Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 96.6%

Time: 19.6s

Alternatives: 17

Speedup: 32.4×

• 39.4% 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: 35.9% 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: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m \cdot k\_m}}{k\_m} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k\_m}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k\_m}{\cos k\_m}}}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}{\frac{2}{\frac{k\_m}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* (/ (* (/ (/ l (* k_m k_m)) k_m) 2.0) t) (/ (* l (cos k_m)) k_m))
   (/
    (/ l (/ k_m (cos k_m)))
    (/ (* t (+ 0.5 (* -0.5 (cos (* k_m 2.0))))) (/ 2.0 (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	} else {
		tmp = (l / (k_m / cos(k_m))) / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (2.0 / (k_m / 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 <= 7d-5) then
        tmp = ((((l / (k_m * k_m)) / k_m) * 2.0d0) / t) * ((l * cos(k_m)) / k_m)
    else
        tmp = (l / (k_m / cos(k_m))) / ((t * (0.5d0 + ((-0.5d0) * cos((k_m * 2.0d0))))) / (2.0d0 / (k_m / 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 <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * Math.cos(k_m)) / k_m);
	} else {
		tmp = (l / (k_m / Math.cos(k_m))) / ((t * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))) / (2.0 / (k_m / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7e-5:
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * math.cos(k_m)) / k_m)
	else:
		tmp = (l / (k_m / math.cos(k_m))) / ((t * (0.5 + (-0.5 * math.cos((k_m * 2.0))))) / (2.0 / (k_m / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / k_m) * 2.0) / t) * Float64(Float64(l * cos(k_m)) / k_m));
	else
		tmp = Float64(Float64(l / Float64(k_m / cos(k_m))) / Float64(Float64(t * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0))))) / Float64(2.0 / Float64(k_m / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7e-5)
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	else
		tmp = (l / (k_m / cos(k_m))) / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (2.0 / (k_m / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.5%

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

   5. Simplified 76.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 79.3%

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

   9. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 79.2%

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

   12. Simplified 79.2%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 65.1%

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

   5. Simplified 65.1%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. frac-times N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

       6. associate-/l* N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

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

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

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

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

       12. clear-num N/A

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

       13. frac-times N/A

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

       14. associate-*l/ N/A

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

       15. clear-num N/A

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

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\cos k}}}{\frac{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{\frac{2}{\frac{k}{\ell}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k \cdot k}}{k} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\cos k}}}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}{\frac{2}{\frac{k}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell \cdot \cos k\_m}{k\_m}\\ \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m \cdot k\_m}}{k\_m} \cdot 2}{t} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{2}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}{\frac{\ell}{k\_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* l (cos k_m)) k_m)))
   (if (<= k_m 7e-5)
     (* (/ (* (/ (/ l (* k_m k_m)) k_m) 2.0) t) t_1)
     (* t_1 (/ 2.0 (/ (* t (+ 0.5 (* -0.5 (cos (* k_m 2.0))))) (/ l k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l * cos(k_m)) / k_m;
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	} else {
		tmp = t_1 * (2.0 / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (l / 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 * cos(k_m)) / k_m
    if (k_m <= 7d-5) then
        tmp = ((((l / (k_m * k_m)) / k_m) * 2.0d0) / t) * t_1
    else
        tmp = t_1 * (2.0d0 / ((t * (0.5d0 + ((-0.5d0) * cos((k_m * 2.0d0))))) / (l / 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.cos(k_m)) / k_m;
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	} else {
		tmp = t_1 * (2.0 / ((t * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))) / (l / k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (l * math.cos(k_m)) / k_m
	tmp = 0
	if k_m <= 7e-5:
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1
	else:
		tmp = t_1 * (2.0 / ((t * (0.5 + (-0.5 * math.cos((k_m * 2.0))))) / (l / k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l * cos(k_m)) / k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / k_m) * 2.0) / t) * t_1);
	else
		tmp = Float64(t_1 * Float64(2.0 / Float64(Float64(t * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0))))) / Float64(l / k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (l * cos(k_m)) / k_m;
	tmp = 0.0;
	if (k_m <= 7e-5)
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	else
		tmp = t_1 * (2.0 / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (l / 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[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 7e-5], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(2.0 / N[(N[(t * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.5%

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

   5. Simplified 76.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 79.3%

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

   9. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 79.2%

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

   12. Simplified 79.2%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 65.1%

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

   5. Simplified 65.1%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. frac-times N/A

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

       3. metadata-eval N/A

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

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

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

       5. clear-num N/A

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

       6. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       13. /-lowering-/.f64 99.1%

