Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 88.6%

Time: 21.0s

Alternatives: 23

Speedup: 3.8×

• Could not converge on a ground truth

  1. t = 1.1991631706528043e-130
  2. l = 2.8735107178176292e+166
  3. k = 1.5046722200920113e+121

• 39.6% 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: 36.2% 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: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_3 \cdot \frac{t}{{\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{2}}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;t\_4 \cdot \frac{\frac{t\_3}{t\_1}}{t\_2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot \frac{2}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \frac{1}{\frac{\frac{t\_2}{t\_3}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \end{array} \]

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 (* t_3 (/ t (pow (* t (* t_1 t_2)) 2.0)))))
   (if (<= (* l l) 5e-309)
     (* t_4 (/ (/ t_3 t_1) t_2))
     (if (<= (* l l) 5e+255)
       (*
        (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
        (/ 2.0 (* t (pow (sin k) 2.0))))
       (* t_4 (/ 1.0 (/ (/ t_2 t_3) (pow (cbrt l) 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 = t_3 * (t / pow((t * (t_1 * t_2)), 2.0));
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = t_4 * ((t_3 / t_1) / t_2);
	} else if ((l * l) <= 5e+255) {
		tmp = ((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) * (2.0 / (t * pow(sin(k), 2.0)));
	} else {
		tmp = t_4 * (1.0 / ((t_2 / t_3) / pow(cbrt(l), 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 = t_3 * (t / Math.pow((t * (t_1 * t_2)), 2.0));
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = t_4 * ((t_3 / t_1) / t_2);
	} else if ((l * l) <= 5e+255) {
		tmp = ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) * (2.0 / (t * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = t_4 * (1.0 / ((t_2 / t_3) / Math.pow(Math.cbrt(l), 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(t_3 * Float64(t / (Float64(t * Float64(t_1 * t_2)) ^ 2.0)))
	tmp = 0.0
	if (Float64(l * l) <= 5e-309)
		tmp = Float64(t_4 * Float64(Float64(t_3 / t_1) / t_2));
	elseif (Float64(l * l) <= 5e+255)
		tmp = Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) * Float64(2.0 / Float64(t * (sin(k) ^ 2.0))));
	else
		tmp = Float64(t_4 * Float64(1.0 / Float64(Float64(t_2 / t_3) / (cbrt(l) ^ 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[(t$95$3 * N[(t / N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[(t$95$4 * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+255], N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(1.0 / N[(N[(t$95$2 / t$95$3), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309

   1. Initial program 19.8%

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

      1. *-commutative 19.8%

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

      2. associate-/r* 19.8%

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

   3. Simplified 35.4%

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

      1. add-sqr-sqrt 35.4%

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

      2. add-cube-cbrt 35.4%

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

      3. times-frac 35.4%

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

   6. Applied egg-rr 76.5%

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

      1. associate-/r/ 76.5%

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

      2. associate-/r* 76.4%

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

      3. associate-/r/ 76.5%

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

   8. Simplified 76.5%

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

      1. frac-times 72.6%

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

      2. div-inv 72.6%

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

      3. pow-flip 72.6%

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

      4. metadata-eval 72.6%

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

      5. div-inv 72.7%

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

      6. pow-flip 72.7%

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

      7. metadata-eval 72.7%

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

   10. Applied egg-rr 72.7%

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

       1. times-frac 76.5%

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

       2. associate-*r/ 76.5%

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

       3. associate-*l* 76.4%

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

       4. associate-/r* 76.5%

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

       5. times-frac 79.2%

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

   12. Simplified 79.2%

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

       1. associate-*r/ 79.2%

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

   14. Applied egg-rr 79.2%

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

       1. associate-/l* 79.2%

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

       2. associate-/l/ 79.2%

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

   16. Applied egg-rr 79.2%

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

       1. associate-*r/ 79.2%

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

       2. associate-*l/ 79.2%

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

       3. associate-/l/ 76.4%

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

       4. associate-/r* 83.6%

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

       5. associate-/l/ 85.2%

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

       6. *-commutative 85.2%

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

       7. times-frac 86.4%

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

       8. *-inverses 86.4%

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

       9. *-rgt-identity 86.4%

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

   18. Simplified 86.4%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 5.0000000000000002e255

   1. Initial program 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/r* 37.1%

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

   3. Simplified 47.1%

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

      1. add-sqr-sqrt 47.1%

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

      2. add-cube-cbrt 47.0%

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

      3. times-frac 47.0%

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

   6. Applied egg-rr 81.6%

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

      1. associate-/r/ 81.6%

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

      2. associate-/r* 81.6%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 79.8%

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

      2. div-inv 79.8%

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

      3. pow-flip 79.9%

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

      4. metadata-eval 79.9%

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

      5. div-inv 79.9%

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

      6. pow-flip 79.9%

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

      7. metadata-eval 79.9%

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

   10. Applied egg-rr 79.9%

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

       1. times-frac 81.7%

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

       2. associate-*r/ 81.7%

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

       3. associate-*l* 81.6%

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

       4. associate-/r* 81.6%

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

       5. times-frac 81.8%

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

   12. Simplified 81.8%

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

   13. Taylor expanded in k around inf 91.7%

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

       1. associate-*r* 91.8%

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

       2. unpow2 91.8%

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

       3. rem-square-sqrt 91.9%

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

       4. times-frac 93.4%

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

   15. Simplified 93.4%

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

   if 5.0000000000000002e255 < (*.f64 l l)

