Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 95.6%

Time: 20.7s

Alternatives: 16

Speedup: 28.1×

• 40.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{2}{\frac{k\_m}{\frac{\ell}{k\_m}} \cdot t}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\frac{k\_m}{2}}}{\sin k\_m}}{\frac{\tan k\_m}{\frac{\frac{\ell}{t}}{k\_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e+155)
   (* (/ (/ 2.0 (* (/ k_m (/ l k_m)) t)) (sin k_m)) (/ l (tan k_m)))
   (/ (/ (/ l (/ k_m 2.0)) (sin k_m)) (/ (tan k_m) (/ (/ l t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e+155) {
		tmp = ((2.0 / ((k_m / (l / k_m)) * t)) / sin(k_m)) * (l / tan(k_m));
	} else {
		tmp = ((l / (k_m / 2.0)) / sin(k_m)) / (tan(k_m) / ((l / t) / 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) :: tmp
    if (k_m <= 1.02d+155) then
        tmp = ((2.0d0 / ((k_m / (l / k_m)) * t)) / sin(k_m)) * (l / tan(k_m))
    else
        tmp = ((l / (k_m / 2.0d0)) / sin(k_m)) / (tan(k_m) / ((l / t) / 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 tmp;
	if (k_m <= 1.02e+155) {
		tmp = ((2.0 / ((k_m / (l / k_m)) * t)) / Math.sin(k_m)) * (l / Math.tan(k_m));
	} else {
		tmp = ((l / (k_m / 2.0)) / Math.sin(k_m)) / (Math.tan(k_m) / ((l / t) / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.02e+155:
		tmp = ((2.0 / ((k_m / (l / k_m)) * t)) / math.sin(k_m)) * (l / math.tan(k_m))
	else:
		tmp = ((l / (k_m / 2.0)) / math.sin(k_m)) / (math.tan(k_m) / ((l / t) / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e+155)
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(k_m / Float64(l / k_m)) * t)) / sin(k_m)) * Float64(l / tan(k_m)));
	else
		tmp = Float64(Float64(Float64(l / Float64(k_m / 2.0)) / sin(k_m)) / Float64(tan(k_m) / Float64(Float64(l / t) / k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.02e+155)
		tmp = ((2.0 / ((k_m / (l / k_m)) * t)) / sin(k_m)) * (l / tan(k_m));
	else
		tmp = ((l / (k_m / 2.0)) / sin(k_m)) / (tan(k_m) / ((l / t) / k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.02e+155], N[(N[(N[(2.0 / N[(N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(k$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] / N[(N[(l / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.02e155

   1. Initial program 34.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 75.7%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.3%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 92.5%

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

       1. associate-*r/ N/A

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

       2. times-frac N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{k \cdot k} \cdot 2}{t}}{\sin k}\right), \color{blue}{\left(\frac{\ell}{\tan k}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k \cdot k} \cdot 2}{t}\right), \sin k\right), \left(\frac{\color{blue}{\ell}}{\tan k}\right)\right) \]

       5. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{k \cdot k}{\ell}} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\frac{k \cdot k}{\ell}} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       8. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot 2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot 2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{1}{\frac{\ell}{k \cdot k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       14. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{\ell}{k}}{k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k}{\frac{\ell}{k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{k}\right)\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

       18. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), t\right)\right), \mathsf{sin.f64}\left(k\right)\right), \left(\frac{\ell}{\tan k}\right)\right) \]

   11. Applied egg-rr 95.0%

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

   if 1.02e155 < k

   1. Initial program 25.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 58.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\tan k\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\sin k}{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\right)}\right) \]

   6. Applied egg-rr 58.9%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k}}{k}}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{t}{\ell} \cdot k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k}\right), \left(\frac{t}{\ell} \cdot k\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{t}{\ell} \cdot k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 81.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 81.3%

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

      1. associate-/r/ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{\tan k}}{-1} \cdot \left(\frac{\frac{2}{k}}{\frac{t}{\ell} \cdot k} \cdot \ell\right)\right), \color{blue}{\sin k}\right) \]

   10. Applied egg-rr 92.7%

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

       1. div-inv N/A

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

       2. frac-times N/A

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

       3. *-lft-identity N/A

          \[\leadsto \frac{\frac{\ell}{\frac{k}{2}}}{\tan k \cdot \frac{k}{\frac{\ell}{t}}} \cdot \frac{1}{\sin k} \]

       4. associate-*l/ N/A

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

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{2}} \cdot \frac{1}{\sin k}\right), \color{blue}{\left(\tan k \cdot \frac{k}{\frac{\ell}{t}}\right)}\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{k}{2}}}{\sin k}\right), \left(\color{blue}{\tan k} \cdot \frac{k}{\frac{\ell}{t}}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{2}}\right), \sin k\right), \left(\color{blue}{\tan k} \cdot \frac{k}{\frac{\ell}{t}}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{2}\right)\right), \sin k\right), \left(\tan \color{blue}{k} \cdot \frac{k}{\frac{\ell}{t}}\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \sin k\right), \left(\tan k \cdot \frac{k}{\frac{\ell}{t}}\right)\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \left(\tan k \cdot \frac{k}{\frac{\ell}{t}}\right)\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \left(\tan k \cdot \frac{1}{\color{blue}{\frac{\frac{\ell}{t}}{k}}}\right)\right) \]

       12. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \left(\frac{\tan k}{\color{blue}{\frac{\frac{\ell}{t}}{k}}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(\tan k, \color{blue}{\left(\frac{\frac{\ell}{t}}{k}\right)}\right)\right) \]

       14. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{\color{blue}{\frac{\ell}{t}}}{k}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{k}\right)\right)\right) \]

       16. /-lowering-/.f64 92.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right)\right)\right) \]