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

   11. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{\frac{2}{\frac{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{\frac{\ell}{k}}}} \cdot \frac{\ell \cdot \cos k}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k \cdot k}}{k} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{2}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}{\frac{\ell}{k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell \cdot \cos k\_m}{k\_m}\\ \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m \cdot k\_m}}{k\_m} \cdot 2}{t} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{k\_m} \cdot \frac{2}{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* l (cos k_m)) k_m)))
   (if (<= k_m 7e-5)
     (* (/ (* (/ (/ l (* k_m k_m)) k_m) 2.0) t) t_1)
     (* t_1 (* (/ l k_m) (/ 2.0 (* t (+ 0.5 (* -0.5 (cos (* k_m 2.0)))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l * cos(k_m)) / k_m;
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	} else {
		tmp = t_1 * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * cos((k_m * 2.0)))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * cos(k_m)) / k_m
    if (k_m <= 7d-5) then
        tmp = ((((l / (k_m * k_m)) / k_m) * 2.0d0) / t) * t_1
    else
        tmp = t_1 * ((l / k_m) * (2.0d0 / (t * (0.5d0 + ((-0.5d0) * cos((k_m * 2.0d0)))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (l * Math.cos(k_m)) / k_m;
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	} else {
		tmp = t_1 * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * Math.cos((k_m * 2.0)))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (l * math.cos(k_m)) / k_m
	tmp = 0
	if k_m <= 7e-5:
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1
	else:
		tmp = t_1 * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * math.cos((k_m * 2.0)))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l * cos(k_m)) / k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / k_m) * 2.0) / t) * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64(l / k_m) * Float64(2.0 / Float64(t * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0))))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (l * cos(k_m)) / k_m;
	tmp = 0.0;
	if (k_m <= 7e-5)
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * t_1;
	else
		tmp = t_1 * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * cos((k_m * 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 7e-5], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(l / k$95$m), $MachinePrecision] * N[(2.0 / N[(t * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.5%

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

   5. Simplified 76.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 79.3%

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

   9. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 79.2%

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

   12. Simplified 79.2%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 65.1%

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

   5. Simplified 65.1%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k \cdot k}}{k} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m \cdot k\_m}}{k\_m} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k\_m}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k\_m}{k\_m}}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}{\frac{2}{\frac{k\_m}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* (/ (* (/ (/ l (* k_m k_m)) k_m) 2.0) t) (/ (* l (cos k_m)) k_m))
   (*
    l
    (/
     (/ (cos k_m) k_m)
     (/ (* t (+ 0.5 (* -0.5 (cos (* k_m 2.0))))) (/ 2.0 (/ k_m l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	} else {
		tmp = l * ((cos(k_m) / k_m) / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (2.0 / (k_m / 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 <= 7d-5) then
        tmp = ((((l / (k_m * k_m)) / k_m) * 2.0d0) / t) * ((l * cos(k_m)) / k_m)
    else
        tmp = l * ((cos(k_m) / k_m) / ((t * (0.5d0 + ((-0.5d0) * cos((k_m * 2.0d0))))) / (2.0d0 / (k_m / 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 <= 7e-5) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * Math.cos(k_m)) / k_m);
	} else {
		tmp = l * ((Math.cos(k_m) / k_m) / ((t * (0.5 + (-0.5 * Math.cos((k_m * 2.0))))) / (2.0 / (k_m / l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7e-5:
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * math.cos(k_m)) / k_m)
	else:
		tmp = l * ((math.cos(k_m) / k_m) / ((t * (0.5 + (-0.5 * math.cos((k_m * 2.0))))) / (2.0 / (k_m / l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / k_m) * 2.0) / t) * Float64(Float64(l * cos(k_m)) / k_m));
	else
		tmp = Float64(l * Float64(Float64(cos(k_m) / k_m) / Float64(Float64(t * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0))))) / Float64(2.0 / Float64(k_m / l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7e-5)
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	else
		tmp = l * ((cos(k_m) / k_m) / ((t * (0.5 + (-0.5 * cos((k_m * 2.0))))) / (2.0 / (k_m / l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(N[(t * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.5%