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-/r* 31.9%

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

   3. Simplified 33.2%

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

      1. add-sqr-sqrt 33.2%

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

      2. add-cube-cbrt 33.1%

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

      3. times-frac 33.1%

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

   6. Applied egg-rr 78.3%

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

      1. associate-/r/ 78.3%

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

      2. associate-/r* 78.3%

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

      3. associate-/r/ 78.4%

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

   8. Simplified 78.4%

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

      1. frac-times 74.4%

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

      2. div-inv 74.4%

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

      3. pow-flip 74.4%

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

      4. metadata-eval 74.4%

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

      5. div-inv 74.4%

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

      6. pow-flip 74.4%

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

      7. metadata-eval 74.4%

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

   10. Applied egg-rr 74.4%

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

       1. times-frac 78.3%

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

       2. associate-*r/ 77.1%

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

       3. associate-*l* 77.2%

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

       4. associate-/r* 77.1%

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

       5. times-frac 79.5%

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

   12. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

   14. Applied egg-rr 79.5%

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

       1. clear-num 79.5%

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

       2. inv-pow 79.5%

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

       3. associate-*l/ 77.2%

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

       4. metadata-eval 77.2%

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

       5. pow-flip 77.2%

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

       6. div-inv 77.2%

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

       7. associate-/r/ 77.2%

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

       8. associate-*l/ 77.2%

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

   16. Applied egg-rr 77.2%

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

       1. unpow-1 77.2%

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

       2. associate-/r* 77.2%

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

       3. associate-/l/ 73.4%

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

       4. *-commutative 73.4%

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

       5. times-frac 80.7%

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

       6. *-inverses 80.7%

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

       7. *-rgt-identity 80.7%

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

   18. Simplified 80.7%

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

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{\sqrt{2}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309} \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+255}\right):\\ \;\;\;\;\left(t\_2 \cdot \frac{t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_1\right)\right)}^{2}}\right) \cdot \frac{t\_2 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot \frac{2}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k)))) (t_2 (/ (sqrt 2.0) k)))
   (if (or (<= (* l l) 5e-309) (not (<= (* l l) 5e+255)))
     (*
      (* t_2 (/ t (pow (* t (* (pow (cbrt l) -2.0) t_1)) 2.0)))
      (/ (* t_2 (pow (cbrt l) 2.0)) t_1))
     (*
      (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
      (/ 2.0 (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = sqrt(2.0) / k;
	double tmp;
	if (((l * l) <= 5e-309) || !((l * l) <= 5e+255)) {
		tmp = (t_2 * (t / pow((t * (pow(cbrt(l), -2.0) * t_1)), 2.0))) * ((t_2 * pow(cbrt(l), 2.0)) / t_1);
	} else {
		tmp = ((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) * (2.0 / (t * pow(sin(k), 2.0)));
	}
	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 = Math.sqrt(2.0) / k;
	double tmp;
	if (((l * l) <= 5e-309) || !((l * l) <= 5e+255)) {
		tmp = (t_2 * (t / Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * t_1)), 2.0))) * ((t_2 * Math.pow(Math.cbrt(l), 2.0)) / t_1);
	} else {
		tmp = ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) * (2.0 / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(sqrt(2.0) / k)
	tmp = 0.0
	if ((Float64(l * l) <= 5e-309) || !(Float64(l * l) <= 5e+255))
		tmp = Float64(Float64(t_2 * Float64(t / (Float64(t * Float64((cbrt(l) ^ -2.0) * t_1)) ^ 2.0))) * Float64(Float64(t_2 * (cbrt(l) ^ 2.0)) / t_1));
	else
		tmp = Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) * Float64(2.0 / Float64(t * (sin(k) ^ 2.0))));
	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[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[Not[LessEqual[N[(l * l), $MachinePrecision], 5e+255]], $MachinePrecision]], N[(N[(t$95$2 * N[(t / N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309 or 5.0000000000000002e255 < (*.f64 l l)

   1. Initial program 26.0%

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

      1. *-commutative 26.0%

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

      2. associate-/r* 26.0%

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

   3. Simplified 34.3%

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

      1. add-sqr-sqrt 34.2%

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

      2. add-cube-cbrt 34.2%

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

      3. times-frac 34.2%

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

   6. Applied egg-rr 77.4%

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

      1. associate-/r/ 77.4%

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

      2. associate-/r* 77.4%

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

      3. associate-/r/ 77.4%

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

   8. Simplified 77.4%

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

      1. frac-times 73.5%

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

      2. div-inv 73.5%

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

      3. pow-flip 73.5%

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

      4. metadata-eval 73.5%

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

      5. div-inv 73.6%

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

      6. pow-flip 73.5%

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

      7. metadata-eval 73.5%

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

   10. Applied egg-rr 73.5%

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

       1. times-frac 77.4%

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

       2. associate-*r/ 76.8%

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

       3. associate-*l* 76.8%

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

       4. associate-/r* 76.8%

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

       5. times-frac 79.4%

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

   12. Simplified 79.4%

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

       1. associate-*r/ 79.4%

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

   14. Applied egg-rr 79.4%

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

       1. div-inv 79.3%

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

       2. associate-*l/ 76.8%

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

       3. metadata-eval 76.8%

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

       4. pow-flip 76.8%

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

       5. div-inv 76.8%

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

       6. associate-/r/ 80.3%

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

       7. associate-*l/ 80.3%

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

   16. Applied egg-rr 80.3%

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

       1. *-commutative 80.3%

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

       2. associate-*l/ 80.3%

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

       3. *-lft-identity 80.3%

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

       4. associate-/l/ 79.1%

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

       5. *-commutative 79.1%

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

       6. times-frac 83.4%

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

       7. *-inverses 83.4%

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

       8. *-rgt-identity 83.4%

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

   18. Simplified 83.4%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 5.0000000000000002e255

   1. Initial program 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/r* 37.1%

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

   3. Simplified 47.1%

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

      1. add-sqr-sqrt 47.1%

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

      2. add-cube-cbrt 47.0%

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

      3. times-frac 47.0%

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

   6. Applied egg-rr 81.6%

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

      1. associate-/r/ 81.6%

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

      2. associate-/r* 81.6%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 79.8%

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

      2. div-inv 79.8%

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

      3. pow-flip 79.9%

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

      4. metadata-eval 79.9%

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

      5. div-inv 79.9%

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

      6. pow-flip 79.9%

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

      7. metadata-eval 79.9%

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

   10. Applied egg-rr 79.9%

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

       1. times-frac 81.7%

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

       2. associate-*r/ 81.7%

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

       3. associate-*l* 81.6%

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

       4. associate-/r* 81.6%

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

       5. times-frac 81.8%

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

   12. Simplified 81.8%

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

   13. Taylor expanded in k around inf 91.7%

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

       1. associate-*r* 91.8%

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

       2. unpow2 91.8%

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

       3. rem-square-sqrt 91.9%

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

       4. times-frac 93.4%

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

   15. Simplified 93.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309} \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+255}\right):\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot \frac{2}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_3 \cdot \frac{t}{{\left(t \cdot \left(t\_1 \cdot t\_2\right)\right)}^{2}}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;t\_4 \cdot \frac{\frac{t\_3}{t\_1}}{t\_2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot \frac{2}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \frac{t\_3 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\\ \end{array} \end{array} \]

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 (* t_3 (/ t (pow (* t (* t_1 t_2)) 2.0)))))
   (if (<= (* l l) 5e-309)
     (* t_4 (/ (/ t_3 t_1) t_2))
     (if (<= (* l l) 5e+255)
       (*
        (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
        (/ 2.0 (* t (pow (sin k) 2.0))))
       (* t_4 (/ (* t_3 (pow (cbrt l) 2.0)) t_2))))))

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 = t_3 * (t / pow((t * (t_1 * t_2)), 2.0));
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = t_4 * ((t_3 / t_1) / t_2);
	} else if ((l * l) <= 5e+255) {
		tmp = ((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) * (2.0 / (t * pow(sin(k), 2.0)));
	} else {
		tmp = t_4 * ((t_3 * pow(cbrt(l), 2.0)) / t_2);
	}
	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 = t_3 * (t / Math.pow((t * (t_1 * t_2)), 2.0));
	double tmp;
	if ((l * l) <= 5e-309) {
		tmp = t_4 * ((t_3 / t_1) / t_2);
	} else if ((l * l) <= 5e+255) {
		tmp = ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) * (2.0 / (t * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = t_4 * ((t_3 * Math.pow(Math.cbrt(l), 2.0)) / t_2);
	}
	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(t_3 * Float64(t / (Float64(t * Float64(t_1 * t_2)) ^ 2.0)))
	tmp = 0.0
	if (Float64(l * l) <= 5e-309)
		tmp = Float64(t_4 * Float64(Float64(t_3 / t_1) / t_2));
	elseif (Float64(l * l) <= 5e+255)
		tmp = Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) * Float64(2.0 / Float64(t * (sin(k) ^ 2.0))));
	else
		tmp = Float64(t_4 * Float64(Float64(t_3 * (cbrt(l) ^ 2.0)) / t_2));
	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[(t$95$3 * N[(t / N[Power[N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-309], N[(t$95$4 * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+255], N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(N[(t$95$3 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4.9999999999999995e-309