   12. Applied egg-rr 92.9%

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

Alternative 2: 93.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k\_m}{\frac{-2}{\frac{k\_m \cdot t}{\frac{\ell}{\frac{k\_m}{\frac{\ell}{k\_m}}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k\_m \cdot \tan k\_m} \cdot \frac{\frac{2}{k\_m}}{k\_m \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-127)
   (/ -1.0 (/ (tan k_m) (/ -2.0 (/ (* k_m t) (/ l (/ k_m (/ l k_m)))))))
   (* (/ l (* (sin k_m) (tan k_m))) (/ (/ 2.0 k_m) (* k_m (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-127) {
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else {
		tmp = (l / (sin(k_m) * tan(k_m))) * ((2.0 / k_m) / (k_m * (t / l)));
	}
	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 <= 3.2d-127) then
        tmp = (-1.0d0) / (tan(k_m) / ((-2.0d0) / ((k_m * t) / (l / (k_m / (l / k_m))))))
    else
        tmp = (l / (sin(k_m) * tan(k_m))) * ((2.0d0 / k_m) / (k_m * (t / l)))
    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 <= 3.2e-127) {
		tmp = -1.0 / (Math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else {
		tmp = (l / (Math.sin(k_m) * Math.tan(k_m))) * ((2.0 / k_m) / (k_m * (t / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.2e-127:
		tmp = -1.0 / (math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))))
	else:
		tmp = (l / (math.sin(k_m) * math.tan(k_m))) * ((2.0 / k_m) / (k_m * (t / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-127)
		tmp = Float64(-1.0 / Float64(tan(k_m) / Float64(-2.0 / Float64(Float64(k_m * t) / Float64(l / Float64(k_m / Float64(l / k_m)))))));
	else
		tmp = Float64(Float64(l / Float64(sin(k_m) * tan(k_m))) * Float64(Float64(2.0 / k_m) / Float64(k_m * Float64(t / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-127)
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	else
		tmp = (l / (sin(k_m) * tan(k_m))) * ((2.0 / k_m) / (k_m * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-127], N[(-1.0 / N[(N[Tan[k$95$m], $MachinePrecision] / N[(-2.0 / N[(N[(k$95$m * t), $MachinePrecision] / N[(l / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.20000000000000017e-127

   1. Initial program 37.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k \cdot k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}}{k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t}\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{t}\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   7. Simplified 79.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)}, \mathsf{tan.f64}\left(k\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)\right), \mathsf{tan.f64}\left(\color{blue}{k}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{{\ell}^{2}}{{k}^{3}}}{t}\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{3}}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2} \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2}}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k \cdot k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{k}}{k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k}\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      13. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   10. Simplified 74.4%

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

       1. clear-num N/A

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

       2. metadata-eval N/A

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

       3. frac-2neg N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{\tan k}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)}\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\tan k\right)}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}\right)\right)\right) \]

       8. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan k\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}}\right)\right) \]

       9. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\tan k}{\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\tan k, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)\right)}\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right)\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{\left(\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}\right)}\right)\right)\right) \]

   12. Applied egg-rr 83.6%

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

   if 3.20000000000000017e-127 < k

   1. Initial program 28.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 69.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 82.9%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2}{k}}{k}}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{\frac{t}{\ell} \cdot k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k}\right), \left(\frac{t}{\ell} \cdot k\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{t}{\ell} \cdot k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 90.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. Applied egg-rr 90.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k}{\frac{-2}{\frac{k \cdot t}{\frac{\ell}{\frac{k}{\frac{\ell}{k}}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{2}{k}}{k \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\sin k\_m \cdot \tan k\_m}\\ \mathbf{if}\;k\_m \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot \frac{\ell}{k\_m \cdot k\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{t} \cdot \frac{2}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* (sin k_m) (tan k_m)))))
   (if (<= k_m 2.6e+153)
     (* t_1 (/ (* 2.0 (/ l (* k_m k_m))) t))
     (* t_1 (/ (* (/ l t) (/ 2.0 k_m)) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (sin(k_m) * tan(k_m));
	double tmp;
	if (k_m <= 2.6e+153) {
		tmp = t_1 * ((2.0 * (l / (k_m * k_m))) / t);
	} else {
		tmp = t_1 * (((l / t) * (2.0 / k_m)) / 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 / (sin(k_m) * tan(k_m))
    if (k_m <= 2.6d+153) then
        tmp = t_1 * ((2.0d0 * (l / (k_m * k_m))) / t)
    else
        tmp = t_1 * (((l / t) * (2.0d0 / k_m)) / 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.sin(k_m) * Math.tan(k_m));
	double tmp;
	if (k_m <= 2.6e+153) {
		tmp = t_1 * ((2.0 * (l / (k_m * k_m))) / t);
	} else {
		tmp = t_1 * (((l / t) * (2.0 / k_m)) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (math.sin(k_m) * math.tan(k_m))
	tmp = 0
	if k_m <= 2.6e+153:
		tmp = t_1 * ((2.0 * (l / (k_m * k_m))) / t)
	else:
		tmp = t_1 * (((l / t) * (2.0 / k_m)) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 2.6e+153)
		tmp = Float64(t_1 * Float64(Float64(2.0 * Float64(l / Float64(k_m * k_m))) / t));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l / t) * Float64(2.0 / k_m)) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (sin(k_m) * tan(k_m));
	tmp = 0.0;
	if (k_m <= 2.6e+153)
		tmp = t_1 * ((2.0 * (l / (k_m * k_m))) / t);
	else
		tmp = t_1 * (((l / t) * (2.0 / k_m)) / 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[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.6e+153], N[(t$95$1 * N[(N[(2.0 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l / t), $MachinePrecision] * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5999999999999999e153

   1. Initial program 35.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 92.9%

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

   if 2.5999999999999999e153 < k

   1. Initial program 24.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 56.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 57.0%