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

   5. Simplified 76.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 79.3%

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

   9. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 79.2%

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

   12. Simplified 79.2%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 65.1%

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

   5. Simplified 65.1%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.1%

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

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

       5. frac-times N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

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

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

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

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

       11. clear-num N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

   11. Applied egg-rr 93.5%

       \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\cos k}{k}}{\frac{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}{\frac{2}{\frac{k}{\ell}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k \cdot k}}{k} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k}}{\frac{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}{\frac{2}{\frac{k}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\frac{1}{t} \cdot \left(2 + \left(k\_m \cdot k\_m\right) \cdot 0.6666666666666666\right)}{k\_m \cdot k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{2}{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.5e+109)
   (*
    (/ (* l (cos k_m)) k_m)
    (*
     (/ l k_m)
     (/ (* (/ 1.0 t) (+ 2.0 (* (* k_m k_m) 0.6666666666666666))) (* k_m k_m))))
   (*
    (/ l k_m)
    (* (/ l k_m) (/ 2.0 (* t (+ 0.5 (* -0.5 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+109) {
		tmp = ((l * cos(k_m)) / k_m) * ((l / k_m) * (((1.0 / t) * (2.0 + ((k_m * k_m) * 0.6666666666666666))) / (k_m * k_m)));
	} else {
		tmp = (l / k_m) * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * cos((k_m * 2.0)))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d+109) then
        tmp = ((l * cos(k_m)) / k_m) * ((l / k_m) * (((1.0d0 / t) * (2.0d0 + ((k_m * k_m) * 0.6666666666666666d0))) / (k_m * k_m)))
    else
        tmp = (l / k_m) * ((l / k_m) * (2.0d0 / (t * (0.5d0 + ((-0.5d0) * cos((k_m * 2.0d0)))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+109) {
		tmp = ((l * Math.cos(k_m)) / k_m) * ((l / k_m) * (((1.0 / t) * (2.0 + ((k_m * k_m) * 0.6666666666666666))) / (k_m * k_m)));
	} else {
		tmp = (l / k_m) * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * Math.cos((k_m * 2.0)))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.5e+109:
		tmp = ((l * math.cos(k_m)) / k_m) * ((l / k_m) * (((1.0 / t) * (2.0 + ((k_m * k_m) * 0.6666666666666666))) / (k_m * k_m)))
	else:
		tmp = (l / k_m) * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * math.cos((k_m * 2.0)))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e+109)
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(l / k_m) * Float64(Float64(Float64(1.0 / t) * Float64(2.0 + Float64(Float64(k_m * k_m) * 0.6666666666666666))) / Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l / k_m) * Float64(2.0 / Float64(t * Float64(0.5 + Float64(-0.5 * cos(Float64(k_m * 2.0))))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e+109)
		tmp = ((l * cos(k_m)) / k_m) * ((l / k_m) * (((1.0 / t) * (2.0 + ((k_m * k_m) * 0.6666666666666666))) / (k_m * k_m)));
	else
		tmp = (l / k_m) * ((l / k_m) * (2.0 / (t * (0.5 + (-0.5 * cos((k_m * 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.5e+109], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(N[(1.0 / t), $MachinePrecision] * N[(2.0 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(2.0 / N[(t * N[(0.5 + N[(-0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e109

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.4%

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

   5. Simplified 76.4%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 81.3%

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

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. +-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. distribute-rgt-out N/A

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

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

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 67.5%

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

   12. Simplified 67.5%

       \[\leadsto \left(\color{blue}{\frac{\frac{1}{t} \cdot \left(2 + \left(k \cdot k\right) \cdot 0.6666666666666666\right)}{k \cdot k}} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell \cdot \cos k}{k} \]

   if 2.5000000000000001e109 < k

   1. Initial program 25.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 56.2%

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

   5. Simplified 56.2%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.3%

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

   9. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, k\right)\right)\right)\right), t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.7%

       \[\leadsto \left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{\ell}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{1}{t} \cdot \left(2 + \left(k \cdot k\right) \cdot 0.6666666666666666\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot \left(0.5 + -0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m \cdot k\_m}}{k\_m} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k\_m}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k\_m}}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.5e+66)
   (* (/ (* (/ (/ l (* k_m k_m)) k_m) 2.0) t) (/ (* l (cos k_m)) k_m))
   (* -0.3333333333333333 (/ (/ (* l (/ l k_m)) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e+66) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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 <= 3.5d+66) then
        tmp = ((((l / (k_m * k_m)) / k_m) * 2.0d0) / t) * ((l * cos(k_m)) / k_m)
    else
        tmp = (-0.3333333333333333d0) * (((l * (l / k_m)) / 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 <= 3.5e+66) {
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * Math.cos(k_m)) / k_m);
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.5e+66:
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * math.cos(k_m)) / k_m)
	else:
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5e+66)
		tmp = Float64(Float64(Float64(Float64(Float64(l / Float64(k_m * k_m)) / k_m) * 2.0) / t) * Float64(Float64(l * cos(k_m)) / k_m));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l * Float64(l / k_m)) / 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 <= 3.5e+66)
		tmp = ((((l / (k_m * k_m)) / k_m) * 2.0) / t) * ((l * cos(k_m)) / k_m);
	else
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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, 3.5e+66], N[(N[(N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999997e66