   1. Initial program 19.8%

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

      1. *-commutative 19.8%

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

      2. associate-/r* 19.8%

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

   3. Simplified 35.4%

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

      1. add-sqr-sqrt 35.4%

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

      2. add-cube-cbrt 35.4%

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

      3. times-frac 35.4%

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

   6. Applied egg-rr 76.5%

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

      1. associate-/r/ 76.5%

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

      2. associate-/r* 76.4%

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

      3. associate-/r/ 76.5%

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

   8. Simplified 76.5%

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

      1. frac-times 72.6%

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

      2. div-inv 72.6%

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

      3. pow-flip 72.6%

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

      4. metadata-eval 72.6%

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

      5. div-inv 72.7%

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

      6. pow-flip 72.7%

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

      7. metadata-eval 72.7%

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

   10. Applied egg-rr 72.7%

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

       1. times-frac 76.5%

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

       2. associate-*r/ 76.5%

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

       3. associate-*l* 76.4%

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

       4. associate-/r* 76.5%

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

       5. times-frac 79.2%

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

   12. Simplified 79.2%

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

       1. associate-*r/ 79.2%

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

   14. Applied egg-rr 79.2%

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

       1. associate-/l* 79.2%

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

       2. associate-/l/ 79.2%

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

   16. Applied egg-rr 79.2%

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

       1. associate-*r/ 79.2%

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

       2. associate-*l/ 79.2%

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

       3. associate-/l/ 76.4%

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

       4. associate-/r* 83.6%

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

       5. associate-/l/ 85.2%

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

       6. *-commutative 85.2%

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

       7. times-frac 86.4%

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

       8. *-inverses 86.4%

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

       9. *-rgt-identity 86.4%

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

   18. Simplified 86.4%

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

   if 4.9999999999999995e-309 < (*.f64 l l) < 5.0000000000000002e255

   1. Initial program 37.1%

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

      1. *-commutative 37.1%

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

      2. associate-/r* 37.1%

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

   3. Simplified 47.1%

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

      1. add-sqr-sqrt 47.1%

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

      2. add-cube-cbrt 47.0%

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

      3. times-frac 47.0%

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

   6. Applied egg-rr 81.6%

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

      1. associate-/r/ 81.6%

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

      2. associate-/r* 81.6%

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

      3. associate-/r/ 81.6%

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

   8. Simplified 81.6%

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

      1. frac-times 79.8%

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

      2. div-inv 79.8%

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

      3. pow-flip 79.9%

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

      4. metadata-eval 79.9%

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

      5. div-inv 79.9%

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

      6. pow-flip 79.9%

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

      7. metadata-eval 79.9%

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

   10. Applied egg-rr 79.9%

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

       1. times-frac 81.7%

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

       2. associate-*r/ 81.7%

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

       3. associate-*l* 81.6%

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

       4. associate-/r* 81.6%

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

       5. times-frac 81.8%

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

   12. Simplified 81.8%

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

   13. Taylor expanded in k around inf 91.7%

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

       1. associate-*r* 91.8%

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

       2. unpow2 91.8%

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

       3. rem-square-sqrt 91.9%

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

       4. times-frac 93.4%

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

   15. Simplified 93.4%

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

   if 5.0000000000000002e255 < (*.f64 l l)

   1. Initial program 31.9%

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

      1. *-commutative 31.9%

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

      2. associate-/r* 31.9%

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

   3. Simplified 33.2%

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

      1. add-sqr-sqrt 33.2%

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

      2. add-cube-cbrt 33.1%

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

      3. times-frac 33.1%

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

   6. Applied egg-rr 78.3%

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

      1. associate-/r/ 78.3%

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

      2. associate-/r* 78.3%

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

      3. associate-/r/ 78.4%

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

   8. Simplified 78.4%

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

      1. frac-times 74.4%

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

      2. div-inv 74.4%

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

      3. pow-flip 74.4%

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

      4. metadata-eval 74.4%

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

      5. div-inv 74.4%

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

      6. pow-flip 74.4%

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

      7. metadata-eval 74.4%

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

   10. Applied egg-rr 74.4%

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

       1. times-frac 78.3%

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

       2. associate-*r/ 77.1%

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

       3. associate-*l* 77.2%

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

       4. associate-/r* 77.1%

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

       5. times-frac 79.5%

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

   12. Simplified 79.5%

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

       1. associate-*r/ 79.5%

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

   14. Applied egg-rr 79.5%

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

       1. div-inv 79.5%

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

       2. associate-*l/ 77.2%

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

       3. metadata-eval 77.2%

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

       4. pow-flip 77.2%

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

       5. div-inv 77.2%

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

       6. associate-/r/ 77.2%

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

       7. associate-*l/ 77.2%

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

   16. Applied egg-rr 77.2%

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

       1. *-commutative 77.2%

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

       2. associate-*l/ 77.2%

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

       3. *-lft-identity 77.2%

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

       4. associate-/l/ 73.4%

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

       5. *-commutative 73.4%

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

       6. times-frac 80.7%

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

       7. *-inverses 80.7%

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

       8. *-rgt-identity 80.7%

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

   18. Simplified 80.7%

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

Alternative 4: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{k}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \tan k}\\ t_4 := t \cdot \frac{t\_3}{t\_2}\\ \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{t\_1 \cdot t\_2}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot t\_3\right)\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(t\_1 \cdot t\right)}^{2}}{t\_4}}{{t\_4}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) k))
        (t_2 (pow (cbrt l) 2.0))
        (t_3 (cbrt (* (sin k) (tan k))))
        (t_4 (* t (/ t_3 t_2))))
   (if (<= l 4.8e-164)
     (*
      (sqrt 2.0)
      (*
       (/ t k)
       (/
        (/ (* t_1 t_2) (pow (cbrt k) 2.0))
        (pow (* (pow (cbrt l) -2.0) (* t t_3)) 2.0))))
     (if (<= l 1.55e+195)
       (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))
       (/ (/ (pow (* t_1 t) 2.0) t_4) (pow t_4 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = sqrt(2.0) / k;
	double t_2 = pow(cbrt(l), 2.0);
	double t_3 = cbrt((sin(k) * tan(k)));
	double t_4 = t * (t_3 / t_2);
	double tmp;
	if (l <= 4.8e-164) {
		tmp = sqrt(2.0) * ((t / k) * (((t_1 * t_2) / pow(cbrt(k), 2.0)) / pow((pow(cbrt(l), -2.0) * (t * t_3)), 2.0)));
	} else if (l <= 1.55e+195) {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = (pow((t_1 * t), 2.0) / t_4) / pow(t_4, 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt(2.0) / k;
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_4 = t * (t_3 / t_2);
	double tmp;
	if (l <= 4.8e-164) {
		tmp = Math.sqrt(2.0) * ((t / k) * (((t_1 * t_2) / Math.pow(Math.cbrt(k), 2.0)) / Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * t_3)), 2.0)));
	} else if (l <= 1.55e+195) {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = (Math.pow((t_1 * t), 2.0) / t_4) / Math.pow(t_4, 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sqrt(2.0) / k)
	t_2 = cbrt(l) ^ 2.0
	t_3 = cbrt(Float64(sin(k) * tan(k)))
	t_4 = Float64(t * Float64(t_3 / t_2))
	tmp = 0.0
	if (l <= 4.8e-164)
		tmp = Float64(sqrt(2.0) * Float64(Float64(t / k) * Float64(Float64(Float64(t_1 * t_2) / (cbrt(k) ^ 2.0)) / (Float64((cbrt(l) ^ -2.0) * Float64(t * t_3)) ^ 2.0))));
	elseif (l <= 1.55e+195)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64((Float64(t_1 * t) ^ 2.0) / t_4) / (t_4 ^ 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.8e-164], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t / k), $MachinePrecision] * N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+195], 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], N[(N[(N[Power[N[(t$95$1 * t), $MachinePrecision], 2.0], $MachinePrecision] / t$95$4), $MachinePrecision] / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.79999999999999966e-164