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

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot k} \cdot \frac{1}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k} \cdot \frac{1}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k} \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. /-lowering-/.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. Applied egg-rr 80.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{2 \cdot \frac{\ell}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell}{t} \cdot \frac{2}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k\_m}{\frac{-2}{\frac{k\_m \cdot t}{\frac{\ell}{\frac{k\_m}{\frac{\ell}{k\_m}}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k\_m \cdot \tan k\_m} \cdot \frac{\frac{\ell}{t} \cdot \frac{2}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-127)
   (/ -1.0 (/ (tan k_m) (/ -2.0 (/ (* k_m t) (/ l (/ k_m (/ l k_m)))))))
   (* (/ l (* (sin k_m) (tan k_m))) (/ (* (/ l t) (/ 2.0 k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-127) {
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else {
		tmp = (l / (sin(k_m) * tan(k_m))) * (((l / t) * (2.0 / k_m)) / 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) :: tmp
    if (k_m <= 3.2d-127) then
        tmp = (-1.0d0) / (tan(k_m) / ((-2.0d0) / ((k_m * t) / (l / (k_m / (l / k_m))))))
    else
        tmp = (l / (sin(k_m) * tan(k_m))) * (((l / t) * (2.0d0 / k_m)) / 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 tmp;
	if (k_m <= 3.2e-127) {
		tmp = -1.0 / (Math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else {
		tmp = (l / (Math.sin(k_m) * Math.tan(k_m))) * (((l / t) * (2.0 / k_m)) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.2e-127:
		tmp = -1.0 / (math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))))
	else:
		tmp = (l / (math.sin(k_m) * math.tan(k_m))) * (((l / t) * (2.0 / k_m)) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-127)
		tmp = Float64(-1.0 / Float64(tan(k_m) / Float64(-2.0 / Float64(Float64(k_m * t) / Float64(l / Float64(k_m / Float64(l / k_m)))))));
	else
		tmp = Float64(Float64(l / Float64(sin(k_m) * tan(k_m))) * Float64(Float64(Float64(l / t) * Float64(2.0 / k_m)) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-127)
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	else
		tmp = (l / (sin(k_m) * tan(k_m))) * (((l / t) * (2.0 / k_m)) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-127], N[(-1.0 / N[(N[Tan[k$95$m], $MachinePrecision] / N[(-2.0 / N[(N[(k$95$m * t), $MachinePrecision] / N[(l / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.20000000000000017e-127

   1. Initial program 37.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k \cdot k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}}{k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t}\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{t}\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   7. Simplified 79.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)}, \mathsf{tan.f64}\left(k\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)\right), \mathsf{tan.f64}\left(\color{blue}{k}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{{\ell}^{2}}{{k}^{3}}}{t}\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{3}}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2} \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2}}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k \cdot k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{k}}{k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k}\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      13. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   10. Simplified 74.4%

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

       1. clear-num N/A

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

       2. metadata-eval N/A

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

       3. frac-2neg N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{\tan k}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)}\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\tan k\right)}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}\right)\right)\right) \]

       8. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan k\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}}\right)\right) \]

       9. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\tan k}{\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\tan k, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)\right)}\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right)\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{\left(\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}\right)}\right)\right)\right) \]

   12. Applied egg-rr 83.6%

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

   if 3.20000000000000017e-127 < k

   1. Initial program 28.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 69.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 82.9%

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

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot k} \cdot \frac{1}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k} \cdot \frac{1}{\frac{t}{\ell}}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k}}{k} \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k} \cdot \frac{\ell}{t}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\frac{\ell}{t}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{\ell}{t}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\ell, t\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. Applied egg-rr 89.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k}{\frac{-2}{\frac{k \cdot t}{\frac{\ell}{\frac{k}{\frac{\ell}{k}}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \tan k} \cdot \frac{\frac{\ell}{t} \cdot \frac{2}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\frac{\ell}{k\_m}}{\frac{t}{2}}}{k\_m} \cdot \frac{\ell}{\sin k\_m \cdot \tan k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ (/ l k_m) (/ t 2.0)) k_m) (/ l (* (sin k_m) (tan k_m)))))

C

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

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 = (((l / k_m) / (t / 2.0d0)) / k_m) * (l / (sin(k_m) * tan(k_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / N[(t / 2.0), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

3. Simplified 73.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

6. Applied egg-rr 86.0%

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

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 88.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

9. Simplified 88.9%

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

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{2}{t}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    3. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \frac{2}{t}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \frac{1}{\frac{t}{2}}\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    6. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{\frac{t}{2}}\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \left(\frac{t}{2}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{t}{2}\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

    9. /-lowering-/.f64 94.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(t, 2\right)\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

11. Applied egg-rr 94.0%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{\frac{t}{2}}}{k}} \cdot \frac{\ell}{\sin k \cdot \tan k} \]
12. Add Preprocessing

Alternative 6: 92.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{2}{\frac{k\_m}{\frac{\ell}{k\_m}} \cdot t}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ 2.0 (* (/ k_m (/ l k_m)) t)) (sin k_m)) (/ l (tan k_m))))

C

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

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 / ((k_m / (l / k_m)) * t)) / sin(k_m)) * (l / tan(k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 / N[(N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

3. Simplified 73.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

6. Applied egg-rr 86.0%

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

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 88.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

9. Simplified 88.9%

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

    1. associate-*r/ N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell}{k \cdot k} \cdot 2}{t}}{\sin k}\right), \color{blue}{\left(\frac{\ell}{\tan k}\right)}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k \cdot k} \cdot 2}{t}\right), \sin k\right), \left(\frac{\color{blue}{\ell}}{\tan k}\right)\right) \]

    5. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    6. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{k \cdot k}{\ell}} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(-1\right)}{\frac{k \cdot k}{\ell}} \cdot \frac{2}{t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    8. frac-times N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1\right)\right) \cdot 2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot 2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    10. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{k \cdot k}{\ell} \cdot t}\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k \cdot k}{\ell} \cdot t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    13. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{1}{\frac{\ell}{k \cdot k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    14. associate-/r* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{\ell}{k}}{k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    15. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{k}{\frac{\ell}{k}}\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    16. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{k}\right)\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    17. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), t\right)\right), \sin k\right), \left(\frac{\ell}{\tan k}\right)\right) \]

