Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 91.4%

Time: 29.4s

Alternatives: 17

Speedup: 2.0×

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

• could not determine a ground truth

  1. t = -2.661217915439458e+70
  2. l = -8.956337413243613e+304
  3. k = -8.472106584479608e+133

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: 55.3% 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := \sqrt[3]{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right)}\\ \mathbf{if}\;k\_m \leq 10^{-152}:\\ \;\;\;\;\frac{2}{{\left(t\_1 \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_1 \cdot \left(\left(t\_2 \cdot t\_2\right) \cdot \sqrt[3]{\tan k\_m \cdot \sin k\_m}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0)))
        (t_2 (cbrt (hypot 1.0 (hypot 1.0 (/ k_m t))))))
   (if (<= k_m 1e-152)
     (/
      2.0
      (pow (* t_1 (* (cbrt 2.0) (* (cbrt (tan k_m)) (cbrt (sin k_m))))) 3.0))
     (/
      2.0
      (pow (* t_1 (* (* t_2 t_2) (cbrt (* (tan k_m) (sin k_m))))) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = cbrt(hypot(1.0, hypot(1.0, (k_m / t))));
	double tmp;
	if (k_m <= 1e-152) {
		tmp = 2.0 / pow((t_1 * (cbrt(2.0) * (cbrt(tan(k_m)) * cbrt(sin(k_m))))), 3.0);
	} else {
		tmp = 2.0 / pow((t_1 * ((t_2 * t_2) * cbrt((tan(k_m) * sin(k_m))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.cbrt(Math.hypot(1.0, Math.hypot(1.0, (k_m / t))));
	double tmp;
	if (k_m <= 1e-152) {
		tmp = 2.0 / Math.pow((t_1 * (Math.cbrt(2.0) * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m))))), 3.0);
	} else {
		tmp = 2.0 / Math.pow((t_1 * ((t_2 * t_2) * Math.cbrt((Math.tan(k_m) * Math.sin(k_m))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = cbrt(hypot(1.0, hypot(1.0, Float64(k_m / t))))
	tmp = 0.0
	if (k_m <= 1e-152)
		tmp = Float64(2.0 / (Float64(t_1 * Float64(cbrt(2.0) * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m))))) ^ 3.0));
	else
		tmp = Float64(2.0 / (Float64(t_1 * Float64(Float64(t_2 * t_2) * cbrt(Float64(tan(k_m) * sin(k_m))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := 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[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k$95$m, 1e-152], N[(2.0 / N[Power[N[(t$95$1 * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$1 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000007e-152

   1. Initial program 62.0%

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

      1. +-commutative 62.0%

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

      2. associate-+r+ 62.0%

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

      3. metadata-eval 62.0%

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

      4. associate-*l* 55.3%

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

      5. associate-/r* 59.4%

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

      6. add-cube-cbrt 59.4%

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

      7. pow3 59.4%

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

   4. Applied egg-rr 71.3%

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

      1. *-commutative 71.3%

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

      2. cbrt-prod 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in k around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. cbrt-prod 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 1.00000000000000007e-152 < k

   1. Initial program 55.7%

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

      1. +-commutative 55.7%

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

      2. associate-+r+ 55.7%

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

      3. metadata-eval 55.7%

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

      4. associate-*l* 55.7%

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

      5. associate-/r* 57.2%

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

      6. add-cube-cbrt 57.1%

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

      7. pow3 57.1%

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

   4. Applied egg-rr 75.1%

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

      1. *-commutative 75.1%

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

      2. cbrt-prod 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. pow1/3 74.4%

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

      2. add-sqr-sqrt 74.4%

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

      3. unpow-prod-down 74.4%

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

   8. Applied egg-rr 74.4%

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

      1. unpow1/3 74.6%

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

      2. unpow1/3 75.0%

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

   10. Simplified 92.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-152}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\left(\sqrt[3]{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \sin k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\sqrt[3]{\frac{2}{\sin k\_m}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_1}^{2}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}} \cdot \frac{t\_1}{\tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (cbrt (/ 2.0 (sin k_m))) (/ t (pow (cbrt l) 2.0)))))
   (if (<= t 1.45e-84)
     (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
     (* (/ (pow t_1 2.0) (+ 2.0 (pow (/ k_m t) 2.0))) (/ t_1 (tan k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cbrt((2.0 / sin(k_m))) / (t / pow(cbrt(l), 2.0));
	double tmp;
	if (t <= 1.45e-84) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else {
		tmp = (pow(t_1, 2.0) / (2.0 + pow((k_m / t), 2.0))) * (t_1 / tan(k_m));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cbrt((2.0 / Math.sin(k_m))) / (t / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (t <= 1.45e-84) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else {
		tmp = (Math.pow(t_1, 2.0) / (2.0 + Math.pow((k_m / t), 2.0))) * (t_1 / Math.tan(k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cbrt(Float64(2.0 / sin(k_m))) / Float64(t / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (t <= 1.45e-84)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	else
		tmp = Float64(Float64((t_1 ^ 2.0) / Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(t_1 / tan(k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.45e-84], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-84

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 1.4500000000000001e-84 < t

   1. Initial program 64.4%

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

   3. Simplified 68.4%

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

      1. associate-/l/ 68.4%

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

      2. add-cube-cbrt 68.3%

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

      3. times-frac 68.4%

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

   6. Applied egg-rr 95.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\frac{2}{\sin k\_m} \cdot \frac{\ell}{{t}^{3}}\right)}{\tan k\_m}}{2 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 t_1))
          (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))))
        1e+270)
     (/
      (/ (* l (* (/ 2.0 (sin k_m)) (/ l (pow t 3.0)))) (tan k_m))
      (+ 2.0 t_1))
     (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))))) <= 1e+270) {
		tmp = ((l * ((2.0 / sin(k_m)) * (l / pow(t, 3.0)))) / tan(k_m)) / (2.0 + t_1);
	} else {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k_m / t) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (1.0d0 + t_1)) * (tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))))) <= 1d+270) then
        tmp = ((l * ((2.0d0 / sin(k_m)) * (l / (t ** 3.0d0)))) / tan(k_m)) / (2.0d0 + t_1)
    else
        tmp = (l / tan(k_m)) * ((l / (k_m ** 2.0d0)) * (2.0d0 / (t * sin(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))))) <= 1e+270) {
		tmp = ((l * ((2.0 / Math.sin(k_m)) * (l / Math.pow(t, 3.0)))) / Math.tan(k_m)) / (2.0 + t_1);
	} else {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow((k_m / t), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (1.0 + t_1)) * (math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))))) <= 1e+270:
		tmp = ((l * ((2.0 / math.sin(k_m)) * (l / math.pow(t, 3.0)))) / math.tan(k_m)) / (2.0 + t_1)
	else:
		tmp = (l / math.tan(k_m)) * ((l / math.pow(k_m, 2.0)) * (2.0 / (t * math.sin(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 1e+270)
		tmp = Float64(Float64(Float64(l * Float64(Float64(2.0 / sin(k_m)) * Float64(l / (t ^ 3.0)))) / tan(k_m)) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))))) <= 1e+270)
		tmp = ((l * ((2.0 / sin(k_m)) * (l / (t ^ 3.0)))) / tan(k_m)) / (2.0 + t_1);
	else
		tmp = (l / tan(k_m)) * ((l / (k_m ^ 2.0)) * (2.0 / (t * sin(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+270], N[(N[(N[(l * N[(N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1e270