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. add-cube-cbrt 39.8%

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

      3. times-frac 39.8%

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

   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/ 80.1%

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

      2. associate-/r* 80.1%

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

      3. associate-/r/ 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in k around 0 69.0%

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

       1. frac-times 66.9%

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

       2. associate-*l/ 66.9%

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

       3. associate-/r/ 66.9%

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

       4. associate-*l/ 66.9%

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

   11. Applied egg-rr 67.5%

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

       1. times-frac 69.6%

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

       2. associate-*l/ 67.4%

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

       3. associate-/l* 68.6%

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

       4. associate-/l* 68.6%

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

       5. associate-*l* 68.6%

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

   13. Simplified 71.6%

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

   if 4.79999999999999966e-164 < l < 1.5500000000000001e195

   1. Initial program 37.7%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-*r/ 89.5%

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

      2. associate-*r* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. associate-/r* 90.1%

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

   10. Simplified 90.1%

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

   if 1.5500000000000001e195 < l

   1. Initial program 29.4%

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

      1. *-commutative 29.4%

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

      2. associate-/r* 29.4%

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

   3. Simplified 29.4%

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

      1. add-sqr-sqrt 29.4%

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

      2. add-cube-cbrt 29.4%

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

      3. times-frac 29.4%

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

   6. Applied egg-rr 78.6%

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

      1. associate-*l/ 71.0%

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

      2. associate-*r/ 58.5%

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

      3. unpow2 58.5%

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

      4. associate-/r/ 58.4%

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

      5. associate-*l/ 58.4%

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

      6. associate-/l* 58.4%

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

      7. associate-*l/ 58.5%

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

      8. associate-/l* 58.5%

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

   8. Simplified 58.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt{2}}{k} \cdot t\right)}^{2}}{t \cdot \frac{\sqrt[3]{\sin k \cdot \tan k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(t \cdot \frac{\sqrt[3]{\sin k \cdot \tan k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := t\_2 \cdot \frac{t}{t\_1}\\ \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t\_1}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot t\_2\right)\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+195}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_3}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0))
        (t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (* t_2 (/ t t_1))))
   (if (<= l 4.8e-164)
     (*
      (sqrt 2.0)
      (*
       (/ t k)
       (/
        (/ (* (/ (sqrt 2.0) k) t_1) (pow (cbrt k) 2.0))
        (pow (* (pow (cbrt l) -2.0) (* t t_2)) 2.0))))
     (if (<= l 2.2e+195)
       (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))
       (* (/ 2.0 (pow t_3 2.0)) (/ (pow (/ k t) -2.0) t_3))))))

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 = t_2 * (t / t_1);
	double tmp;
	if (l <= 4.8e-164) {
		tmp = sqrt(2.0) * ((t / k) * ((((sqrt(2.0) / k) * t_1) / pow(cbrt(k), 2.0)) / pow((pow(cbrt(l), -2.0) * (t * t_2)), 2.0)));
	} else if (l <= 2.2e+195) {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = (2.0 / pow(t_3, 2.0)) * (pow((k / t), -2.0) / t_3);
	}
	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 = t_2 * (t / t_1);
	double tmp;
	if (l <= 4.8e-164) {
		tmp = Math.sqrt(2.0) * ((t / k) * ((((Math.sqrt(2.0) / k) * t_1) / Math.pow(Math.cbrt(k), 2.0)) / Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * t_2)), 2.0)));
	} else if (l <= 2.2e+195) {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = (2.0 / Math.pow(t_3, 2.0)) * (Math.pow((k / t), -2.0) / t_3);
	}
	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(t_2 * Float64(t / t_1))
	tmp = 0.0
	if (l <= 4.8e-164)
		tmp = Float64(sqrt(2.0) * Float64(Float64(t / k) * Float64(Float64(Float64(Float64(sqrt(2.0) / k) * t_1) / (cbrt(k) ^ 2.0)) / (Float64((cbrt(l) ^ -2.0) * Float64(t * t_2)) ^ 2.0))));
	elseif (l <= 2.2e+195)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / (t_3 ^ 2.0)) * Float64((Float64(k / t) ^ -2.0) / t_3));
	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[(t$95$2 * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.8e-164], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t / k), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+195], 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], N[(N[(2.0 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.79999999999999966e-164

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. add-cube-cbrt 39.8%

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

      3. times-frac 39.8%

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

   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/ 80.1%

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

      2. associate-/r* 80.1%

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

      3. associate-/r/ 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in k around 0 69.0%

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

       1. frac-times 66.9%

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

       2. associate-*l/ 66.9%

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

       3. associate-/r/ 66.9%

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

       4. associate-*l/ 66.9%

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

   11. Applied egg-rr 67.5%

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

       1. times-frac 69.6%

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

       2. associate-*l/ 67.4%

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

       3. associate-/l* 68.6%

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

       4. associate-/l* 68.6%

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

       5. associate-*l* 68.6%

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

   13. Simplified 71.6%

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

   if 4.79999999999999966e-164 < l < 2.2e195

   1. Initial program 37.7%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-*r/ 89.5%

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

      2. associate-*r* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. associate-/r* 90.1%

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

   10. Simplified 90.1%

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

   if 2.2e195 < l

   1. Initial program 29.4%

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

      1. *-commutative 29.4%

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

      2. associate-/r* 29.4%

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

   3. Simplified 29.4%

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

      1. add-cube-cbrt 29.4%

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

      2. div-inv 29.4%

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

      3. times-frac 29.4%

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

   6. Applied egg-rr 58.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+195}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.8e-164)
   (*
    (sqrt 2.0)
    (*
     (/ t k)
     (/
      (/ (* (/ (sqrt 2.0) k) (pow (cbrt l) 2.0)) (pow (cbrt k) 2.0))
      (pow (* (pow (cbrt l) -2.0) (* t (cbrt (* (sin k) (tan k))))) 2.0))))
   (* (* 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) {
	double tmp;
	if (l <= 4.8e-164) {
		tmp = sqrt(2.0) * ((t / k) * ((((sqrt(2.0) / k) * pow(cbrt(l), 2.0)) / pow(cbrt(k), 2.0)) / pow((pow(cbrt(l), -2.0) * (t * cbrt((sin(k) * tan(k))))), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.8e-164) {
		tmp = Math.sqrt(2.0) * ((t / k) * ((((Math.sqrt(2.0) / k) * Math.pow(Math.cbrt(l), 2.0)) / Math.pow(Math.cbrt(k), 2.0)) / Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt((Math.sin(k) * Math.tan(k))))), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.8e-164)
		tmp = Float64(sqrt(2.0) * Float64(Float64(t / k) * Float64(Float64(Float64(Float64(sqrt(2.0) / k) * (cbrt(l) ^ 2.0)) / (cbrt(k) ^ 2.0)) / (Float64((cbrt(l) ^ -2.0) * Float64(t * cbrt(Float64(sin(k) * tan(k))))) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.8e-164], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t / k), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if l < 4.79999999999999966e-164

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. add-cube-cbrt 39.8%

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

      3. times-frac 39.8%

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

   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/ 80.1%

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

      2. associate-/r* 80.1%

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

      3. associate-/r/ 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in k around 0 69.0%