    18. sin-lowering-sin.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), t\right)\right), \mathsf{sin.f64}\left(k\right)\right), \left(\frac{\ell}{\tan k}\right)\right) \]

11. Applied egg-rr 93.0%

    \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\frac{\ell}{k}} \cdot t}}{\sin k} \cdot \frac{\ell}{\tan k}} \]
12. Add Preprocessing

Alternative 7: 75.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 200000000000:\\ \;\;\;\;\frac{\frac{-1 + \left(k\_m \cdot k\_m\right) \cdot 0.3333333333333333}{\frac{\frac{t}{\frac{\ell \cdot \frac{-2}{k\_m}}{k\_m}}}{\frac{\ell}{\sin k\_m}}}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot k\_m}}{k\_m \cdot \frac{\frac{k\_m}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 200000000000.0)
   (/
    (/
     (+ -1.0 (* (* k_m k_m) 0.3333333333333333))
     (/ (/ t (/ (* l (/ -2.0 k_m)) k_m)) (/ l (sin k_m))))
    k_m)
   (/ (/ 2.0 (* k_m k_m)) (* k_m (/ (/ k_m (/ l t)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 200000000000.0) {
		tmp = ((-1.0 + ((k_m * k_m) * 0.3333333333333333)) / ((t / ((l * (-2.0 / k_m)) / k_m)) / (l / sin(k_m)))) / k_m;
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	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 <= 200000000000.0d0) then
        tmp = (((-1.0d0) + ((k_m * k_m) * 0.3333333333333333d0)) / ((t / ((l * ((-2.0d0) / k_m)) / k_m)) / (l / sin(k_m)))) / k_m
    else
        tmp = (2.0d0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
    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 <= 200000000000.0) {
		tmp = ((-1.0 + ((k_m * k_m) * 0.3333333333333333)) / ((t / ((l * (-2.0 / k_m)) / k_m)) / (l / Math.sin(k_m)))) / k_m;
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 200000000000.0:
		tmp = ((-1.0 + ((k_m * k_m) * 0.3333333333333333)) / ((t / ((l * (-2.0 / k_m)) / k_m)) / (l / math.sin(k_m)))) / k_m
	else:
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 200000000000.0)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(k_m * k_m) * 0.3333333333333333)) / Float64(Float64(t / Float64(Float64(l * Float64(-2.0 / k_m)) / k_m)) / Float64(l / sin(k_m)))) / k_m);
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * k_m)) / Float64(k_m * Float64(Float64(k_m / Float64(l / t)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 200000000000.0)
		tmp = ((-1.0 + ((k_m * k_m) * 0.3333333333333333)) / ((t / ((l * (-2.0 / k_m)) / k_m)) / (l / sin(k_m)))) / k_m;
	else
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 200000000000.0], N[(N[(N[(-1.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[(t / N[(N[(l * N[(-2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e11

   1. Initial program 37.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 77.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\tan k\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\sin k}{\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{\frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\right)}\right) \]

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{3} \cdot {k}^{2} - 1}{k}\right)}, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{3} \cdot {k}^{2} - 1\right), k\right), \mathsf{/.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{3} \cdot {k}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{3} \cdot {k}^{2} + -1\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 + \frac{1}{3} \cdot {k}^{2}\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \left(\frac{1}{3} \cdot {k}^{2}\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \left({k}^{2} \cdot \frac{1}{3}\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({k}^{2}\right), \frac{1}{3}\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(k \cdot k\right), \frac{1}{3}\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{1}{3}\right)\right), k\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right)\right)\right)\right) \]

   9. Simplified 70.7%

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

       1. div-inv N/A

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

       2. clear-num N/A

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

       3. associate-*l/ N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 + \left(k \cdot k\right) \cdot \frac{1}{3}\right) \cdot \frac{\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k}}{-1}\right), \color{blue}{k}\right) \]

   11. Applied egg-rr 75.7%

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

   if 2e11 < k

   1. Initial program 23.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 44.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 44.2%

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

      1. associate-*l* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \color{blue}{\left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\color{blue}{t} \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{t}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{1}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{\ell \cdot \ell}{t}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\color{blue}{\frac{t}{\ell \cdot \ell}}}\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}}\right)\right)\right) \]

      20. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\ell}{\color{blue}{\frac{t}{\ell}}}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      22. /-lowering-/.f64 48.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   7. Applied egg-rr 48.1%

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

      1. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(k \cdot \frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(k \cdot \left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(k\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(0 - k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(0 - \color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\right)\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}\right)\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{k}{\frac{\ell}{t}}}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{1}{\frac{\ell}{t}}}{\ell}\right)\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell} \cdot k}{\ell}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot k\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{t}{\ell}\right), \ell\right)\right)\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{1}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{k}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right), \ell\right)\right)\right)\right) \]

      20. /-lowering-/.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right), \ell\right)\right)\right)\right) \]