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.4%

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

   5. Simplified 76.4%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   7. Applied egg-rr 78.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 80.7%

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

   9. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\left(\frac{2}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \cdot \frac{\ell}{k}\right)} \cdot \frac{\ell \cdot \cos k}{k} \]

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. unpow3 N/A

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

       7. unpow2 N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 76.8%

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

   12. Simplified 76.8%

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

   if 3.4999999999999997e66 < k

   1. Initial program 25.7%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 45.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

   7. Simplified 9.0%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.8%

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

   10. Simplified 47.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

       4. associate-/r* N/A

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

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

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

       6. associate-/l* N/A

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

       7. associate-/l* N/A

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

       8. associate-/r* N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 53.0%

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

   12. Applied egg-rr 53.0%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

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

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

       6. /-lowering-/.f64 53.4%

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

   14. Applied egg-rr 53.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{t}}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k \cdot k}}{k} \cdot 2}{t} \cdot \frac{\ell \cdot \cos k}{k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k}}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k\_m \cdot t}}{k\_m} \cdot \left(2 + k\_m \cdot \left(k\_m \cdot -0.3333333333333333\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k\_m}}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.5e+40)
   (/
    (*
     (/ (* l (/ l (* k_m t))) k_m)
     (+ 2.0 (* k_m (* k_m -0.3333333333333333))))
    (* k_m k_m))
   (* -0.3333333333333333 (/ (/ (* l (/ l k_m)) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e+40) {
		tmp = (((l * (l / (k_m * t))) / k_m) * (2.0 + (k_m * (k_m * -0.3333333333333333)))) / (k_m * k_m);
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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 <= 3.5d+40) then
        tmp = (((l * (l / (k_m * t))) / k_m) * (2.0d0 + (k_m * (k_m * (-0.3333333333333333d0))))) / (k_m * k_m)
    else
        tmp = (-0.3333333333333333d0) * (((l * (l / k_m)) / 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 <= 3.5e+40) {
		tmp = (((l * (l / (k_m * t))) / k_m) * (2.0 + (k_m * (k_m * -0.3333333333333333)))) / (k_m * k_m);
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.5e+40:
		tmp = (((l * (l / (k_m * t))) / k_m) * (2.0 + (k_m * (k_m * -0.3333333333333333)))) / (k_m * k_m)
	else:
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5e+40)
		tmp = Float64(Float64(Float64(Float64(l * Float64(l / Float64(k_m * t))) / k_m) * Float64(2.0 + Float64(k_m * Float64(k_m * -0.3333333333333333)))) / Float64(k_m * k_m));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l * Float64(l / k_m)) / 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 <= 3.5e+40)
		tmp = (((l * (l / (k_m * t))) / k_m) * (2.0 + (k_m * (k_m * -0.3333333333333333)))) / (k_m * k_m);
	else
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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, 3.5e+40], N[(N[(N[(N[(l * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(2.0 + N[(k$95$m * N[(k$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999999e40

   1. Initial program 36.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + 2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right)}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot 2}{{k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot 2\right)}{{k}^{4}} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right) \cdot 2\right)}{{k}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{6}\right) \cdot 2\right)}{{k}^{4}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{6} \cdot 2\right)\right)}{{k}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

   8. Simplified 53.1%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(\color{blue}{k} \cdot k\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \mathsf{+.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right)\right), \left(k \cdot k\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \mathsf{+.f64}\left(2, \left(k \cdot \left(k \cdot \frac{-1}{3}\right)\right)\right)\right), \left(k \cdot k\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \frac{-1}{3}\right)\right)\right)\right), \left(k \cdot k\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right)\right), \left(k \cdot k\right)\right) \]

      18. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Applied egg-rr 66.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{k \cdot t}}{k} \cdot \left(2 + k \cdot \left(k \cdot -0.3333333333333333\right)\right)}{k \cdot k}} \]

   if 3.4999999999999999e40 < k

   1. Initial program 23.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 47.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 47.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 51.6%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 51.6%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t} \cdot \ell\right), k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t} \cdot \ell\right), k\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k} \cdot \ell}{t}\right), k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \ell\right), t\right), k\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \ell\right), t\right), k\right)\right) \]

       6. /-lowering-/.f64 51.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), t\right), k\right)\right) \]