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

       1. frac-times 66.9%

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

       2. associate-*l/ 66.9%

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

       3. associate-/r/ 66.9%

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

       4. associate-*l/ 66.9%

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

   11. Applied egg-rr 67.5%

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

       1. times-frac 69.6%

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

       2. associate-*l/ 67.4%

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

       3. associate-/l* 68.6%

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

       4. associate-/l* 68.6%

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

       5. associate-*l* 68.6%

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

   13. Simplified 71.6%

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

   if 4.79999999999999966e-164 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{k} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := \sqrt[3]{{k}^{2}}\\ t_3 := \frac{\sqrt{2}}{k} \cdot t\\ \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t\_3}{{\left(t\_1 \cdot t\_2\right)}^{2}} \cdot \frac{\frac{t\_3}{t\_1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0)))
        (t_2 (cbrt (pow k 2.0)))
        (t_3 (* (/ (sqrt 2.0) k) t)))
   (if (<= l 2.6e-164)
     (* (/ t_3 (pow (* t_1 t_2) 2.0)) (/ (/ t_3 t_1) t_2))
     (* (* 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) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = cbrt(pow(k, 2.0));
	double t_3 = (sqrt(2.0) / k) * t;
	double tmp;
	if (l <= 2.6e-164) {
		tmp = (t_3 / pow((t_1 * t_2), 2.0)) * ((t_3 / t_1) / t_2);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.cbrt(Math.pow(k, 2.0));
	double t_3 = (Math.sqrt(2.0) / k) * t;
	double tmp;
	if (l <= 2.6e-164) {
		tmp = (t_3 / Math.pow((t_1 * t_2), 2.0)) * ((t_3 / t_1) / t_2);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = cbrt((k ^ 2.0))
	t_3 = Float64(Float64(sqrt(2.0) / k) * t)
	tmp = 0.0
	if (l <= 2.6e-164)
		tmp = Float64(Float64(t_3 / (Float64(t_1 * t_2) ^ 2.0)) * Float64(Float64(t_3 / t_1) / t_2));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[l, 2.6e-164], N[(N[(t$95$3 / N[Power[N[(t$95$1 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if l < 2.6000000000000002e-164

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 39.8%

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

      2. add-cube-cbrt 39.8%

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

      3. times-frac 39.8%

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

   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/ 80.1%

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

      2. associate-/r* 80.1%

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

      3. associate-/r/ 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in k around 0 69.0%

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

   10. Taylor expanded in k around 0 69.1%

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

   if 2.6000000000000002e-164 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot t}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot t}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{{k}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\log \left({\left(e^{\frac{{t}^{3}}{{\ell}^{2}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot \frac{2}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.1e-165)
   (/
    (/ 2.0 (* (/ k t) (/ k t)))
    (log (pow (exp (/ (pow t 3.0) (pow l 2.0))) (* (sin k) (tan k)))))
   (*
    (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
    (/ 2.0 (* t (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-165) {
		tmp = (2.0 / ((k / t) * (k / t))) / log(pow(exp((pow(t, 3.0) / pow(l, 2.0))), (sin(k) * tan(k))));
	} else {
		tmp = ((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) * (2.0 / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-165) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / log((exp(((t ** 3.0d0) / (l ** 2.0d0))) ** (sin(k) * tan(k))))
    else
        tmp = (((l ** 2.0d0) * cos(k)) / (k ** 2.0d0)) * (2.0d0 / (t * (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.1e-165) {
		tmp = (2.0 / ((k / t) * (k / t))) / Math.log(Math.pow(Math.exp((Math.pow(t, 3.0) / Math.pow(l, 2.0))), (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) * (2.0 / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.1e-165:
		tmp = (2.0 / ((k / t) * (k / t))) / math.log(math.pow(math.exp((math.pow(t, 3.0) / math.pow(l, 2.0))), (math.sin(k) * math.tan(k))))
	else:
		tmp = ((math.pow(l, 2.0) * math.cos(k)) / math.pow(k, 2.0)) * (2.0 / (t * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.1e-165)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / log((exp(Float64((t ^ 3.0) / (l ^ 2.0))) ^ Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) * Float64(2.0 / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.1e-165)
		tmp = (2.0 / ((k / t) * (k / t))) / log((exp(((t ^ 3.0) / (l ^ 2.0))) ^ (sin(k) * tan(k))));
	else
		tmp = (((l ^ 2.0) * cos(k)) / (k ^ 2.0)) * (2.0 / (t * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.1e-165], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Exp[N[(N[Power[t, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e-165

   1. Initial program 29.2%

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

      1. *-commutative 29.2%

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

      2. associate-/r* 29.2%

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

   3. Simplified 38.3%

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

      1. unpow3 38.3%

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

      2. times-frac 47.8%

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

      3. pow2 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. +-rgt-identity 47.8%

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

      2. unpow2 47.8%

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

   8. Applied egg-rr 47.8%

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

      1. add-log-exp 28.4%

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

      2. exp-prod 35.7%

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

      3. frac-times 34.8%

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

      4. unpow2 34.8%

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

      5. unpow3 34.8%

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

      6. pow2 34.8%

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

   10. Applied egg-rr 34.8%

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

   if 2.09999999999999995e-165 < k

   1. Initial program 33.6%

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

      1. *-commutative 33.6%

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

      2. associate-/r* 33.6%

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

   3. Simplified 42.4%

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

      1. add-sqr-sqrt 42.4%

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

      2. add-cube-cbrt 42.4%

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

      3. times-frac 42.4%

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

   6. Applied egg-rr 71.3%

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

      1. associate-/r/ 71.3%

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

      2. associate-/r* 71.3%

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

      3. associate-/r/ 71.3%

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

   8. Simplified 71.3%

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

      1. frac-times 68.2%

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

      2. div-inv 68.2%

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

      3. pow-flip 68.2%

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

      4. metadata-eval 68.2%

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

      5. div-inv 68.2%

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

      6. pow-flip 68.2%

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

      7. metadata-eval 68.2%

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

   10. Applied egg-rr 68.2%

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

       1. times-frac 71.4%

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

       2. associate-*r/ 71.4%

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

       3. associate-*l* 71.4%

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

       4. associate-/r* 71.4%

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

       5. times-frac 74.3%

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

   12. Simplified 74.3%

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

   13. Taylor expanded in k around inf 74.9%

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

       1. associate-*r* 74.7%

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

       2. unpow2 74.7%

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

       3. rem-square-sqrt 74.8%

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

       4. times-frac 74.4%

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

   15. Simplified 74.4%

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

Alternative 9: 61.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-259}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.02e-259)
   (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))
   (if (<= l 2.5e-170)
     (pow
      (/
       (/ (sqrt 2.0) (/ k t))
       (* (/ (pow t 1.5) l) (sqrt (* (sin k) (tan k)))))
      2.0)
     (* (* 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) {
	double tmp;
	if (l <= 1.02e-259) {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	} else if (l <= 2.5e-170) {
		tmp = pow(((sqrt(2.0) / (k / t)) / ((pow(t, 1.5) / l) * sqrt((sin(k) * tan(k))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.02d-259) then
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    else if (l <= 2.5d-170) then
        tmp = ((sqrt(2.0d0) / (k / t)) / (((t ** 1.5d0) / l) * sqrt((sin(k) * tan(k))))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.02e-259) {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	} else if (l <= 2.5e-170) {
		tmp = Math.pow(((Math.sqrt(2.0) / (k / t)) / ((Math.pow(t, 1.5) / l) * Math.sqrt((Math.sin(k) * Math.tan(k))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.02e-259:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	elif l <= 2.5e-170:
		tmp = math.pow(((math.sqrt(2.0) / (k / t)) / ((math.pow(t, 1.5) / l) * math.sqrt((math.sin(k) * math.tan(k))))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.02e-259)
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	elseif (l <= 2.5e-170)
		tmp = Float64(Float64(sqrt(2.0) / Float64(k / t)) / Float64(Float64((t ^ 1.5) / l) * sqrt(Float64(sin(k) * tan(k))))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.02e-259)
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	elseif (l <= 2.5e-170)
		tmp = ((sqrt(2.0) / (k / t)) / (((t ^ 1.5) / l) * sqrt((sin(k) * tan(k))))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.02e-259], 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], If[LessEqual[l, 2.5e-170], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 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. Split input into 3 regimes