   1. Initial program 80.4%

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

   3. Simplified 82.5%

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

      1. associate-/r/ 87.0%

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

      2. div-inv 87.0%

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

      3. clear-num 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 1e270 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 31.8%

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

   3. Simplified 37.9%

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

      1. add-cube-cbrt 37.9%

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

      2. pow3 37.9%

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

      3. cbrt-div 37.9%

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

      4. rem-cbrt-cube 56.1%

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

   6. Applied egg-rr 56.1%

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

      1. *-un-lft-identity 56.1%

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

      2. associate-/r/ 54.2%

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

      3. cube-div 37.9%

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

      4. pow3 37.9%

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

      5. add-cube-cbrt 37.9%

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

   8. Applied egg-rr 37.9%

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

      1. *-lft-identity 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-/l/ 37.9%

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

   10. Simplified 37.9%

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

       1. *-un-lft-identity 37.9%

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

       2. associate-/l/ 37.9%

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

       3. associate-*r/ 37.9%

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

       4. associate-*l/ 37.9%

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

   12. Applied egg-rr 37.9%

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

       1. *-lft-identity 37.9%

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

       2. associate-/l* 37.9%

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

       3. *-commutative 37.9%

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

       4. times-frac 37.9%

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

       5. associate-/r/ 37.9%

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

       6. *-commutative 37.9%

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

   14. Simplified 37.9%

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

   15. Taylor expanded in k around inf 74.7%

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

       1. associate-*r/ 74.7%

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

       2. *-commutative 74.7%

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

       3. times-frac 75.6%

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

   17. Simplified 75.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 10^{+270}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k\_m}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3}}{\tan k\_m}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9.6e-85)
   (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
   (/
    (/
     (pow (/ (cbrt (/ 2.0 (sin k_m))) (/ t (pow (cbrt l) 2.0))) 3.0)
     (tan k_m))
    (+ 2.0 (pow (/ k_m t) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.6e-85) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else {
		tmp = (pow((cbrt((2.0 / sin(k_m))) / (t / pow(cbrt(l), 2.0))), 3.0) / tan(k_m)) / (2.0 + pow((k_m / t), 2.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.6e-85) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else {
		tmp = (Math.pow((Math.cbrt((2.0 / Math.sin(k_m))) / (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) / Math.tan(k_m)) / (2.0 + Math.pow((k_m / t), 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9.6e-85)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	else
		tmp = Float64(Float64((Float64(cbrt(Float64(2.0 / sin(k_m))) / Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) / tan(k_m)) / Float64(2.0 + (Float64(k_m / t) ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9.6e-85], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[Power[N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.6000000000000002e-85

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 9.6000000000000002e-85 < t

   1. Initial program 64.4%

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

   3. Simplified 68.4%

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

      1. add-cube-cbrt 68.3%

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

      2. pow2 68.3%

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

      3. cbrt-div 68.2%

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

      4. associate-/r* 65.0%

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

      5. cbrt-div 65.1%

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

      6. rem-cbrt-cube 65.1%

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

      7. cbrt-prod 68.2%

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

      8. pow2 68.2%

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

      9. cbrt-div 68.2%

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

      10. associate-/r* 65.0%

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

      11. cbrt-div 65.0%

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

      12. rem-cbrt-cube 73.9%

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

      13. cbrt-prod 90.2%

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

      14. pow2 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. unpow2 90.2%

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

      2. unpow3 90.2%

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

   8. Simplified 90.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{{t}^{1.5}}{\ell}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_1} \cdot \frac{\frac{1}{\sin k\_m}}{t\_1}}{\tan k\_m}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (pow t 1.5) l)))
   (if (<= t 1.6e-84)
     (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
     (if (<= t 5.8e+203)
       (/
        (/ (* (/ 2.0 t_1) (/ (/ 1.0 (sin k_m)) t_1)) (tan k_m))
        (+ 2.0 (pow (/ k_m t) 2.0)))
       (/ 1.0 (pow (* (pow (cbrt k_m) 2.0) (* t (pow (cbrt l) -2.0))) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(t, 1.5) / l;
	double tmp;
	if (t <= 1.6e-84) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 5.8e+203) {
		tmp = (((2.0 / t_1) * ((1.0 / sin(k_m)) / t_1)) / tan(k_m)) / (2.0 + pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / pow((pow(cbrt(k_m), 2.0) * (t * pow(cbrt(l), -2.0))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(t, 1.5) / l;
	double tmp;
	if (t <= 1.6e-84) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 5.8e+203) {
		tmp = (((2.0 / t_1) * ((1.0 / Math.sin(k_m)) / t_1)) / Math.tan(k_m)) / (2.0 + Math.pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / Math.pow((Math.pow(Math.cbrt(k_m), 2.0) * (t * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((t ^ 1.5) / l)
	tmp = 0.0
	if (t <= 1.6e-84)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 5.8e+203)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_1) * Float64(Float64(1.0 / sin(k_m)) / t_1)) / tan(k_m)) / Float64(2.0 + (Float64(k_m / t) ^ 2.0)));
	else
		tmp = Float64(1.0 / (Float64((cbrt(k_m) ^ 2.0) * Float64(t * (cbrt(l) ^ -2.0))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.6e-84], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+203], N[(N[(N[(N[(2.0 / t$95$1), $MachinePrecision] * N[(N[(1.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.6e-84

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 1.6e-84 < t < 5.80000000000000021e203

   1. Initial program 60.3%

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

   3. Simplified 63.9%

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

      1. div-inv 63.9%

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

      2. add-sqr-sqrt 63.9%

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

      3. times-frac 63.9%

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

      4. associate-/r* 61.4%

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

      5. sqrt-div 61.3%

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

      6. sqrt-pow1 61.3%

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

      7. metadata-eval 61.3%

         \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. sqrt-prod 33.0%

         \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      9. add-sqr-sqrt 45.3%

         \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\color{blue}{\ell}}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      10. associate-/r* 42.8%

          \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      11. sqrt-div 42.8%

          \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      12. sqrt-pow1 49.8%

          \[\leadsto \frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      13. metadata-eval 49.8%

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

      14. sqrt-prod 53.4%

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

      15. add-sqr-sqrt 91.0%

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

   6. Applied egg-rr 91.0%

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

   if 5.80000000000000021e203 < t

   1. Initial program 76.0%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. *-commutative 60.3%

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

   7. Simplified 60.3%

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

      1. clear-num 60.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 60.3%

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

   9. Applied egg-rr 60.3%

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

       1. unpow-1 60.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 60.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 60.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 60.3%

