (* (sqrt t_1) (* (/ k t) (pow t 1.5))))) 2.0)
     (if (<= k 4e+154)
       (/
        2.0
        (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k)))))
       (if (<= k 2.4e+203)
         (/ (/ 2.0 (pow (/ k t) 2.0)) (* t_1 (* (/ (pow t 2.0) l) (/ t l))))
         (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.9e-166) {
		tmp = pow((l * (sqrt(2.0) / (sqrt(t_1) * ((k / t) * pow(t, 1.5))))), 2.0);
	} else if (k <= 4e+154) {
		tmp = 2.0 / (sin(k) * ((pow(k, 2.0) / pow(l, 2.0)) * ((t * sin(k)) / cos(k))));
	} else if (k <= 2.4e+203) {
		tmp = (2.0 / pow((k / t), 2.0)) / (t_1 * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 1.9d-166) then
        tmp = (l * (sqrt(2.0d0) / (sqrt(t_1) * ((k / t) * (t ** 1.5d0))))) ** 2.0d0
    else if (k <= 4d+154) then
        tmp = 2.0d0 / (sin(k) * (((k ** 2.0d0) / (l ** 2.0d0)) * ((t * sin(k)) / cos(k))))
    else if (k <= 2.4d+203) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / (t_1 * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.9e-166) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sqrt(t_1) * ((k / t) * Math.pow(t, 1.5))))), 2.0);
	} else if (k <= 4e+154) {
		tmp = 2.0 / (Math.sin(k) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.sin(k)) / Math.cos(k))));
	} else if (k <= 2.4e+203) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (t_1 * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 1.9e-166:
		tmp = math.pow((l * (math.sqrt(2.0) / (math.sqrt(t_1) * ((k / t) * math.pow(t, 1.5))))), 2.0)
	elif k <= 4e+154:
		tmp = 2.0 / (math.sin(k) * ((math.pow(k, 2.0) / math.pow(l, 2.0)) * ((t * math.sin(k)) / math.cos(k))))
	elif k <= 2.4e+203:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (t_1 * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.9e-166)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sqrt(t_1) * Float64(Float64(k / t) * (t ^ 1.5))))) ^ 2.0;
	elseif (k <= 4e+154)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))));
	elseif (k <= 2.4e+203)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(t_1 * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 1.9e-166)
		tmp = (l * (sqrt(2.0) / (sqrt(t_1) * ((k / t) * (t ^ 1.5))))) ^ 2.0;
	elseif (k <= 4e+154)
		tmp = 2.0 / (sin(k) * (((k ^ 2.0) / (l ^ 2.0)) * ((t * sin(k)) / cos(k))));
	elseif (k <= 2.4e+203)
		tmp = (2.0 / ((k / t) ^ 2.0)) / (t_1 * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.9e-166], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 4e+154], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e+203], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.89999999999999991e-166

   1. Initial program 38.8%

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

   3. Simplified 40.3%

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

      1. add-sqr-sqrt 25.7%

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

      2. pow2 25.7%

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

   6. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l* 28.2%

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

   8. Simplified 28.2%

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

   if 1.89999999999999991e-166 < k < 4.00000000000000015e154

   1. Initial program 23.1%

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

      1. *-commutative 23.1%

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

      2. associate-/r* 23.2%

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

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 31.5%

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

      2. associate-/l/ 31.5%

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

      3. div-inv 31.5%

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

      4. *-commutative 31.5%

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

      5. associate-*l/ 31.4%

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

      6. associate-/l* 31.4%

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

      7. pow2 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-*r/ 31.4%

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

      2. metadata-eval 31.4%

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

      3. *-commutative 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-/r/ 31.5%

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

      6. associate-/l* 31.5%

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

      7. associate-*l* 31.5%

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

   8. Simplified 31.5%

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

   9. Taylor expanded in k around inf 77.9%

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

       1. times-frac 82.6%

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

   11. Simplified 82.6%

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

   if 4.00000000000000015e154 < k < 2.4000000000000001e203

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

      2. associate-/r* 20.0%

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

   3. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 40.0%

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

      2. times-frac 79.5%

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

      3. pow2 79.5%

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

   6. Applied egg-rr 79.5%

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

   if 2.4000000000000001e203 < k

   1. Initial program 52.0%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in k around 0 71.9%

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

      1. add-log-exp 71.9%

         \[\leadsto \color{blue}{\log \left(e^{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)} \]