2. if l < 1.01999999999999995e-259

   1. Initial program 30.1%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 62.1%

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

      1. pow2 62.1%

         \[\leadsto \frac{2}{{k}^{4} \cdot t} \cdot \color{blue}{{\ell}^{2}} \]

      2. add-log-exp 57.8%

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

      3. *-commutative 57.8%

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

      4. exp-prod 52.5%

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

      5. *-commutative 52.5%

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

   7. Applied egg-rr 52.5%

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

   if 1.01999999999999995e-259 < l < 2.50000000000000005e-170

   1. Initial program 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/r* 15.9%

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

   3. Simplified 21.6%

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

      1. add-sqr-sqrt 21.6%

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

      2. pow2 21.6%

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

   6. Applied egg-rr 16.7%

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

   if 2.50000000000000005e-170 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-259}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-260}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{{t}^{1.5}}{\ell}}}{\sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 9e-260)
   (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))
   (if (<= l 2.9e-171)
     (pow
      (/
       (* (/ (sqrt 2.0) k) (/ t (/ (pow t 1.5) l)))
       (sqrt (* (sin k) (tan k))))
      2.0)
     (* (* 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) {
	double tmp;
	if (l <= 9e-260) {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	} else if (l <= 2.9e-171) {
		tmp = pow((((sqrt(2.0) / k) * (t / (pow(t, 1.5) / l))) / sqrt((sin(k) * tan(k)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 9d-260) then
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    else if (l <= 2.9d-171) then
        tmp = (((sqrt(2.0d0) / k) * (t / ((t ** 1.5d0) / l))) / sqrt((sin(k) * tan(k)))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 9e-260) {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	} else if (l <= 2.9e-171) {
		tmp = Math.pow((((Math.sqrt(2.0) / k) * (t / (Math.pow(t, 1.5) / l))) / Math.sqrt((Math.sin(k) * Math.tan(k)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 9e-260:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	elif l <= 2.9e-171:
		tmp = math.pow((((math.sqrt(2.0) / k) * (t / (math.pow(t, 1.5) / l))) / math.sqrt((math.sin(k) * math.tan(k)))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 9e-260)
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	elseif (l <= 2.9e-171)
		tmp = Float64(Float64(Float64(sqrt(2.0) / k) * Float64(t / Float64((t ^ 1.5) / l))) / sqrt(Float64(sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 9e-260)
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	elseif (l <= 2.9e-171)
		tmp = (((sqrt(2.0) / k) * (t / ((t ^ 1.5) / l))) / sqrt((sin(k) * tan(k)))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 9e-260], 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], If[LessEqual[l, 2.9e-171], N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t / N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 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. Split input into 3 regimes

2. if l < 8.9999999999999995e-260

   1. Initial program 30.1%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 62.1%

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

      1. pow2 62.1%

         \[\leadsto \frac{2}{{k}^{4} \cdot t} \cdot \color{blue}{{\ell}^{2}} \]

      2. add-log-exp 57.8%

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

      3. *-commutative 57.8%

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

      4. exp-prod 52.5%

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

      5. *-commutative 52.5%

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

   7. Applied egg-rr 52.5%

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

   if 8.9999999999999995e-260 < l < 2.8999999999999999e-171

   1. Initial program 15.9%

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

      1. *-commutative 15.9%

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

      2. associate-/r* 15.9%

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

   3. Simplified 21.6%

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

      1. add-sqr-sqrt 21.6%

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

      2. sqrt-div 5.4%

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

      3. sqrt-div 5.4%

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

   6. Applied egg-rr 16.7%

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

      1. unpow2 16.7%

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

      2. associate-/r* 16.6%

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

      3. associate-/r/ 16.6%

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

      4. associate-/l* 16.6%

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

   8. Simplified 16.6%

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

   if 2.8999999999999999e-171 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-260}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{k} \cdot \frac{t}{\frac{{t}^{1.5}}{\ell}}}{\sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-258}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-171}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\frac{k}{t} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.05e-258)
   (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))
   (if (<= l 6.2e-171)
     (pow
      (*
       l
       (/ (sqrt (/ 2.0 (pow t 3.0))) (* (/ k t) (sqrt (* (sin k) (tan k))))))
      2.0)
     (* (* 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) {
	double tmp;
	if (l <= 1.05e-258) {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	} else if (l <= 6.2e-171) {
		tmp = pow((l * (sqrt((2.0 / pow(t, 3.0))) / ((k / t) * sqrt((sin(k) * tan(k)))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.05d-258) then
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    else if (l <= 6.2d-171) then
        tmp = (l * (sqrt((2.0d0 / (t ** 3.0d0))) / ((k / t) * sqrt((sin(k) * tan(k)))))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.05e-258) {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	} else if (l <= 6.2e-171) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / Math.pow(t, 3.0))) / ((k / t) * Math.sqrt((Math.sin(k) * Math.tan(k)))))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.05e-258:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	elif l <= 6.2e-171:
		tmp = math.pow((l * (math.sqrt((2.0 / math.pow(t, 3.0))) / ((k / t) * math.sqrt((math.sin(k) * math.tan(k)))))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.05e-258)
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	elseif (l <= 6.2e-171)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / (t ^ 3.0))) / Float64(Float64(k / t) * sqrt(Float64(sin(k) * tan(k)))))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.05e-258)
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	elseif (l <= 6.2e-171)
		tmp = (l * (sqrt((2.0 / (t ^ 3.0))) / ((k / t) * sqrt((sin(k) * tan(k)))))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.05e-258], 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], If[LessEqual[l, 6.2e-171], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(k / t), $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 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. Split input into 3 regimes

2. if l < 1.05e-258

   1. Initial program 30.1%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 62.1%

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

      1. pow2 62.1%

         \[\leadsto \frac{2}{{k}^{4} \cdot t} \cdot \color{blue}{{\ell}^{2}} \]

      2. add-log-exp 57.8%

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

      3. *-commutative 57.8%

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

      4. exp-prod 52.5%

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

      5. *-commutative 52.5%

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

   7. Applied egg-rr 52.5%

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

   if 1.05e-258 < l < 6.2000000000000001e-171

   1. Initial program 15.9%

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

   3. Simplified 21.1%

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

      1. add-sqr-sqrt 21.1%

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

      2. pow2 21.1%

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

   6. Applied egg-rr 27.1%

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

      1. *-commutative 27.1%

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

   8. Simplified 27.1%

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

   if 6.2000000000000001e-171 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-258}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-171}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{\frac{k}{t} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.7e-170)
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (* (* (sin k) (tan k)) (pow (/ (pow t 1.5) l) 2.0)))
   (* (* 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) {
	double tmp;
	if (l <= 4.7e-170) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.7d-170) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.7e-170) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 4.7e-170:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.7e-170)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 4.7e-170)
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.7e-170], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if l < 4.7000000000000002e-170