   9. Applied egg-rr 50.0%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\color{blue}{\left(0 - k\right) \cdot \left(0 - \frac{\frac{k}{\frac{\ell}{t}}}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 200000000000:\\ \;\;\;\;\frac{\frac{-1 + \left(k \cdot k\right) \cdot 0.3333333333333333}{\frac{\frac{t}{\frac{\ell \cdot \frac{-2}{k}}{k}}}{\frac{\ell}{\sin k}}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{k \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k\_m}{\frac{-2}{\frac{k\_m \cdot t}{\frac{\ell}{\frac{k\_m}{\frac{\ell}{k\_m}}}}}}}\\ \mathbf{elif}\;k\_m \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot k\_m}}{\frac{t}{\ell}} \cdot \frac{\ell \cdot \left(\left(k\_m \cdot k\_m\right) \cdot -0.16666666666666666 + 1\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m}}{k\_m} \cdot \frac{\frac{2}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-127)
   (/ -1.0 (/ (tan k_m) (/ -2.0 (/ (* k_m t) (/ l (/ k_m (/ l k_m)))))))
   (if (<= k_m 4.8e+106)
     (*
      (/ (/ 2.0 (* k_m k_m)) (/ t l))
      (/ (* l (+ (* (* k_m k_m) -0.16666666666666666) 1.0)) (* k_m k_m)))
     (* (/ (/ l k_m) k_m) (/ (/ 2.0 k_m) (/ k_m (/ l t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-127) {
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else if (k_m <= 4.8e+106) {
		tmp = ((2.0 / (k_m * k_m)) / (t / l)) * ((l * (((k_m * k_m) * -0.16666666666666666) + 1.0)) / (k_m * k_m));
	} else {
		tmp = ((l / k_m) / k_m) * ((2.0 / k_m) / (k_m / (l / t)));
	}
	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 <= 3.2d-127) then
        tmp = (-1.0d0) / (tan(k_m) / ((-2.0d0) / ((k_m * t) / (l / (k_m / (l / k_m))))))
    else if (k_m <= 4.8d+106) then
        tmp = ((2.0d0 / (k_m * k_m)) / (t / l)) * ((l * (((k_m * k_m) * (-0.16666666666666666d0)) + 1.0d0)) / (k_m * k_m))
    else
        tmp = ((l / k_m) / k_m) * ((2.0d0 / k_m) / (k_m / (l / t)))
    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 <= 3.2e-127) {
		tmp = -1.0 / (Math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	} else if (k_m <= 4.8e+106) {
		tmp = ((2.0 / (k_m * k_m)) / (t / l)) * ((l * (((k_m * k_m) * -0.16666666666666666) + 1.0)) / (k_m * k_m));
	} else {
		tmp = ((l / k_m) / k_m) * ((2.0 / k_m) / (k_m / (l / t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.2e-127:
		tmp = -1.0 / (math.tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))))
	elif k_m <= 4.8e+106:
		tmp = ((2.0 / (k_m * k_m)) / (t / l)) * ((l * (((k_m * k_m) * -0.16666666666666666) + 1.0)) / (k_m * k_m))
	else:
		tmp = ((l / k_m) / k_m) * ((2.0 / k_m) / (k_m / (l / t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-127)
		tmp = Float64(-1.0 / Float64(tan(k_m) / Float64(-2.0 / Float64(Float64(k_m * t) / Float64(l / Float64(k_m / Float64(l / k_m)))))));
	elseif (k_m <= 4.8e+106)
		tmp = Float64(Float64(Float64(2.0 / Float64(k_m * k_m)) / Float64(t / l)) * Float64(Float64(l * Float64(Float64(Float64(k_m * k_m) * -0.16666666666666666) + 1.0)) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l / k_m) / k_m) * Float64(Float64(2.0 / k_m) / Float64(k_m / Float64(l / t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-127)
		tmp = -1.0 / (tan(k_m) / (-2.0 / ((k_m * t) / (l / (k_m / (l / k_m))))));
	elseif (k_m <= 4.8e+106)
		tmp = ((2.0 / (k_m * k_m)) / (t / l)) * ((l * (((k_m * k_m) * -0.16666666666666666) + 1.0)) / (k_m * k_m));
	else
		tmp = ((l / k_m) / k_m) * ((2.0 / k_m) / (k_m / (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-127], N[(-1.0 / N[(N[Tan[k$95$m], $MachinePrecision] / N[(-2.0 / N[(N[(k$95$m * t), $MachinePrecision] / N[(l / N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.8e+106], N[(N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.20000000000000017e-127

   1. Initial program 37.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k \cdot k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}}{k}\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{t}}{k}\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t}\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{t}\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), k\right), k\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   7. Simplified 79.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)}, \mathsf{tan.f64}\left(k\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{\ell}^{2}}{{k}^{3} \cdot t}\right)\right), \mathsf{tan.f64}\left(\color{blue}{k}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\frac{{\ell}^{2}}{{k}^{3}}}{t}\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{3}}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      4. unpow3 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2} \cdot k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{k}\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{{k}^{2}}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k \cdot k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{{\ell}^{2}}{k}}{k}\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{k}\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

      13. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), k\right), k\right), k\right), t\right)\right), \mathsf{tan.f64}\left(k\right)\right) \]

   10. Simplified 74.4%

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

       1. clear-num N/A

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

       2. metadata-eval N/A

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

       3. frac-2neg N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\frac{\tan k}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)}\right) \]

       7. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\tan k\right)}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}\right)\right)\right) \]

       8. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan k\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)}}\right)\right) \]

       9. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\tan k}{\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\tan k, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}\right)\right)}\right)\right) \]

       11. tan-lowering-tan.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right)\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}{t}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(k\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2\right)\right), \color{blue}{\left(\frac{t}{\frac{\frac{\frac{\ell \cdot \ell}{k}}{k}}{k}}\right)}\right)\right)\right) \]

   12. Applied egg-rr 83.6%

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

   if 3.20000000000000017e-127 < k < 4.8000000000000001e106

   1. Initial program 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 80.3%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(\frac{\ell + \frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\ell + \frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right)\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\ell + \left(\frac{-1}{6} \cdot {k}^{2}\right) \cdot \ell\right), \left({k}^{2}\right)\right)\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot {k}^{2} + 1\right) \cdot \ell\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot {k}^{2} + 1\right), \ell\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {k}^{2}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left({k}^{2} \cdot \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left({k}^{2}\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      11. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \frac{-1}{6}\right), 1\right), \ell\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \color{blue}{\frac{\left(\left(k \cdot k\right) \cdot -0.16666666666666666 + 1\right) \cdot \ell}{k \cdot k}} \]

   if 4.8000000000000001e106 < k

   1. Initial program 21.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 48.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 48.7%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

      20. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 51.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 51.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. associate-/r/ N/A

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

      8. associate-/l* N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l* N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

   9. Applied egg-rr 54.5%

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

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, k\right)}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, k\right)}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right)\right) \]