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

       2. pow3 60.3%

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

   13. Applied egg-rr 99.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\frac{{t}^{1.5}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k\_m \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k\_m}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.45e-84)
   (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
   (if (<= t 5.4e+140)
     (/
      (/ (* l (/ 2.0 (* (sin k_m) (* t (/ (pow t 2.0) l))))) (tan k_m))
      (+ 2.0 (pow (/ k_m t) 2.0)))
     (/ 1.0 (pow (* (pow (cbrt k_m) 2.0) (* t (pow (cbrt l) -2.0))) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.45e-84) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 5.4e+140) {
		tmp = ((l * (2.0 / (sin(k_m) * (t * (pow(t, 2.0) / l))))) / tan(k_m)) / (2.0 + pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / pow((pow(cbrt(k_m), 2.0) * (t * pow(cbrt(l), -2.0))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.45e-84) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 5.4e+140) {
		tmp = ((l * (2.0 / (Math.sin(k_m) * (t * (Math.pow(t, 2.0) / l))))) / Math.tan(k_m)) / (2.0 + Math.pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / Math.pow((Math.pow(Math.cbrt(k_m), 2.0) * (t * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.45e-84)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 5.4e+140)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(sin(k_m) * Float64(t * Float64((t ^ 2.0) / l))))) / tan(k_m)) / Float64(2.0 + (Float64(k_m / t) ^ 2.0)));
	else
		tmp = Float64(1.0 / (Float64((cbrt(k_m) ^ 2.0) * Float64(t * (cbrt(l) ^ -2.0))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.45e-84], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e+140], N[(N[(N[(l * N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4500000000000001e-84

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 1.4500000000000001e-84 < t < 5.40000000000000036e140

   1. Initial program 63.6%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.3%

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

      2. pow3 67.3%

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

      3. cbrt-div 67.3%

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

      4. rem-cbrt-cube 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. *-un-lft-identity 77.5%

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

      2. associate-/r/ 89.4%

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

      3. cube-div 79.5%

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

      4. pow3 79.4%

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

      5. add-cube-cbrt 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. *-lft-identity 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-/l/ 79.8%

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

   10. Simplified 79.8%

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

       1. cube-mult 79.7%

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

       2. *-un-lft-identity 79.7%

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

       3. times-frac 89.8%

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

       4. pow2 89.8%

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

   12. Applied egg-rr 89.8%

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

   if 5.40000000000000036e140 < t

   1. Initial program 65.9%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in k around 0 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

      1. clear-num 51.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 51.0%

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

   9. Applied egg-rr 51.0%

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

       1. unpow-1 51.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 51.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 51.0%

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

       2. pow3 51.0%

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

   13. Applied egg-rr 89.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k\_m \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k\_m}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2}}}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.8e-85)
   (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
   (if (<= t 9e+140)
     (/
      (/ (* l (/ 2.0 (* (sin k_m) (* t (/ (pow t 2.0) l))))) (tan k_m))
      (+ 2.0 (pow (/ k_m t) 2.0)))
     (pow (/ (cbrt (pow l 2.0)) (* t (pow (cbrt k_m) 2.0))) 3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.8e-85) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 9e+140) {
		tmp = ((l * (2.0 / (sin(k_m) * (t * (pow(t, 2.0) / l))))) / tan(k_m)) / (2.0 + pow((k_m / t), 2.0));
	} else {
		tmp = pow((cbrt(pow(l, 2.0)) / (t * pow(cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.8e-85) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 9e+140) {
		tmp = ((l * (2.0 / (Math.sin(k_m) * (t * (Math.pow(t, 2.0) / l))))) / Math.tan(k_m)) / (2.0 + Math.pow((k_m / t), 2.0));
	} else {
		tmp = Math.pow((Math.cbrt(Math.pow(l, 2.0)) / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.8e-85)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 9e+140)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(sin(k_m) * Float64(t * Float64((t ^ 2.0) / l))))) / tan(k_m)) / Float64(2.0 + (Float64(k_m / t) ^ 2.0)));
	else
		tmp = Float64(cbrt((l ^ 2.0)) / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.8e-85], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+140], N[(N[(N[(l * N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[Power[l, 2.0], $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999999e-85

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 3.7999999999999999e-85 < t < 9.0000000000000003e140

   1. Initial program 63.6%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.3%

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

      2. pow3 67.3%

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

      3. cbrt-div 67.3%

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

      4. rem-cbrt-cube 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. *-un-lft-identity 77.5%

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

      2. associate-/r/ 89.4%

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

      3. cube-div 79.5%

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

      4. pow3 79.4%

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

      5. add-cube-cbrt 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. *-lft-identity 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-/l/ 79.8%

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

   10. Simplified 79.8%

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

       1. cube-mult 79.7%

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

       2. *-un-lft-identity 79.7%

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

       3. times-frac 89.8%

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

       4. pow2 89.8%

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

   12. Applied egg-rr 89.8%

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

   if 9.0000000000000003e140 < t

   1. Initial program 65.9%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in k around 0 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

      1. add-cube-cbrt 51.0%

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

      2. pow3 51.0%

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

      3. cbrt-div 51.0%

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

      4. cbrt-prod 51.0%

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

      5. unpow3 51.0%

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

      6. add-cbrt-cube 61.5%

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

      7. unpow2 61.5%

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

      8. cbrt-prod 85.8%

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

      9. pow2 85.8%

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

   9. Applied egg-rr 85.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2}}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k\_m}\\ t_2 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 8.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+100}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{2}{\sin k\_m} \cdot \frac{\ell}{{t}^{3}}}{2 + t\_2}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (tan k_m))) (t_2 (pow (/ k_m t) 2.0)))
   (if (<= t 8.2e-85)
     (* t_1 (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
     (if (<= t 2e+100)
       (* t_1 (/ (* (/ 2.0 (sin k_m)) (/ l (pow t 3.0))) (+ 2.0 t_2)))
       (if (<= t 8.2e+141)
         (/
          2.0
          (*
           (* (tan k_m) (* (sin k_m) (* (/ (pow t 2.0) l) (/ t l))))
           (+ 1.0 (+ 1.0 t_2))))
         (/ 1.0 (/ (pow (* t (pow (cbrt k_m) 2.0)) 3.0) (pow l 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / tan(k_m);
	double t_2 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 8.2e-85) {
		tmp = t_1 * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 2e+100) {
		tmp = t_1 * (((2.0 / sin(k_m)) * (l / pow(t, 3.0))) / (2.0 + t_2));
	} else if (t <= 8.2e+141) {
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * ((pow(t, 2.0) / l) * (t / l)))) * (1.0 + (1.0 + t_2)));
	} else {
		tmp = 1.0 / (pow((t * pow(cbrt(k_m), 2.0)), 3.0) / pow(l, 2.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.tan(k_m);
	double t_2 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 8.2e-85) {
		tmp = t_1 * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 2e+100) {
		tmp = t_1 * (((2.0 / Math.sin(k_m)) * (l / Math.pow(t, 3.0))) / (2.0 + t_2));
	} else if (t <= 8.2e+141) {
		tmp = 2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * ((Math.pow(t, 2.0) / l) * (t / l)))) * (1.0 + (1.0 + t_2)));
	} else {
		tmp = 1.0 / (Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0) / Math.pow(l, 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / tan(k_m))
	t_2 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 8.2e-85)
		tmp = Float64(t_1 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 2e+100)
		tmp = Float64(t_1 * Float64(Float64(Float64(2.0 / sin(k_m)) * Float64(l / (t ^ 3.0))) / Float64(2.0 + t_2)));
	elseif (t <= 8.2e+141)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))) * Float64(1.0 + Float64(1.0 + t_2))));
	else
		tmp = Float64(1.0 / Float64((Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0) / (l ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 8.2e-85], N[(t$95$1 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+100], N[(t$95$1 * N[(N[(N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.2e+141], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 8.19999999999999987e-85