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 19.3%

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

      2. pow2 19.3%

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

      3. sqrt-div 19.3%

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

      4. sqrt-pow1 22.9%

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

      5. metadata-eval 22.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 4.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 24.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 24.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 24.4%

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

      2. unpow2 24.4%

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

      3. clear-num 24.4%

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

      4. un-div-inv 24.4%

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

   8. Applied egg-rr 24.4%

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

   if 4.7000000000000002e-170 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 3.2e-171)
   (/
    (/ 2.0 (* (/ k t) (/ k t)))
    (* (* (sin k) (tan k)) (pow (/ (pow t 1.5) l) 2.0)))
   (* (* 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) {
	double tmp;
	if (l <= 3.2e-171) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.2d-171) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 3.2e-171) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 3.2e-171:
		tmp = (2.0 / ((k / t) * (k / t))) / ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 3.2e-171)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(sin(k) * tan(k)) * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 3.2e-171)
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 3.2e-171], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if l < 3.2000000000000001e-171

   1. Initial program 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-/r* 28.5%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 19.3%

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

      2. pow2 19.3%

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

      3. sqrt-div 19.3%

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

      4. sqrt-pow1 22.9%

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

      5. metadata-eval 22.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 4.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 24.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 24.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 50.7%

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

      2. unpow2 50.7%

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

   8. Applied egg-rr 24.4%

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

   if 3.2000000000000001e-171 < l

   1. Initial program 35.3%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 78.4%

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

      1. associate-*r/ 78.4%

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

      2. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in k around inf 78.4%

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

      1. associate-/r* 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.6% 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 30.7%

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

3. Simplified 40.1%

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

5. Taylor expanded in t around 0 71.1%

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

   1. associate-*r/ 71.1%

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

   2. associate-*r* 71.1%

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

7. Simplified 71.1%

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

8. Taylor expanded in k around inf 71.1%

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

   1. associate-/r* 71.3%

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

10. Simplified 71.3%

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

11. Final simplification 71.3%

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

Alternative 15: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

3. Simplified 40.1%

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

   1. add-log-exp 37.1%

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

   2. exp-prod 40.8%

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

   3. associate-/r* 40.8%

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

   4. associate-*r* 40.8%

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

   5. *-commutative 40.8%

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

   6. associate-*l* 40.8%

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

   7. pow2 40.8%

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

6. Applied egg-rr 40.8%

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

   1. associate-/r* 40.8%

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

   2. associate-/r* 40.8%

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

   3. associate-/l/ 40.8%

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

8. Simplified 40.8%

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

9. Taylor expanded in k around 0 71.1%

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

    1. associate-*r* 71.1%

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

11. Simplified 71.1%

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

12. Final simplification 71.1%

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

Alternative 16: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * (pow(l, 2.0) / ((sin(k) * tan(k)) * (t * pow(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 = 2.0d0 * ((l ** 2.0d0) / ((sin(k) * tan(k)) * (t * (k ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

3. Simplified 40.1%

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

   1. add-log-exp 37.1%

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

   2. exp-prod 40.8%

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

   3. associate-/r* 40.8%

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

   4. associate-*r* 40.8%

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

   5. *-commutative 40.8%

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

   6. associate-*l* 40.8%

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

   7. pow2 40.8%

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

6. Applied egg-rr 40.8%

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

   1. associate-/r* 40.8%

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

   2. associate-/r* 40.8%

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

   3. associate-/l/ 40.8%

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

8. Simplified 40.8%

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

9. Taylor expanded in k around 0 71.1%

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

    1. associate-*r* 71.1%

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

    2. *-commutative 71.1%

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

11. Simplified 71.1%

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

12. Final simplification 71.1%

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

Alternative 17: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-168} \lor \neg \left(t \leq 1.12 \cdot 10^{+52}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t 3.5e-168) (not (<= t 1.12e+52)))
   (* (* l l) (/ (* 2.0 (cos k)) (* (pow k 2.0) (* t (pow k 2.0)))))
   (/
    (/ 2.0 (* (/ k t) (/ k t)))
    (* (* (sin k) (tan k)) (* (/ (* t t) l) (/ t l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= 3.5e-168) || !(t <= 1.12e+52)) {
		tmp = (l * l) * ((2.0 * cos(k)) / (pow(k, 2.0) * (t * pow(k, 2.0))));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)));
	}
	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 ((t <= 3.5d-168) .or. (.not. (t <= 1.12d+52))) then
        tmp = (l * l) * ((2.0d0 * cos(k)) / ((k ** 2.0d0) * (t * (k ** 2.0d0))))
    else
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= 3.5e-168) || !(t <= 1.12e+52)) {
		tmp = (l * l) * ((2.0 * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(k, 2.0))));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / ((Math.sin(k) * Math.tan(k)) * (((t * t) / l) * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= 3.5e-168) or not (t <= 1.12e+52):
		tmp = (l * l) * ((2.0 * math.cos(k)) / (math.pow(k, 2.0) * (t * math.pow(k, 2.0))))
	else:
		tmp = (2.0 / ((k / t) * (k / t))) / ((math.sin(k) * math.tan(k)) * (((t * t) / l) * (t / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= 3.5e-168) || !(t <= 1.12e+52))
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * cos(k)) / Float64((k ^ 2.0) * Float64(t * (k ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(t * t) / l) * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= 3.5e-168) || ~((t <= 1.12e+52)))
		tmp = (l * l) * ((2.0 * cos(k)) / ((k ^ 2.0) * (t * (k ^ 2.0))));
	else
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, 3.5e-168], N[Not[LessEqual[t, 1.12e+52]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999982e-168 or 1.12000000000000002e52 < t

   1. Initial program 26.9%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in t around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. associate-*r* 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in k around 0 65.6%

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

   if 3.49999999999999982e-168 < t < 1.12000000000000002e52

   1. Initial program 51.4%

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

      1. *-commutative 51.4%

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

      2. associate-/r* 51.4%

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

   3. Simplified 55.2%

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

      1. unpow3 55.3%

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

      2. times-frac 60.8%

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

      3. pow2 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. +-rgt-identity 60.8%

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

      2. unpow2 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. unpow2 60.8%

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

   10. Applied egg-rr 60.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-168} \lor \neg \left(t \leq 1.12 \cdot 10^{+52}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot {k}^{2}\\ \mathbf{if}\;k \leq 3.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\sin k}^{2} \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{t\_1 \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow k 2.0))))
   (if (<= k 3.4e-7)
     (* (* l l) (/ 2.0 (* (pow (sin k) 2.0) t_1)))
     (* (* l l) (/ (* 2.0 (cos k)) (* t_1 (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = t * pow(k, 2.0);
	double tmp;
	if (k <= 3.4e-7) {
		tmp = (l * l) * (2.0 / (pow(sin(k), 2.0) * t_1));
	} else {
		tmp = (l * l) * ((2.0 * cos(k)) / (t_1 * (0.5 - (cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (k ** 2.0d0)
    if (k <= 3.4d-7) then
        tmp = (l * l) * (2.0d0 / ((sin(k) ** 2.0d0) * t_1))
    else
        tmp = (l * l) * ((2.0d0 * cos(k)) / (t_1 * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(k, 2.0);
	double tmp;
	if (k <= 3.4e-7) {
		tmp = (l * l) * (2.0 / (Math.pow(Math.sin(k), 2.0) * t_1));
	} else {
		tmp = (l * l) * ((2.0 * Math.cos(k)) / (t_1 * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t * math.pow(k, 2.0)
	tmp = 0
	if k <= 3.4e-7:
		tmp = (l * l) * (2.0 / (math.pow(math.sin(k), 2.0) * t_1))
	else:
		tmp = (l * l) * ((2.0 * math.cos(k)) / (t_1 * (0.5 - (math.cos((2.0 * k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t * (k ^ 2.0))
	tmp = 0.0
	if (k <= 3.4e-7)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((sin(k) ^ 2.0) * t_1)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * cos(k)) / Float64(t_1 * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t * (k ^ 2.0);
	tmp = 0.0;
	if (k <= 3.4e-7)
		tmp = (l * l) * (2.0 / ((sin(k) ^ 2.0) * t_1));
	else
		tmp = (l * l) * ((2.0 * cos(k)) / (t_1 * (0.5 - (cos((2.0 * k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.4e-7], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999974e-7