       3. /-lowering-/.f64 54.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right)\right) \]

   11. Applied egg-rr 54.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-1}{\frac{\tan k}{\frac{-2}{\frac{k \cdot t}{\frac{\ell}{\frac{k}{\frac{\ell}{k}}}}}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{\ell \cdot \left(\left(k \cdot k\right) \cdot -0.16666666666666666 + 1\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{2}{k}}{\frac{k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot \sin k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot k\_m}}{k\_m \cdot \frac{\frac{k\_m}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.9e+41)
   (* (/ (* 2.0 (/ l (* k_m k_m))) t) (/ l (* k_m (sin k_m))))
   (/ (/ 2.0 (* k_m k_m)) (* k_m (/ (/ k_m (/ l t)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.9e+41) {
		tmp = ((2.0 * (l / (k_m * k_m))) / t) * (l / (k_m * sin(k_m)));
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	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 <= 2.9d+41) then
        tmp = ((2.0d0 * (l / (k_m * k_m))) / t) * (l / (k_m * sin(k_m)))
    else
        tmp = (2.0d0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
    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 <= 2.9e+41) {
		tmp = ((2.0 * (l / (k_m * k_m))) / t) * (l / (k_m * Math.sin(k_m)));
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.9e+41:
		tmp = ((2.0 * (l / (k_m * k_m))) / t) * (l / (k_m * math.sin(k_m)))
	else:
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.9e+41)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64(k_m * k_m))) / t) * Float64(l / Float64(k_m * sin(k_m))));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * k_m)) / Float64(k_m * Float64(Float64(k_m / Float64(l / t)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.9e+41)
		tmp = ((2.0 * (l / (k_m * k_m))) / t) * (l / (k_m * sin(k_m)));
	else
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.9e+41], N[(N[(N[(2.0 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.89999999999999988e41

   1. Initial program 37.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 92.5%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \color{blue}{k}\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 82.5%

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

   if 2.89999999999999988e41 < k

   1. Initial program 22.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 47.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 47.3%

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

      1. associate-*l* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \color{blue}{\left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\color{blue}{t} \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{t}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{1}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{\ell \cdot \ell}{t}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\color{blue}{\frac{t}{\ell \cdot \ell}}}\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}}\right)\right)\right) \]

      20. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\ell}{\color{blue}{\frac{t}{\ell}}}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      22. /-lowering-/.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   7. Applied egg-rr 51.6%

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

      1. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(k \cdot \frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(k \cdot \left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(k\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(0 - k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(0 - \color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\right)\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}\right)\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{k}{\frac{\ell}{t}}}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{1}{\frac{\ell}{t}}}{\ell}\right)\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell} \cdot k}{\ell}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot k\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{t}{\ell}\right), \ell\right)\right)\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{1}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{k}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right), \ell\right)\right)\right)\right) \]

      20. /-lowering-/.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right), \ell\right)\right)\right)\right) \]

   9. Applied egg-rr 55.6%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\color{blue}{\left(0 - k\right) \cdot \left(0 - \frac{\frac{k}{\frac{\ell}{t}}}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{k \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 14.2:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot t\_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k\_m \cdot k\_m}}{k\_m \cdot \frac{\frac{k\_m}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 14.2)
     (* t_1 (/ (* 2.0 t_1) t))
     (/ (/ 2.0 (* k_m k_m)) (* k_m (/ (/ k_m (/ l t)) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 14.2) {
		tmp = t_1 * ((2.0 * t_1) / t);
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	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 / (k_m * k_m)
    if (k_m <= 14.2d0) then
        tmp = t_1 * ((2.0d0 * t_1) / t)
    else
        tmp = (2.0d0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
    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 / (k_m * k_m);
	double tmp;
	if (k_m <= 14.2) {
		tmp = t_1 * ((2.0 * t_1) / t);
	} else {
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * k_m)
	tmp = 0
	if k_m <= 14.2:
		tmp = t_1 * ((2.0 * t_1) / t)
	else:
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 14.2)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_1) / t));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * k_m)) / Float64(k_m * Float64(Float64(k_m / Float64(l / t)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m * k_m);
	tmp = 0.0;
	if (k_m <= 14.2)
		tmp = t_1 * ((2.0 * t_1) / t);
	else
		tmp = (2.0 / (k_m * k_m)) / (k_m * ((k_m / (l / t)) / l));
	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[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 14.2], N[(t$95$1 * N[(N[(2.0 * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 14.199999999999999

   1. Initial program 37.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.1%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 92.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       3. *-lowering-*.f64 84.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   12. Simplified 84.4%

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

   if 14.199999999999999 < k

   1. Initial program 24.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 43.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 43.2%

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

      1. associate-*l* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \color{blue}{\left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\color{blue}{t} \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{t}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right) \]

      14. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{1}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      15. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{k \cdot k}{\color{blue}{\frac{\ell \cdot \ell}{t}}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{\ell \cdot \ell}{t}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\color{blue}{\frac{t}{\ell \cdot \ell}}}\right)\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{1}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}}\right)\right)\right) \]

      20. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\ell}{\color{blue}{\frac{t}{\ell}}}\right)\right)\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

      22. /-lowering-/.f64 46.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

   7. Applied egg-rr 46.9%

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

      1. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(k \cdot \frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\mathsf{neg}\left(k \cdot \left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\left(\mathsf{neg}\left(k\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(k\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(0 - k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \left(0 - \color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\right)\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}\right)\right)\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{k}{\frac{\ell}{t}}}{\color{blue}{\ell}}\right)\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{1}{\frac{\ell}{t}}}{\ell}\right)\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell} \cdot k}{\ell}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot k\right), \color{blue}{\ell}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{t}{\ell}\right), \ell\right)\right)\right)\right) \]

      17. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot \frac{1}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      18. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{k}{\frac{\ell}{t}}\right), \ell\right)\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right), \ell\right)\right)\right)\right) \]