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 8.19999999999999987e-85 < t < 2.00000000000000003e100

   1. Initial program 72.8%

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

   3. Simplified 77.6%

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

      1. associate-/l/ 77.5%

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

      2. associate-/r/ 92.5%

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

      3. times-frac 92.5%

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

      4. div-inv 92.5%

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

      5. clear-num 92.5%

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

   6. Applied egg-rr 92.5%

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

   if 2.00000000000000003e100 < t < 8.20000000000000044e141

   1. Initial program 23.6%

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

      1. unpow3 23.6%

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

      2. times-frac 78.4%

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

      3. pow2 78.4%

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

   4. Applied egg-rr 78.4%

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

   if 8.20000000000000044e141 < t

   1. Initial program 65.9%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in k around 0 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

      1. clear-num 51.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 51.0%

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

   9. Applied egg-rr 51.0%

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

       1. unpow-1 51.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 51.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 51.0%

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

       2. pow3 51.0%

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

       3. *-commutative 51.0%

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

       4. cbrt-prod 51.0%

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

       5. unpow3 51.0%

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

       6. add-cbrt-cube 54.9%

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

       7. unpow2 54.9%

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

       8. cbrt-prod 76.2%

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

       9. pow2 76.2%

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

   13. Applied egg-rr 76.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k\_m \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k\_m}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.65e-84)
   (* (/ l (tan k_m)) (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
   (if (<= t 7e+139)
     (/
      (/ (* l (/ 2.0 (* (sin k_m) (* t (/ (pow t 2.0) l))))) (tan k_m))
      (+ 2.0 (pow (/ k_m t) 2.0)))
     (/ 1.0 (/ (pow (* t (pow (cbrt k_m) 2.0)) 3.0) (pow l 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.65e-84) {
		tmp = (l / tan(k_m)) * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 7e+139) {
		tmp = ((l * (2.0 / (sin(k_m) * (t * (pow(t, 2.0) / l))))) / tan(k_m)) / (2.0 + pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / (pow((t * pow(cbrt(k_m), 2.0)), 3.0) / pow(l, 2.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.65e-84) {
		tmp = (l / Math.tan(k_m)) * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 7e+139) {
		tmp = ((l * (2.0 / (Math.sin(k_m) * (t * (Math.pow(t, 2.0) / l))))) / Math.tan(k_m)) / (2.0 + Math.pow((k_m / t), 2.0));
	} else {
		tmp = 1.0 / (Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0) / Math.pow(l, 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.65e-84)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 7e+139)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(sin(k_m) * Float64(t * Float64((t ^ 2.0) / l))))) / tan(k_m)) / Float64(2.0 + (Float64(k_m / t) ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64((Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0) / (l ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.65e-84], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e+139], N[(N[(N[(l * N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999992e-84

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 1.64999999999999992e-84 < t < 6.99999999999999957e139

   1. Initial program 63.6%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.3%

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

      2. pow3 67.3%

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

      3. cbrt-div 67.3%

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

      4. rem-cbrt-cube 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. *-un-lft-identity 77.5%

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

      2. associate-/r/ 89.4%

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

      3. cube-div 79.5%

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

      4. pow3 79.4%

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

      5. add-cube-cbrt 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. *-lft-identity 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-/l/ 79.8%

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

   10. Simplified 79.8%

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

       1. cube-mult 79.7%

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

       2. *-un-lft-identity 79.7%

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

       3. times-frac 89.8%

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

       4. pow2 89.8%

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

   12. Applied egg-rr 89.8%

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

   if 6.99999999999999957e139 < t

   1. Initial program 65.9%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in k around 0 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

      1. clear-num 51.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 51.0%

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

   9. Applied egg-rr 51.0%

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

       1. unpow-1 51.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 51.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 51.0%

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

       2. pow3 51.0%

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

       3. *-commutative 51.0%

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

       4. cbrt-prod 51.0%

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

       5. unpow3 51.0%

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

       6. add-cbrt-cube 54.9%

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

       7. unpow2 54.9%

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

       8. cbrt-prod 76.2%

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

       9. pow2 76.2%

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

   13. Applied egg-rr 76.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\sin k \cdot \left(t \cdot \frac{{t}^{2}}{\ell}\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k\_m}\\ \mathbf{if}\;t \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_1 \cdot \frac{\ell \cdot \frac{2}{\sin k\_m \cdot {t}^{3}}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (tan k_m))))
   (if (<= t 1.8e-33)
     (* t_1 (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
     (if (<= t 5.6e+102)
       (*
        t_1
        (/
         (* l (/ 2.0 (* (sin k_m) (pow t 3.0))))
         (+ 2.0 (pow (/ k_m t) 2.0))))
       (/ 1.0 (/ (pow (* t (pow (cbrt k_m) 2.0)) 3.0) (pow l 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / tan(k_m);
	double tmp;
	if (t <= 1.8e-33) {
		tmp = t_1 * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 5.6e+102) {
		tmp = t_1 * ((l * (2.0 / (sin(k_m) * pow(t, 3.0)))) / (2.0 + pow((k_m / t), 2.0)));
	} else {
		tmp = 1.0 / (pow((t * pow(cbrt(k_m), 2.0)), 3.0) / pow(l, 2.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.tan(k_m);
	double tmp;
	if (t <= 1.8e-33) {
		tmp = t_1 * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 5.6e+102) {
		tmp = t_1 * ((l * (2.0 / (Math.sin(k_m) * Math.pow(t, 3.0)))) / (2.0 + Math.pow((k_m / t), 2.0)));
	} else {
		tmp = 1.0 / (Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0) / Math.pow(l, 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / tan(k_m))
	tmp = 0.0
	if (t <= 1.8e-33)
		tmp = Float64(t_1 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 5.6e+102)
		tmp = Float64(t_1 * Float64(Float64(l * Float64(2.0 / Float64(sin(k_m) * (t ^ 3.0)))) / Float64(2.0 + (Float64(k_m / t) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64((Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0) / (l ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.8e-33], N[(t$95$1 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], N[(t$95$1 * N[(N[(l * N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.80000000000000017e-33