      2. *-commutative 71.9%

         \[\leadsto \log \left(e^{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot t}}}\right) \]

      3. exp-prod 76.4%

         \[\leadsto \log \color{blue}{\left({\left(e^{\ell \cdot \ell}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right)} \]

      4. pow2 76.4%

         \[\leadsto \log \left({\left(e^{\color{blue}{{\ell}^{2}}}\right)}^{\left(\frac{2}{{k}^{4} \cdot t}\right)}\right) \]

      5. *-commutative 76.4%

         \[\leadsto \log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{\color{blue}{t \cdot {k}^{4}}}\right)}\right) \]

   7. Applied egg-rr 76.4%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-166}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}\right)}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (sin k) (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (sin k)) (cos k))))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (sin(k) * (((k ** 2.0d0) / (l ** 2.0d0)) * ((t * sin(k)) / cos(k))))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(sin(k) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * sin(k)) / cos(k)))))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. *-commutative 36.0%

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

   2. associate-/r* 36.0%

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

3. Simplified 39.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-rgt-identity 39.7%

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

   2. associate-/l/ 39.7%

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

   3. div-inv 39.7%

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

   4. *-commutative 39.7%

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

   5. associate-*l/ 39.9%

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

   6. associate-/l* 39.9%

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

   7. pow2 39.9%

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

6. Applied egg-rr 39.9%

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

   1. associate-*r/ 39.9%

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

   2. metadata-eval 39.9%

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

   3. *-commutative 39.9%

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

   4. *-commutative 39.9%

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

   5. associate-/r/ 39.8%

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

   6. associate-/l* 39.8%

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

   7. associate-*l* 39.8%

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

8. Simplified 39.8%

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

9. Taylor expanded in k around inf 70.2%

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

    1. times-frac 72.4%

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

11. Simplified 72.4%

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

Alternative 12: 74.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (pow l 2.0) (pow k 2.0)) (* (/ (cos k) t) (/ 2.0 (pow (sin k) 2.0)))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l ** 2.0d0) / (k ** 2.0d0)) * ((cos(k) / t) * (2.0d0 / (sin(k) ** 2.0d0)))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(cos(k) / t) * Float64(2.0 / (sin(k) ^ 2.0))))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. *-commutative 36.0%

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

   2. associate-/r* 36.0%

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

3. Simplified 39.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 39.7%

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

   2. add-cube-cbrt 39.6%

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

   3. times-frac 39.6%

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

6. Applied egg-rr 79.6%

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

   1. associate-/r/ 79.6%

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

   2. associate-/r* 79.6%

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

   3. associate-/r/ 79.6%

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

8. Simplified 79.6%

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

   1. associate-/l* 79.7%

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

   2. div-inv 79.7%

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

   3. pow-flip 79.7%

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

   4. metadata-eval 79.7%

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

10. Applied egg-rr 79.7%

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

    1. associate-*l/ 79.7%

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

    2. associate-/r* 79.7%

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

    3. *-inverses 79.7%

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

    4. associate-*r/ 79.7%

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

12. Simplified 79.7%

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

13. Taylor expanded in k around inf 70.1%

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

    1. times-frac 71.6%

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

    2. *-commutative 71.6%

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

    3. unpow2 71.6%

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

    4. rem-square-sqrt 71.7%

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

    5. rem-cube-cbrt 71.5%

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

    6. rem-cube-cbrt 71.7%

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

    7. *-commutative 71.7%

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

    8. times-frac 71.7%

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

15. Simplified 71.7%

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

Alternative 13: 73.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0))))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in t around 0 70.2%

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

   1. associate-/r* 70.4%

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

7. Simplified 70.4%

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

8. Final simplification 70.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right) \]
9. Add Preprocessing

Alternative 14: 73.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* t (pow k 2.0)) (* (sin k) (/ (tan k) (pow l 2.0))))))

C

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

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 * (k ** 2.0d0)) * (sin(k) * (tan(k) / (l ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. *-commutative 36.0%