      20. /-lowering-/.f64 48.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right), \ell\right)\right)\right)\right) \]

   9. Applied egg-rr 48.7%

      \[\leadsto \frac{\frac{2}{k \cdot k}}{\color{blue}{\left(0 - k\right) \cdot \left(0 - \frac{\frac{k}{\frac{\ell}{t}}}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14.2:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \frac{\ell}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot k}}{k \cdot \frac{\frac{k}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 14.2:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot t\_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m \cdot k\_m}}{k\_m}}{\frac{\frac{k\_m}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 14.2)
     (* t_1 (/ (* 2.0 t_1) t))
     (/ (/ (/ 2.0 (* k_m k_m)) k_m) (/ (/ k_m (/ l t)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 14.2) {
		tmp = t_1 * ((2.0 * t_1) / t);
	} else {
		tmp = ((2.0 / (k_m * k_m)) / k_m) / ((k_m / (l / t)) / l);
	}
	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 / (k_m * k_m)
    if (k_m <= 14.2d0) then
        tmp = t_1 * ((2.0d0 * t_1) / t)
    else
        tmp = ((2.0d0 / (k_m * k_m)) / k_m) / ((k_m / (l / t)) / l)
    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 / (k_m * k_m);
	double tmp;
	if (k_m <= 14.2) {
		tmp = t_1 * ((2.0 * t_1) / t);
	} else {
		tmp = ((2.0 / (k_m * k_m)) / k_m) / ((k_m / (l / t)) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * k_m)
	tmp = 0
	if k_m <= 14.2:
		tmp = t_1 * ((2.0 * t_1) / t)
	else:
		tmp = ((2.0 / (k_m * k_m)) / k_m) / ((k_m / (l / t)) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 14.2)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_1) / t));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(k_m * k_m)) / k_m) / Float64(Float64(k_m / Float64(l / t)) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m * k_m);
	tmp = 0.0;
	if (k_m <= 14.2)
		tmp = t_1 * ((2.0 * t_1) / t);
	else
		tmp = ((2.0 / (k_m * k_m)) / k_m) / ((k_m / (l / t)) / l);
	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[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 14.2], N[(t$95$1 * N[(N[(2.0 * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 14.199999999999999

   1. Initial program 37.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

   3. Simplified 76.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

   6. Applied egg-rr 89.1%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 92.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       3. *-lowering-*.f64 84.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   12. Simplified 84.4%

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

   if 14.199999999999999 < k

   1. Initial program 24.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 43.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 43.2%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

      20. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 44.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 44.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/r/ N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. frac-2neg N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{2}{k \cdot k}}{k}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)}\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{k}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), k\right)\right), \left(\mathsf{neg}\left(\frac{\color{blue}{k}}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), k\right)\right), \left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \left(\mathsf{neg}\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \left(0 - \color{blue}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{k}{\frac{\ell}{t}}}{\color{blue}{\ell}}\right)\right)\right) \]

      17. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{1}{\frac{\ell}{t}}}{\ell}\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \frac{t}{\ell}}{\ell}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell} \cdot k}{\ell}\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), k\right)\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot k\right), \color{blue}{\ell}\right)\right)\right) \]

   9. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{\frac{-\frac{\frac{2}{k \cdot k}}{k}}{0 - \frac{\frac{k}{\frac{\ell}{t}}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14.2:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \frac{\ell}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{k}}{\frac{\frac{k}{\frac{\ell}{t}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;t \leq 10^{-63}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{2}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{2}{t \cdot \left(k\_m \cdot k\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= t 1e-63)
     (* t_1 (/ (/ 2.0 k_m) (/ k_m (/ l t))))
     (* t_1 (* l (/ 2.0 (* t (* k_m k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (t <= 1e-63) {
		tmp = t_1 * ((2.0 / k_m) / (k_m / (l / t)));
	} else {
		tmp = t_1 * (l * (2.0 / (t * (k_m * 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 / (k_m * k_m)
    if (t <= 1d-63) then
        tmp = t_1 * ((2.0d0 / k_m) / (k_m / (l / t)))
    else
        tmp = t_1 * (l * (2.0d0 / (t * (k_m * 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 / (k_m * k_m);
	double tmp;
	if (t <= 1e-63) {
		tmp = t_1 * ((2.0 / k_m) / (k_m / (l / t)));
	} else {
		tmp = t_1 * (l * (2.0 / (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * k_m)
	tmp = 0
	if t <= 1e-63:
		tmp = t_1 * ((2.0 / k_m) / (k_m / (l / t)))
	else:
		tmp = t_1 * (l * (2.0 / (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (t <= 1e-63)
		tmp = Float64(t_1 * Float64(Float64(2.0 / k_m) / Float64(k_m / Float64(l / t))));
	else
		tmp = Float64(t_1 * Float64(l * Float64(2.0 / Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m * k_m);
	tmp = 0.0;
	if (t <= 1e-63)
		tmp = t_1 * ((2.0 / k_m) / (k_m / (l / t)));
	else
		tmp = t_1 * (l * (2.0 / (t * (k_m * 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[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1e-63], N[(t$95$1 * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l * N[(2.0 / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000007e-63

   1. Initial program 30.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 63.7%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

      20. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. associate-/r/ N/A

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

      8. associate-/l* N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l* N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

   9. Applied egg-rr 74.8%

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

   if 1.00000000000000007e-63 < t

   1. Initial program 41.6%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 64.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      18. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

      20. cube-unmult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 65.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 65.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. associate-/r/ N/A

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

      8. associate-/l* N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

      13. associate-/r* N/A

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

      14. clear-num N/A

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

      15. associate-/l* N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

   9. Applied egg-rr 70.1%

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

       1. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k}}{k} \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{2}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{t}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k \cdot k} \cdot \ell}{\color{blue}{t}}\right)\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \color{blue}{\ell}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{t}\right), \color{blue}{\ell}\right)\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(k \cdot k\right)}\right), \ell\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \ell\right)\right) \]