   1. Initial program 59.3%

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

   3. Simplified 63.0%

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

      1. add-cube-cbrt 62.9%

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

      2. pow3 62.9%

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

      3. cbrt-div 62.9%

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

      4. rem-cbrt-cube 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. *-un-lft-identity 70.6%

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

      2. associate-/r/ 71.6%

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

      3. cube-div 64.9%

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

      4. pow3 64.9%

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

      5. add-cube-cbrt 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. *-lft-identity 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-/l/ 65.0%

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

   10. Simplified 65.0%

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

       1. *-un-lft-identity 65.0%

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

       2. associate-/l/ 65.0%

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

       3. associate-*r/ 65.0%

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

       4. associate-*l/ 62.5%

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

   12. Applied egg-rr 62.5%

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

       1. *-lft-identity 62.5%

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

       2. associate-/l* 62.5%

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

       3. *-commutative 62.5%

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

       4. times-frac 63.0%

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

       5. associate-/r/ 62.6%

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

       6. *-commutative 62.6%

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

   14. Simplified 62.6%

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

   15. Taylor expanded in k around inf 75.5%

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

       1. associate-*r/ 75.5%

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

       2. *-commutative 75.5%

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

       3. times-frac 76.1%

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

   17. Simplified 76.1%

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

   if 1.80000000000000017e-33 < t < 5.60000000000000037e102

   1. Initial program 70.0%

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

   3. Simplified 74.9%

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

      1. add-cube-cbrt 74.5%

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

      2. pow3 74.5%

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

      3. cbrt-div 74.4%

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

      4. rem-cbrt-cube 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. *-un-lft-identity 74.5%

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

      2. associate-/r/ 85.2%

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

      3. cube-div 85.3%

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

      4. pow3 85.2%

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

      5. add-cube-cbrt 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. *-lft-identity 85.8%

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

      2. *-commutative 85.8%

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

      3. associate-/l/ 85.8%

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

   10. Simplified 85.8%

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

       1. *-un-lft-identity 85.8%

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

       2. associate-/l/ 85.7%

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

       3. associate-*r/ 85.7%

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

       4. associate-*l/ 82.2%

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

   12. Applied egg-rr 82.2%

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

       1. *-lft-identity 82.2%

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

       2. associate-/l* 82.2%

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

       3. *-commutative 82.2%

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

       4. times-frac 85.8%

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

       5. associate-/r/ 85.6%

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

       6. *-commutative 85.6%

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

   14. Simplified 85.6%

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

   if 5.60000000000000037e102 < t

   1. Initial program 57.0%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in k around 0 45.2%

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

      1. *-commutative 45.2%

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

   7. Simplified 45.2%

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

      1. clear-num 45.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 45.2%

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

   9. Applied egg-rr 45.2%

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

       1. unpow-1 45.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 45.2%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 45.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 45.2%

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

       2. pow3 45.2%

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

       3. *-commutative 45.2%

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

       4. cbrt-prod 45.2%

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

       5. unpow3 45.2%

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

       6. add-cbrt-cube 48.8%

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

       7. unpow2 48.8%

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

       8. cbrt-prod 70.4%

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

       9. pow2 70.4%

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

   13. Applied egg-rr 70.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell \cdot \frac{2}{\sin k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k\_m}\\ \mathbf{if}\;t \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{2}{\sin k\_m} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (tan k_m))))
   (if (<= t 5.2e-85)
     (* t_1 (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m)))))
     (if (<= t 1.75e+102)
       (*
        t_1
        (/
         (* (/ 2.0 (sin k_m)) (/ l (pow t 3.0)))
         (+ 2.0 (pow (/ k_m t) 2.0))))
       (/ 1.0 (/ (pow (* t (pow (cbrt k_m) 2.0)) 3.0) (pow l 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / tan(k_m);
	double tmp;
	if (t <= 5.2e-85) {
		tmp = t_1 * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	} else if (t <= 1.75e+102) {
		tmp = t_1 * (((2.0 / sin(k_m)) * (l / pow(t, 3.0))) / (2.0 + pow((k_m / t), 2.0)));
	} else {
		tmp = 1.0 / (pow((t * pow(cbrt(k_m), 2.0)), 3.0) / pow(l, 2.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.tan(k_m);
	double tmp;
	if (t <= 5.2e-85) {
		tmp = t_1 * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	} else if (t <= 1.75e+102) {
		tmp = t_1 * (((2.0 / Math.sin(k_m)) * (l / Math.pow(t, 3.0))) / (2.0 + Math.pow((k_m / t), 2.0)));
	} else {
		tmp = 1.0 / (Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0) / Math.pow(l, 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / tan(k_m))
	tmp = 0.0
	if (t <= 5.2e-85)
		tmp = Float64(t_1 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	elseif (t <= 1.75e+102)
		tmp = Float64(t_1 * Float64(Float64(Float64(2.0 / sin(k_m)) * Float64(l / (t ^ 3.0))) / Float64(2.0 + (Float64(k_m / t) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64((Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0) / (l ^ 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5.2e-85], N[(t$95$1 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e+102], N[(t$95$1 * N[(N[(N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.20000000000000023e-85

   1. Initial program 58.2%

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

   3. Simplified 61.9%

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

      1. add-cube-cbrt 61.9%

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

      2. pow3 61.9%

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

      3. cbrt-div 61.9%

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

      4. rem-cbrt-cube 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. associate-/r/ 69.6%

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

      3. cube-div 62.4%

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

      4. pow3 62.4%

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

      5. add-cube-cbrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. *-lft-identity 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-/l/ 62.5%

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

   10. Simplified 62.5%

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

       1. *-un-lft-identity 62.5%

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

       2. associate-/l/ 62.5%

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

       3. associate-*r/ 62.5%

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

       4. associate-*l/ 61.5%

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

   12. Applied egg-rr 61.5%

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

       1. *-lft-identity 61.5%

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

       2. associate-/l* 61.5%

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

       3. *-commutative 61.5%

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

       4. times-frac 62.1%

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

       5. associate-/r/ 62.0%

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

       6. *-commutative 62.0%

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

   14. Simplified 62.0%

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

   15. Taylor expanded in k around inf 75.9%

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

       1. associate-*r/ 75.9%

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

       2. *-commutative 75.9%

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

       3. times-frac 76.5%

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

   17. Simplified 76.5%

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

   if 5.20000000000000023e-85 < t < 1.75000000000000005e102

   1. Initial program 72.8%

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

   3. Simplified 77.6%

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

      1. associate-/l/ 77.5%

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

      2. associate-/r/ 92.5%

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

      3. times-frac 92.5%

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

      4. div-inv 92.5%

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

      5. clear-num 92.5%

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

   6. Applied egg-rr 92.5%

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

   if 1.75000000000000005e102 < t

   1. Initial program 55.6%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in k around 0 44.0%

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

      1. *-commutative 44.0%

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

   7. Simplified 44.0%

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

      1. clear-num 44.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. inv-pow 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. unpow-1 44.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}} \]