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

   2. associate-/r* 36.0%

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

3. Simplified 39.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-rgt-identity 39.7%

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

   2. associate-/l/ 39.7%

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

   3. div-inv 39.7%

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

   4. *-commutative 39.7%

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

   5. associate-*l/ 39.9%

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

   6. associate-/l* 39.9%

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

   7. pow2 39.9%

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

6. Applied egg-rr 39.9%

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

   1. associate-*r/ 39.9%

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

   2. metadata-eval 39.9%

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

   3. associate-*r* 39.6%

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

   4. associate-/l* 39.6%

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

8. Simplified 39.6%

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

9. Taylor expanded in k around 0 70.2%

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

10. Final simplification 70.2%

    \[\leadsto \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(\sin k \cdot \frac{\tan k}{{\ell}^{2}}\right)} \]
11. Add Preprocessing

Alternative 15: 54.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+166}:\\ \;\;\;\;{\left({\left({\ell}^{2} \cdot \frac{-0.11666666666666667}{t}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.2e+45)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   (if (<= k 2.1e+166)
     (pow
      (pow (* (pow l 2.0) (/ -0.11666666666666667 t)) 3.0)
      0.3333333333333333)
     (* (* l l) (/ 2.0 (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e+45) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else if (k <= 2.1e+166) {
		tmp = pow(pow((pow(l, 2.0) * (-0.11666666666666667 / t)), 3.0), 0.3333333333333333);
	} else {
		tmp = (l * l) * (2.0 / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.2d+45) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else if (k <= 2.1d+166) then
        tmp = (((l ** 2.0d0) * ((-0.11666666666666667d0) / t)) ** 3.0d0) ** 0.3333333333333333d0
    else
        tmp = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e+45) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else if (k <= 2.1e+166) {
		tmp = Math.pow(Math.pow((Math.pow(l, 2.0) * (-0.11666666666666667 / t)), 3.0), 0.3333333333333333);
	} else {
		tmp = (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.2e+45:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	elif k <= 2.1e+166:
		tmp = math.pow(math.pow((math.pow(l, 2.0) * (-0.11666666666666667 / t)), 3.0), 0.3333333333333333)
	else:
		tmp = (l * l) * (2.0 / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.2e+45)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	elseif (k <= 2.1e+166)
		tmp = (Float64((l ^ 2.0) * Float64(-0.11666666666666667 / t)) ^ 3.0) ^ 0.3333333333333333;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.2e+45)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	elseif (k <= 2.1e+166)
		tmp = (((l ^ 2.0) * (-0.11666666666666667 / t)) ^ 3.0) ^ 0.3333333333333333;
	else
		tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.2e+45], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e+166], N[Power[N[Power[N[(N[Power[l, 2.0], $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.20000000000000014e45

   1. Initial program 35.5%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in k around 0 48.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.20000000000000014e45 < k < 2.1000000000000001e166

   1. Initial program 22.1%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in k around 0 28.0%

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   6. Taylor expanded in k around inf 62.7%

      \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. add-cbrt-cube 73.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right)}} \]

      2. pow1/3 61.5%

         \[\leadsto \color{blue}{{\left(\left(\left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right)\right)}^{0.3333333333333333}} \]

      3. pow3 61.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. pow2 61.5%

         \[\leadsto {\left({\left(\frac{-0.11666666666666667}{t} \cdot \color{blue}{{\ell}^{2}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 61.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{-0.11666666666666667}{t} \cdot {\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

   if 2.1000000000000001e166 < k

   1. Initial program 48.3%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in k around 0 65.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+166}:\\ \;\;\;\;{\left({\left({\ell}^{2} \cdot \frac{-0.11666666666666667}{t}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (* (/ 2.0 t) (/ (cos k) (pow (* k (sin k)) 2.0)))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((2.0d0 / t) * (cos(k) / ((k * sin(k)) ** 2.0d0)))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(Float64(2.0 / t) * Float64(cos(k) / (Float64(k * sin(k)) ^ 2.0))))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in t around 0 70.2%