       9. *-lowering-*.f64 71.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right)\right) \]

   11. Applied egg-rr 71.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-63}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{k}}{\frac{k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\ell \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ t\_1 \cdot \frac{2 \cdot t\_1}{t} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m)))) (* t_1 (/ (* 2.0 t_1) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	return t_1 * ((2.0 * t_1) / t);
}

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
    t_1 = l / (k_m * k_m)
    code = t_1 * ((2.0d0 * t_1) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	return t_1 * ((2.0 * t_1) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * k_m)
	return t_1 * ((2.0 * t_1) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	return Float64(t_1 * Float64(Float64(2.0 * t_1) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	t_1 = l / (k_m * k_m);
	tmp = t_1 * ((2.0 * t_1) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * N[(N[(2.0 * t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right), \color{blue}{\tan k}\right) \]

3. Simplified 73.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r/ N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot \frac{t}{t}}{t \cdot \frac{t}{\ell}}\right), \color{blue}{\left(\frac{\ell}{\tan k \cdot \sin k}\right)}\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\frac{2}{k \cdot k} \cdot t\right) \cdot 1}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{t \cdot \frac{t}{\ell}}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot \left(t \cdot 1\right)}{\frac{t}{\ell} \cdot t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t \cdot 1}{t}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot \frac{t}{t}\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}} \cdot 1\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{\ell}}{\tan k \cdot \sin k}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{\ell}{\tan k \cdot \sin k}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\tan k \cdot \sin k\right)}\right)\right) \]

6. Applied egg-rr 86.0%

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

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2} \cdot t} \cdot 2\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}}}{t} \cdot 2\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{{k}^{2}} \cdot 2}{t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}} \cdot 2\right), t\right), \mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 88.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right)\right) \]

9. Simplified 88.9%

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

10. Taylor expanded in k around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \color{blue}{\left(\frac{\ell}{{k}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    3. *-lowering-*.f64 74.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), 2\right), t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

12. Simplified 74.9%

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

13. Final simplification 74.9%

    \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \frac{\ell}{k \cdot k}}{t} \]
14. Add Preprocessing

Alternative 14: 73.6% accurate, 28.1× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (/ (/ l (/ k_m 2.0)) (* k_m t))))

C

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

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 = (l / (k_m * k_m)) * ((l / (k_m / 2.0d0)) / (k_m * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 63.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

   18. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

   20. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   22. *-lowering-*.f64 67.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 67.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. associate-/r/ N/A

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

   8. associate-/l* N/A

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

   9. unpow-prod-down N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. div-inv N/A

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

   13. associate-/r* N/A

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

   14. clear-num N/A

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

   15. associate-/l* N/A

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

   16. inv-pow N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 73.2%

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

    1. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k}}{k} \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

    2. frac-times N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k} \cdot \ell}{\color{blue}{k \cdot t}}\right)\right) \]

    3. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{1}{\frac{k}{2}} \cdot \ell}{k \cdot t}\right)\right) \]

    4. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{1}{\frac{\frac{k}{2}}{\ell}}}{\color{blue}{k} \cdot t}\right)\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{\frac{k}{2}}}{\color{blue}{k} \cdot t}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{2}}\right), \color{blue}{\left(k \cdot t\right)}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{2}\right)\right), \left(\color{blue}{k} \cdot t\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \left(k \cdot t\right)\right)\right) \]

    9. *-lowering-*.f64 74.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, 2\right)\right), \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

11. Applied egg-rr 74.2%

    \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{\frac{k}{2}}}{k \cdot t}} \]
12. Add Preprocessing

Alternative 15: 73.4% accurate, 28.1× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (/ (/ 2.0 k_m) (/ (* k_m t) l))))

C

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

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 = (l / (k_m * k_m)) * ((2.0d0 / k_m) / ((k_m * t) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 63.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

   18. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

   20. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   22. *-lowering-*.f64 67.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 67.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. associate-/r/ N/A

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

   8. associate-/l* N/A

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

   9. unpow-prod-down N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. div-inv N/A

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

   13. associate-/r* N/A

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

   14. clear-num N/A

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

   15. associate-/l* N/A

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

   16. inv-pow N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 73.2%

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

    1. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{k}{\ell} \cdot \color{blue}{t}\right)\right)\right) \]

    2. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\frac{k \cdot t}{\color{blue}{\ell}}\right)\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\left(k \cdot t\right), \color{blue}{\ell}\right)\right)\right) \]

    4. *-lowering-*.f64 74.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, t\right), \ell\right)\right)\right) \]

11. Applied egg-rr 74.2%

    \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}} \]
12. Add Preprocessing

Alternative 16: 72.7% accurate, 28.1× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (* l (/ 2.0 (* t (* k_m k_m))))))

C

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

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 = (l / (k_m * k_m)) * (l * (2.0d0 / (t * (k_m * k_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 63.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \color{blue}{\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{t}{\ell \cdot \ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\frac{t}{\ell}}{\ell}}\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{2}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

   18. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot {k}^{\color{blue}{3}}\right)\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left({k}^{3}\right)}\right)\right)\right) \]

   20. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   22. *-lowering-*.f64 67.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 67.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. associate-/r/ N/A

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

   8. associate-/l* N/A

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

   9. unpow-prod-down N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. div-inv N/A

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

   13. associate-/r* N/A

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

   14. clear-num N/A

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

   15. associate-/l* N/A

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

   16. inv-pow N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 73.2%

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

    1. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k}}{k} \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{2}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{t}\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k \cdot k} \cdot \ell}{\color{blue}{t}}\right)\right) \]

    4. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{2}{k \cdot k}}{t} \cdot \color{blue}{\ell}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{t}\right), \color{blue}{\ell}\right)\right) \]

    6. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(k \cdot k\right)}\right), \ell\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \ell\right)\right) \]

    9. *-lowering-*.f64 72.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right)\right) \]

11. Applied egg-rr 72.7%

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

12. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024192 
(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))))