       2. *-commutative 44.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} \cdot {t}^{3}}}{{\ell}^{2}}} \]

   11. Simplified 44.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-cube-cbrt 44.0%

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

       2. pow3 44.0%

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

       3. *-commutative 44.0%

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

       4. cbrt-prod 44.0%

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

       5. unpow3 44.0%

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

       6. add-cbrt-cube 47.5%

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

       7. unpow2 47.5%

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

       8. cbrt-prod 68.6%

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

       9. pow2 68.6%

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

   13. Applied egg-rr 68.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k\_m}\\ \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \frac{\ell}{{k\_m}^{2} \cdot \left(t \cdot \sin k\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (tan k_m))))
   (if (<= k_m 2.65e-49)
     (* t_1 (/ (/ l k_m) (pow t 3.0)))
     (* t_1 (* 2.0 (/ l (* (pow k_m 2.0) (* t (sin k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / tan(k_m);
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = t_1 * ((l / k_m) / pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (pow(k_m, 2.0) * (t * sin(k_m)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k_m)
    if (k_m <= 2.65d-49) then
        tmp = t_1 * ((l / k_m) / (t ** 3.0d0))
    else
        tmp = t_1 * (2.0d0 * (l / ((k_m ** 2.0d0) * (t * sin(k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.tan(k_m);
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = t_1 * ((l / k_m) / Math.pow(t, 3.0));
	} else {
		tmp = t_1 * (2.0 * (l / (Math.pow(k_m, 2.0) * (t * Math.sin(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / math.tan(k_m)
	tmp = 0
	if k_m <= 2.65e-49:
		tmp = t_1 * ((l / k_m) / math.pow(t, 3.0))
	else:
		tmp = t_1 * (2.0 * (l / (math.pow(k_m, 2.0) * (t * math.sin(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / tan(k_m))
	tmp = 0.0
	if (k_m <= 2.65e-49)
		tmp = Float64(t_1 * Float64(Float64(l / k_m) / (t ^ 3.0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(l / Float64((k_m ^ 2.0) * Float64(t * sin(k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / tan(k_m);
	tmp = 0.0;
	if (k_m <= 2.65e-49)
		tmp = t_1 * ((l / k_m) / (t ^ 3.0));
	else
		tmp = t_1 * (2.0 * (l / ((k_m ^ 2.0) * (t * sin(k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.65e-49], N[(t$95$1 * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000001e-49

   1. Initial program 64.3%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. cbrt-div 68.4%

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

      4. rem-cbrt-cube 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-/r/ 78.5%

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

      3. cube-div 71.7%

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

      4. pow3 71.7%

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

      5. add-cube-cbrt 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. *-lft-identity 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-/l/ 71.9%

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

   10. Simplified 71.9%

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

       1. *-un-lft-identity 71.9%

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

       2. associate-/l/ 71.9%

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

       3. associate-*r/ 71.8%

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

       4. associate-*l/ 69.0%

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

   12. Applied egg-rr 69.0%

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

       1. *-lft-identity 69.0%

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

       2. associate-/l* 69.1%

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

       3. *-commutative 69.1%

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

       4. times-frac 69.9%

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

       5. associate-/r/ 69.5%

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

       6. *-commutative 69.5%

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

   14. Simplified 69.5%

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

   15. Taylor expanded in k around 0 67.5%

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

       1. associate-/r* 69.8%

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

   17. Simplified 69.8%

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

   if 2.6500000000000001e-49 < k

   1. Initial program 43.9%

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

   3. Simplified 46.0%

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

      1. add-cube-cbrt 45.9%

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

      2. pow3 45.9%

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

      3. cbrt-div 45.9%

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

      4. rem-cbrt-cube 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. *-un-lft-identity 55.1%

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

      2. associate-/r/ 55.1%

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

      3. cube-div 46.0%

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

      4. pow3 46.0%

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

      5. add-cube-cbrt 46.0%

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

   8. Applied egg-rr 46.0%

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

      1. *-lft-identity 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-/l/ 46.0%

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

   10. Simplified 46.0%

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

       1. *-un-lft-identity 46.0%

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

       2. associate-/l/ 46.0%

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

       3. associate-*r/ 46.0%

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

       4. associate-*l/ 46.0%

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

   12. Applied egg-rr 46.0%

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

       1. *-lft-identity 46.0%

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

       2. associate-/l* 46.0%

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

       3. *-commutative 46.0%

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

       4. times-frac 46.2%

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

       5. associate-/r/ 46.1%

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

       6. *-commutative 46.1%

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

   14. Simplified 46.1%

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

   15. Taylor expanded in k around inf 67.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k\_m}\\ \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{2}{t \cdot \sin k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (tan k_m))))
   (if (<= k_m 1.2e-50)
     (* t_1 (/ (/ l k_m) (pow t 3.0)))
     (* t_1 (* (/ l (pow k_m 2.0)) (/ 2.0 (* t (sin k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / tan(k_m);
	double tmp;
	if (k_m <= 1.2e-50) {
		tmp = t_1 * ((l / k_m) / pow(t, 3.0));
	} else {
		tmp = t_1 * ((l / pow(k_m, 2.0)) * (2.0 / (t * sin(k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k_m)
    if (k_m <= 1.2d-50) then
        tmp = t_1 * ((l / k_m) / (t ** 3.0d0))
    else
        tmp = t_1 * ((l / (k_m ** 2.0d0)) * (2.0d0 / (t * sin(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.tan(k_m);
	double tmp;
	if (k_m <= 1.2e-50) {
		tmp = t_1 * ((l / k_m) / Math.pow(t, 3.0));
	} else {
		tmp = t_1 * ((l / Math.pow(k_m, 2.0)) * (2.0 / (t * Math.sin(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / math.tan(k_m)
	tmp = 0
	if k_m <= 1.2e-50:
		tmp = t_1 * ((l / k_m) / math.pow(t, 3.0))
	else:
		tmp = t_1 * ((l / math.pow(k_m, 2.0)) * (2.0 / (t * math.sin(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / tan(k_m))
	tmp = 0.0
	if (k_m <= 1.2e-50)
		tmp = Float64(t_1 * Float64(Float64(l / k_m) / (t ^ 3.0)));
	else
		tmp = Float64(t_1 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(2.0 / Float64(t * sin(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / tan(k_m);
	tmp = 0.0;
	if (k_m <= 1.2e-50)
		tmp = t_1 * ((l / k_m) / (t ^ 3.0));
	else
		tmp = t_1 * ((l / (k_m ^ 2.0)) * (2.0 / (t * sin(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.2e-50], N[(t$95$1 * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000001e-50

   1. Initial program 64.3%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. cbrt-div 68.4%

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

      4. rem-cbrt-cube 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-/r/ 78.5%

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

      3. cube-div 71.7%

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

      4. pow3 71.7%

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

      5. add-cube-cbrt 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. *-lft-identity 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-/l/ 71.9%