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

   1. *-commutative 70.2%

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

   2. *-commutative 70.2%

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

   3. associate-*l* 70.2%

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

   4. *-commutative 70.2%

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

7. Simplified 70.2%

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

8. Taylor expanded in k around inf 70.2%

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

   1. associate-*r/ 70.2%

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

   2. *-commutative 70.2%

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

   3. associate-*l* 69.3%

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

   4. unpow2 69.3%

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

   5. unpow2 69.3%

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

   6. swap-sqr 69.3%

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

   7. unpow2 69.3%

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

   8. *-commutative 69.3%

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

   9. *-commutative 69.3%

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

   10. times-frac 69.5%

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

   11. *-commutative 69.5%

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

10. Simplified 69.5%

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

11. Final simplification 69.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot \frac{\cos k}{{\left(k \cdot \sin k\right)}^{2}}\right) \]
12. Add Preprocessing

Alternative 17: 63.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\sin k \cdot \frac{t \cdot {k}^{3}}{{\ell}^{2}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (sin k) (/ (* t (pow k 3.0)) (pow l 2.0)))))

C

double code(double t, double l, double k) {
	return 2.0 / (sin(k) * ((t * pow(k, 3.0)) / pow(l, 2.0)));
}

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 / (sin(k) * ((t * (k ** 3.0d0)) / (l ** 2.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (Math.sin(k) * ((t * Math.pow(k, 3.0)) / Math.pow(l, 2.0)));
}

Python

def code(t, l, k):
	return 2.0 / (math.sin(k) * ((t * math.pow(k, 3.0)) / math.pow(l, 2.0)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(sin(k) * Float64(Float64(t * (k ^ 3.0)) / (l ^ 2.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (sin(k) * ((t * (k ^ 3.0)) / (l ^ 2.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(t * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. *-commutative 36.0%

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

   2. associate-/r* 36.0%

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

3. Simplified 39.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-rgt-identity 39.7%

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

   2. associate-/l/ 39.7%

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

   3. div-inv 39.7%

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

   4. *-commutative 39.7%

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

   5. associate-*l/ 39.9%

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

   6. associate-/l* 39.9%

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

   7. pow2 39.9%

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

6. Applied egg-rr 39.9%

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

   1. associate-*r/ 39.9%

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

   2. metadata-eval 39.9%

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

   3. *-commutative 39.9%

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

   4. *-commutative 39.9%

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

   5. associate-/r/ 39.8%

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

   6. associate-/l* 39.8%

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

   7. associate-*l* 39.8%

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

8. Simplified 39.8%

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

9. Taylor expanded in k around 0 62.1%

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

10. Final simplification 62.1%

    \[\leadsto \frac{2}{\sin k \cdot \frac{t \cdot {k}^{3}}{{\ell}^{2}}} \]
11. Add Preprocessing

Alternative 18: 53.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.2e+45)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   (/ 2.0 (* t (/ (pow k 4.0) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e+45) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else {
		tmp = 2.0 / (t * (pow(k, 4.0) / pow(l, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.2d+45) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else
        tmp = 2.0d0 / (t * ((k ** 4.0d0) / (l ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e+45) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / (t * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.2e+45:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	else:
		tmp = 2.0 / (t * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.2e+45)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(t * Float64((k ^ 4.0) / (l ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.2e+45)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	else
		tmp = 2.0 / (t * ((k ^ 4.0) / (l ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.2e+45], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.20000000000000014e45

   1. Initial program 35.5%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in k around 0 48.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.20000000000000014e45 < k

   1. Initial program 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/r* 38.3%

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

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 46.9%

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

      2. associate-/l/ 46.8%

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

      3. div-inv 46.8%

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

      4. *-commutative 46.8%

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

      5. associate-*l/ 46.8%

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

      6. associate-/l* 46.8%

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

      7. pow2 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. associate-*r/ 46.8%

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

      2. metadata-eval 46.8%

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

      3. *-commutative 46.8%

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

      4. *-commutative 46.8%

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

      5. associate-/r/ 46.8%

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

      6. associate-/l* 46.8%

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

      7. associate-*l* 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in k around 0 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   10. Step-by-step derivation

       1. *-commutative 64.0%

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

       2. associate-/l* 64.0%

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

   11. Simplified 64.0%

       \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* t (/ (pow k 4.0) (pow l 2.0)))))