   1. Initial program 31.1%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in t around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. associate-*r* 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in k around 0 65.0%

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

   if 3.39999999999999974e-7 < k

   1. Initial program 29.5%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in t around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. associate-*r* 73.9%

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

   7. Simplified 73.9%

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

      1. unpow2 73.9%

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

      2. sin-mult 73.9%

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

   9. Applied egg-rr 73.9%

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

       1. div-sub 73.9%

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

       2. +-inverses 73.9%

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

       3. cos-0 73.9%

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

       4. metadata-eval 73.9%

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

       5. count-2 73.9%

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

   11. Simplified 73.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-171}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.1e-171)
   (* (* l l) (/ (* 2.0 (cos k)) (* t (pow k 4.0))))
   (if (<= t 1.9e+136)
     (/
      (/ 2.0 (* (/ k t) (/ k t)))
      (* (* (sin k) (tan k)) (* (/ (* t t) l) (/ t l))))
     (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.1e-171) {
		tmp = (l * l) * ((2.0 * cos(k)) / (t * pow(k, 4.0)));
	} else if (t <= 1.9e+136) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	}
	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 (t <= 1.1d-171) then
        tmp = (l * l) * ((2.0d0 * cos(k)) / (t * (k ** 4.0d0)))
    else if (t <= 1.9d+136) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.1e-171) {
		tmp = (l * l) * ((2.0 * Math.cos(k)) / (t * Math.pow(k, 4.0)));
	} else if (t <= 1.9e+136) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((Math.sin(k) * Math.tan(k)) * (((t * t) / l) * (t / l)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.1e-171:
		tmp = (l * l) * ((2.0 * math.cos(k)) / (t * math.pow(k, 4.0)))
	elif t <= 1.9e+136:
		tmp = (2.0 / ((k / t) * (k / t))) / ((math.sin(k) * math.tan(k)) * (((t * t) / l) * (t / l)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.1e-171)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * cos(k)) / Float64(t * (k ^ 4.0))));
	elseif (t <= 1.9e+136)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(t * t) / l) * Float64(t / l))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.1e-171)
		tmp = (l * l) * ((2.0 * cos(k)) / (t * (k ^ 4.0)));
	elseif (t <= 1.9e+136)
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t * t) / l) * (t / l)));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.1e-171], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e+136], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1000000000000001e-171

   1. Initial program 29.9%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in t around 0 70.8%

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

      1. associate-*r/ 70.8%

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

      2. associate-*r* 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in k around 0 60.5%

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

   if 1.1000000000000001e-171 < t < 1.90000000000000007e136

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-/r* 50.2%

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

   3. Simplified 55.1%

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

      1. unpow3 55.2%

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

      2. times-frac 61.8%

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

      3. pow2 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. +-rgt-identity 61.8%

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

      2. unpow2 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. unpow2 61.8%

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

   10. Applied egg-rr 61.8%

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

   if 1.90000000000000007e136 < t

   1. Initial program 7.7%

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

      1. *-commutative 7.7%

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

      2. associate-/r* 7.7%

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

   3. Simplified 23.2%

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

      1. add-sqr-sqrt 23.2%

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

      2. add-cube-cbrt 23.2%

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

      3. times-frac 23.2%

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

   6. Applied egg-rr 77.2%

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

      1. associate-/r/ 77.2%

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

      2. associate-/r* 77.2%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in k around 0 76.2%

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

       1. *-commutative 76.2%

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

       2. unpow2 76.2%

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

       3. rem-square-sqrt 76.1%

          \[\leadsto \frac{\color{blue}{2} \cdot {\ell}^{2}}{{k}^{4} \cdot t} \]

       4. associate-*r/ 76.1%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

       5. associate-/r* 76.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   11. Simplified 76.1%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-171}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ (* 2.0 (cos k)) (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * ((2.0 * cos(k)) / (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 * cos(k)) / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * ((2.0 * Math.cos(k)) / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return (l * l) * ((2.0 * math.cos(k)) / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(Float64(2.0 * cos(k)) / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * ((2.0 * cos(k)) / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

3. Simplified 40.1%

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

5. Taylor expanded in t around 0 71.1%

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

   1. associate-*r/ 71.1%

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

   2. associate-*r* 71.1%

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

7. Simplified 71.1%

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

8. Taylor expanded in k around 0 62.1%

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

9. Final simplification 62.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \cos k}{t \cdot {k}^{4}} \]
10. Add Preprocessing

Alternative 21: 62.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{2} \cdot \left(2 \cdot {k}^{-4}\right)}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* (pow l 2.0) (* 2.0 (pow k -4.0))) t))

C

double code(double t, double l, double k) {
	return (pow(l, 2.0) * (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 ** 2.0d0) * (2.0d0 * (k ** (-4.0d0)))) / t
end function

Java

public static double code(double t, double l, double k) {
	return (Math.pow(l, 2.0) * (2.0 * Math.pow(k, -4.0))) / t;
}

Python

def code(t, l, k):
	return (math.pow(l, 2.0) * (2.0 * math.pow(k, -4.0))) / t

Julia

function code(t, l, k)
	return Float64(Float64((l ^ 2.0) * Float64(2.0 * (k ^ -4.0))) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l ^ 2.0) * (2.0 * (k ^ -4.0))) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 60.4%

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

   1. *-commutative 60.4%

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

   2. associate-/r* 60.4%

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

7. Simplified 60.4%

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

   1. div-inv 60.4%

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 60.8%

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

   3. metadata-eval 60.8%

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

9. Applied egg-rr 60.8%

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

    1. associate-*l/ 60.8%

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

11. Simplified 60.8%

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

    1. associate-*l/ 60.9%

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

    2. pow2 60.9%

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

13. Applied egg-rr 60.9%

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

14. Final simplification 60.9%

    \[\leadsto \frac{{\ell}^{2} \cdot \left(2 \cdot {k}^{-4}\right)}{t} \]
15. Add Preprocessing

Alternative 22: 62.7% 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 30.7%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 60.4%

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

   1. *-commutative 60.4%

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

   2. associate-/r* 60.4%

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

7. Simplified 60.4%

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

   1. div-inv 60.4%

      \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow-flip 60.8%

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

   3. metadata-eval 60.8%

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

9. Applied egg-rr 60.8%

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

    1. associate-*l/ 60.8%

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

11. Simplified 60.8%

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

12. Final simplification 60.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t} \]
13. Add Preprocessing

Alternative 23: 62.7% 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 30.7%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 60.4%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))))