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

   10. Simplified 71.9%

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

       1. *-un-lft-identity 71.9%

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

       2. associate-/l/ 71.9%

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

       3. associate-*r/ 71.8%

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

       4. associate-*l/ 69.0%

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

   12. Applied egg-rr 69.0%

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

       1. *-lft-identity 69.0%

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

       2. associate-/l* 69.1%

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

       3. *-commutative 69.1%

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

       4. times-frac 69.9%

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

       5. associate-/r/ 69.5%

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

       6. *-commutative 69.5%

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

   14. Simplified 69.5%

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

   15. Taylor expanded in k around 0 67.5%

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

       1. associate-/r* 69.8%

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

   17. Simplified 69.8%

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

   if 1.20000000000000001e-50 < k

   1. Initial program 43.9%

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

   3. Simplified 46.0%

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

      1. add-cube-cbrt 45.9%

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

      2. pow3 45.9%

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

      3. cbrt-div 45.9%

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

      4. rem-cbrt-cube 55.1%

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

   6. Applied egg-rr 55.1%

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

      1. *-un-lft-identity 55.1%

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

      2. associate-/r/ 55.1%

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

      3. cube-div 46.0%

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

      4. pow3 46.0%

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

      5. add-cube-cbrt 46.0%

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

   8. Applied egg-rr 46.0%

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

      1. *-lft-identity 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-/l/ 46.0%

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

   10. Simplified 46.0%

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

       1. *-un-lft-identity 46.0%

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

       2. associate-/l/ 46.0%

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

       3. associate-*r/ 46.0%

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

       4. associate-*l/ 46.0%

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

   12. Applied egg-rr 46.0%

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

       1. *-lft-identity 46.0%

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

       2. associate-/l* 46.0%

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

       3. *-commutative 46.0%

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

       4. times-frac 46.2%

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

       5. associate-/r/ 46.1%

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

       6. *-commutative 46.1%

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

   14. Simplified 46.1%

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

   15. Taylor expanded in k around inf 67.2%

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

       1. associate-*r/ 67.2%

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

       2. *-commutative 67.2%

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

       3. times-frac 70.7%

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

   17. Simplified 70.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{2}{t \cdot \sin k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \frac{\frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{t \cdot {k\_m}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-49)
   (* (/ l (tan k_m)) (/ (/ l k_m) (pow t 3.0)))
   (* 2.0 (* (pow l 2.0) (/ (cos k_m) (* t (pow k_m 4.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = (l / tan(k_m)) * ((l / k_m) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / (t * pow(k_m, 4.0))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d-49) then
        tmp = (l / tan(k_m)) * ((l / k_m) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / (t * (k_m ** 4.0d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = (l / Math.tan(k_m)) * ((l / k_m) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / (t * Math.pow(k_m, 4.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-49:
		tmp = (l / math.tan(k_m)) * ((l / k_m) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k_m) / (t * math.pow(k_m, 4.0))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-49)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / k_m) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(t * (k_m ^ 4.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e-49)
		tmp = (l / tan(k_m)) * ((l / k_m) / (t ^ 3.0));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / (t * (k_m ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.65e-49], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000001e-49

   1. Initial program 64.3%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. cbrt-div 68.4%

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

      4. rem-cbrt-cube 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. associate-/r/ 78.5%

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

      3. cube-div 71.7%

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

      4. pow3 71.7%

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

      5. add-cube-cbrt 71.9%

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

   8. Applied egg-rr 71.9%

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

      1. *-lft-identity 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-/l/ 71.9%

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

   10. Simplified 71.9%

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

       1. *-un-lft-identity 71.9%

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

       2. associate-/l/ 71.9%

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

       3. associate-*r/ 71.8%

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

       4. associate-*l/ 69.0%

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

   12. Applied egg-rr 69.0%

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

       1. *-lft-identity 69.0%

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

       2. associate-/l* 69.1%

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

       3. *-commutative 69.1%

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

       4. times-frac 69.9%

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

       5. associate-/r/ 69.5%

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

       6. *-commutative 69.5%

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

   14. Simplified 69.5%

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

   15. Taylor expanded in k around 0 67.5%

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

       1. associate-/r* 69.8%

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

   17. Simplified 69.8%

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

   if 2.6500000000000001e-49 < k

   1. Initial program 43.9%

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

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right) \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

      3. times-frac 46.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}} \]

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k}} \]
   7. Step-by-step derivation

      1. pow1/3 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. add-sqr-sqrt 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left({\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      3. unpow-prod-down 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   8. Applied egg-rr 32.7%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]
   9. Step-by-step derivation

      1. unpow1/3 33.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. unpow1/3 33.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \color{blue}{\sqrt[3]{\sqrt{\ell}}}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   10. Simplified 33.8%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   11. Taylor expanded in k around inf 59.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 59.9%

          \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       2. *-commutative 59.9%

          \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}}\right) \]

   13. Simplified 59.9%

       \[\leadsto \color{blue}{2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]

   14. Taylor expanded in k around 0 47.0%

       \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\color{blue}{{k}^{4} \cdot t}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \frac{\ell}{k\_m \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-49)
   (* (/ l (tan k_m)) (/ l (* k_m (pow t 3.0))))
   (* 2.0 (/ (pow l 2.0) (* t (pow k_m 4.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-49) {
		tmp = (l / tan(k_m)) * (l / (k_m * pow(t, 3.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k_m, 4.0)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3d-49) then
        tmp = (l / tan(k_m)) * (l / (k_m * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 4.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-49) {
		tmp = (l / Math.tan(k_m)) * (l / (k_m * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 4.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e-49:
		tmp = (l / math.tan(k_m)) * (l / (k_m * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 4.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-49)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(l / Float64(k_m * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 4.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e-49)
		tmp = (l / tan(k_m)) * (l / (k_m * (t ^ 3.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k_m ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e-49], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3e-49

   1. Initial program 64.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. pow3 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{3}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. cbrt-div 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{3}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. rem-cbrt-cube 76.2%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 76.2%

      \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 76.2%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{\frac{2}{\sin k}}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 78.5%

         \[\leadsto \frac{\frac{1 \cdot \color{blue}{\left(\frac{\frac{2}{\sin k}}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}} \cdot \ell\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. cube-div 71.7%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\color{blue}{\frac{{t}^{3}}{{\left(\sqrt[3]{\ell}\right)}^{3}}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. pow3 71.7%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. add-cube-cbrt 71.9%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\color{blue}{\ell}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 71.9%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}} \cdot \ell\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   9. Step-by-step derivation

      1. *-lft-identity 71.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-commutative 71.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l/ 71.9%

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Simplified 71.9%

       \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 71.9%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       2. associate-/l/ 71.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

       3. associate-*r/ 71.8%

          \[\leadsto 1 \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

       4. associate-*l/ 69.0%

          \[\leadsto 1 \cdot \frac{\frac{\ell \cdot 2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