C

double code(double t, double l, double k) {
	return 2.0 / (t * (pow(k, 4.0) / pow(l, 2.0)));
}

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 * ((k ** 4.0d0) / (l ** 2.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (t * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
}

Python

def code(t, l, k):
	return 2.0 / (t * (math.pow(k, 4.0) / math.pow(l, 2.0)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(t * Float64((k ^ 4.0) / (l ^ 2.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (t * ((k ^ 4.0) / (l ^ 2.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. *-commutative 36.0%

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

   2. associate-/r* 36.0%

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

3. Simplified 39.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-rgt-identity 39.7%

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

   2. associate-/l/ 39.7%

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

   3. div-inv 39.7%

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

   4. *-commutative 39.7%

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

   5. associate-*l/ 39.9%

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

   6. associate-/l* 39.9%

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

   7. pow2 39.9%

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

6. Applied egg-rr 39.9%

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

   1. associate-*r/ 39.9%

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

   2. metadata-eval 39.9%

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

   3. *-commutative 39.9%

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

   4. *-commutative 39.9%

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

   5. associate-/r/ 39.8%

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

   6. associate-/l* 39.8%

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

   7. associate-*l* 39.8%

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

8. Simplified 39.8%

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

9. Taylor expanded in k around 0 60.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
10. Step-by-step derivation

    1. *-commutative 60.0%

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

    2. associate-/l* 60.2%

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

11. Simplified 60.2%

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

Alternative 20: 62.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
}

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 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in k around 0 60.0%

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

6. Final simplification 60.0%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 21: 61.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ (* 2.0 (pow k -4.0)) t)))

C

double code(double t, double l, double k) {
	return (l * l) * ((2.0 * pow(k, -4.0)) / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((2.0d0 * (k ** (-4.0d0))) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * ((2.0 * Math.pow(k, -4.0)) / t);
}

Python

def code(t, l, k):
	return (l * l) * ((2.0 * math.pow(k, -4.0)) / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -4.0)) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * ((2.0 * (k ^ -4.0)) / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in t around 0 70.2%

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

   1. *-commutative 70.2%

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

   2. *-commutative 70.2%

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

   3. associate-*l* 70.2%

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

   4. *-commutative 70.2%

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

7. Simplified 70.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k}}} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in k around 0 60.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. *-commutative 60.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. associate-/r* 60.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

10. Simplified 60.0%

    \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
11. Step-by-step derivation

    1. *-un-lft-identity 60.0%

       \[\leadsto \color{blue}{\left(1 \cdot \frac{\frac{2}{t}}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

    2. div-inv 60.0%

       \[\leadsto \left(1 \cdot \frac{\color{blue}{2 \cdot \frac{1}{t}}}{{k}^{4}}\right) \cdot \left(\ell \cdot \ell\right) \]

    3. div-inv 60.0%

       \[\leadsto \left(1 \cdot \color{blue}{\left(\left(2 \cdot \frac{1}{t}\right) \cdot \frac{1}{{k}^{4}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

    4. div-inv 60.0%

       \[\leadsto \left(1 \cdot \left(\color{blue}{\frac{2}{t}} \cdot \frac{1}{{k}^{4}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

    5. pow-flip 60.0%

       \[\leadsto \left(1 \cdot \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

    6. metadata-eval 60.0%

       \[\leadsto \left(1 \cdot \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right)\right) \cdot \left(\ell \cdot \ell\right) \]

12. Applied egg-rr 60.0%

    \[\leadsto \color{blue}{\left(1 \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\right)} \cdot \left(\ell \cdot \ell\right) \]
13. Step-by-step derivation

    1. *-lft-identity 60.0%

       \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]

    2. associate-*l/ 60.0%

       \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-4}}{t}} \cdot \left(\ell \cdot \ell\right) \]

14. Simplified 60.0%

    \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-4}}{t}} \cdot \left(\ell \cdot \ell\right) \]

15. Final simplification 60.0%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t} \]
16. Add Preprocessing

Alternative 22: 61.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ 2.0 (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.3%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 60.0%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 23: 20.0% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((-0.11666666666666667d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.11666666666666667 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.3%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 40.3%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 22.1%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 22.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))))