   12. Applied egg-rr 69.0%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{\ell \cdot 2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]
   13. Step-by-step derivation

       1. *-lft-identity 69.0%

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot 2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

       2. associate-/l* 69.1%

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

       3. *-commutative 69.1%

          \[\leadsto \frac{\ell \cdot \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\color{blue}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

       4. times-frac 69.9%

          \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       5. associate-/r/ 69.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 69.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\frac{2}{\color{blue}{\sin k \cdot {t}^{3}}} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   14. Simplified 69.5%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k \cdot {t}^{3}} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   15. Taylor expanded in k around 0 67.5%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]

   if 3e-49 < k

   1. Initial program 43.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right) \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

      3. times-frac 46.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}} \]

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k}} \]
   7. Step-by-step derivation

      1. pow1/3 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. add-sqr-sqrt 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left({\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      3. unpow-prod-down 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   8. Applied egg-rr 32.7%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]
   9. Step-by-step derivation

      1. unpow1/3 33.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. unpow1/3 33.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \color{blue}{\sqrt[3]{\sqrt{\ell}}}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   10. Simplified 33.8%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   11. Taylor expanded in k around inf 59.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 59.9%

          \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       2. *-commutative 59.9%

          \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}}\right) \]

   13. Simplified 59.9%

       \[\leadsto \color{blue}{2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]

   14. Taylor expanded in k around 0 47.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k\_m} \cdot \frac{\frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.65e-49)
   (* (/ l (tan k_m)) (/ (/ l k_m) (pow t 3.0)))
   (* 2.0 (/ (pow l 2.0) (* t (pow k_m 4.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = (l / tan(k_m)) * ((l / k_m) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t * pow(k_m, 4.0)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.65d-49) then
        tmp = (l / tan(k_m)) * ((l / k_m) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 4.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.65e-49) {
		tmp = (l / Math.tan(k_m)) * ((l / k_m) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 4.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.65e-49:
		tmp = (l / math.tan(k_m)) * ((l / k_m) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 4.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.65e-49)
		tmp = Float64(Float64(l / tan(k_m)) * Float64(Float64(l / k_m) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 4.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.65e-49)
		tmp = (l / tan(k_m)) * ((l / k_m) / (t ^ 3.0));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t * (k_m ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.65e-49], N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6500000000000001e-49

   1. Initial program 64.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. pow3 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{3}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. cbrt-div 68.4%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{3}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. rem-cbrt-cube 76.2%

         \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 76.2%

      \[\leadsto \frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 76.2%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{\frac{2}{\sin k}}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-/r/ 78.5%

         \[\leadsto \frac{\frac{1 \cdot \color{blue}{\left(\frac{\frac{2}{\sin k}}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}} \cdot \ell\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. cube-div 71.7%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\color{blue}{\frac{{t}^{3}}{{\left(\sqrt[3]{\ell}\right)}^{3}}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. pow3 71.7%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. add-cube-cbrt 71.9%

         \[\leadsto \frac{\frac{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\color{blue}{\ell}}} \cdot \ell\right)}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 71.9%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}} \cdot \ell\right)}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   9. Step-by-step derivation

      1. *-lft-identity 71.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-commutative 71.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{\frac{2}{\sin k}}{\frac{{t}^{3}}{\ell}}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l/ 71.9%

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Simplified 71.9%

       \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 71.9%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       2. associate-/l/ 71.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

       3. associate-*r/ 71.8%

          \[\leadsto 1 \cdot \frac{\color{blue}{\frac{\ell \cdot 2}{\frac{{t}^{3}}{\ell} \cdot \sin k}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

       4. associate-*l/ 69.0%

          \[\leadsto 1 \cdot \frac{\frac{\ell \cdot 2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

   12. Applied egg-rr 69.0%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{\ell \cdot 2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]
   13. Step-by-step derivation

       1. *-lft-identity 69.0%

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot 2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

       2. associate-/l* 69.1%

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

       3. *-commutative 69.1%

          \[\leadsto \frac{\ell \cdot \frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{\color{blue}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

       4. times-frac 69.9%

          \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       5. associate-/r/ 69.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

       6. *-commutative 69.5%

          \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\frac{2}{\color{blue}{\sin k \cdot {t}^{3}}} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   14. Simplified 69.5%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k \cdot {t}^{3}} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   15. Taylor expanded in k around 0 67.5%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]
   16. Step-by-step derivation

       1. associate-/r* 69.8%

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   17. Simplified 69.8%

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 2.6500000000000001e-49 < k

   1. Initial program 43.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 45.9%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right) \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

      3. times-frac 46.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}} \]

   6. Applied egg-rr 67.9%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k}} \]
   7. Step-by-step derivation

      1. pow1/3 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. add-sqr-sqrt 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left({\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      3. unpow-prod-down 32.7%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   8. Applied egg-rr 32.7%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]
   9. Step-by-step derivation

      1. unpow1/3 33.0%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

      2. unpow1/3 33.8%

         \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \color{blue}{\sqrt[3]{\sqrt{\ell}}}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   10. Simplified 33.8%

       \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   11. Taylor expanded in k around inf 59.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   12. Step-by-step derivation

       1. associate-/l* 59.9%

          \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

       2. *-commutative 59.9%

          \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}}\right) \]

   13. Simplified 59.9%

       \[\leadsto \color{blue}{2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]

   14. Taylor expanded in k around 0 47.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{4}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (/ (pow l 2.0) (* t (pow k_m 4.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * (pow(l, 2.0) / (t * pow(k_m, 4.0)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 4.0d0)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 4.0)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 4.0)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 4.0))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * ((l ^ 2.0) / (t * (k_m ^ 4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 63.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ 63.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \]

   2. add-cube-cbrt 63.8%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right) \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \]

   3. times-frac 63.8%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \cdot \sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{\sin k}}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\tan k}} \]

6. Applied egg-rr 83.8%

   \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k}} \]
7. Step-by-step derivation

   1. pow1/3 41.4%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   2. add-sqr-sqrt 41.4%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left({\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   3. unpow-prod-down 41.4%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

8. Applied egg-rr 41.4%

   \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left({\left(\sqrt{\ell}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]
9. Step-by-step derivation

   1. unpow1/3 41.6%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\color{blue}{\sqrt[3]{\sqrt{\ell}}} \cdot {\left(\sqrt{\ell}\right)}^{0.3333333333333333}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

   2. unpow1/3 42.4%

      \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \color{blue}{\sqrt[3]{\sqrt{\ell}}}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

10. Simplified 42.4%

    \[\leadsto \frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\color{blue}{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\tan k} \]

11. Taylor expanded in k around inf 64.0%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
12. Step-by-step derivation

    1. associate-/l* 63.6%

       \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

    2. *-commutative 63.6%

       \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}}\right) \]

13. Simplified 63.6%

    \[\leadsto \color{blue}{2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]

14. Taylor expanded in k around 0 55.0%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

15. Final simplification 55.0%

    \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))))