   14. Applied egg-rr 51.9%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{t}}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot t}}{k} \cdot \left(2 + k \cdot \left(k \cdot -0.3333333333333333\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k}}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k\_m \cdot t}}{k\_m} \cdot \frac{2 + k\_m \cdot \left(k\_m \cdot -0.3333333333333333\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k\_m}}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.2e+41)
   (*
    (/ (* l (/ l (* k_m t))) k_m)
    (/ (+ 2.0 (* k_m (* k_m -0.3333333333333333))) (* k_m k_m)))
   (* -0.3333333333333333 (/ (/ (* l (/ l k_m)) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e+41) {
		tmp = ((l * (l / (k_m * t))) / k_m) * ((2.0 + (k_m * (k_m * -0.3333333333333333))) / (k_m * k_m));
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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 <= 8.2d+41) then
        tmp = ((l * (l / (k_m * t))) / k_m) * ((2.0d0 + (k_m * (k_m * (-0.3333333333333333d0)))) / (k_m * k_m))
    else
        tmp = (-0.3333333333333333d0) * (((l * (l / k_m)) / 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 <= 8.2e+41) {
		tmp = ((l * (l / (k_m * t))) / k_m) * ((2.0 + (k_m * (k_m * -0.3333333333333333))) / (k_m * k_m));
	} else {
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.2e+41:
		tmp = ((l * (l / (k_m * t))) / k_m) * ((2.0 + (k_m * (k_m * -0.3333333333333333))) / (k_m * k_m))
	else:
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / t) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.2e+41)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k_m * t))) / k_m) * Float64(Float64(2.0 + Float64(k_m * Float64(k_m * -0.3333333333333333))) / Float64(k_m * k_m)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l * Float64(l / k_m)) / 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 <= 8.2e+41)
		tmp = ((l * (l / (k_m * t))) / k_m) * ((2.0 + (k_m * (k_m * -0.3333333333333333))) / (k_m * k_m));
	else
		tmp = -0.3333333333333333 * (((l * (l / k_m)) / 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, 8.2e+41], N[(N[(N[(l * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 + N[(k$95$m * N[(k$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000007e41

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right), \color{blue}{\left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot \cos k\right)\right), \left(\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \cos k\right)\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \cos k\right)\right)\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left({k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \left(\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right), \color{blue}{k}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {\sin k}^{2}\right), k\right), k\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({\sin k}^{2}\right)\right), k\right), k\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\sin k, 2\right)\right), k\right), k\right)\right) \]

      16. sin-lowering-sin.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right), k\right), k\right)\right) \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + 2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right)}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot 2}{{k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot 2\right)}{{k}^{4}} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{2} - \frac{-1}{3}\right)\right) \cdot 2\right)}{{k}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{6}\right) \cdot 2\right)}{{k}^{4}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \left(\frac{-1}{6} \cdot 2\right)\right)}{{k}^{4}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{\left(2 \cdot \color{blue}{2}\right)}} \]

      10. pow-sqr N/A

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

   8. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \left(2 + \left(k \cdot k\right) \cdot -0.3333333333333333\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right) \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)} \]

      2. times-frac N/A

         \[\leadsto \frac{2 + \left(k \cdot k\right) \cdot \frac{-1}{3}}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 + \left(k \cdot k\right) \cdot \frac{-1}{3}}{k \cdot k}\right), \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 + \left(k \cdot k\right) \cdot \frac{-1}{3}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{k \cdot k}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{-1}{3}\right)\right), \left(k \cdot k\right)\right), \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{k \cdot k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(k \cdot \left(k \cdot \frac{-1}{3}\right)\right)\right), \left(k \cdot k\right)\right), \left(\frac{\frac{\ell \cdot \color{blue}{\ell}}{t}}{k \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \frac{-1}{3}\right)\right)\right), \left(k \cdot k\right)\right), \left(\frac{\frac{\ell \cdot \color{blue}{\ell}}{t}}{k \cdot k}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \left(k \cdot k\right)\right), \left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell \cdot \ell}{\color{blue}{t}}}{k \cdot k}\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

      14. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

      18. *-lowering-*.f64 66.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \frac{-1}{3}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   10. Applied egg-rr 66.3%

       \[\leadsto \color{blue}{\frac{2 + k \cdot \left(k \cdot -0.3333333333333333\right)}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]

   if 8.2000000000000007e41 < k

   1. Initial program 24.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 44.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 48.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 48.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 52.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t} \cdot \ell\right), k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k}}{t} \cdot \ell\right), k\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{k} \cdot \ell}{t}\right), k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \ell\right), t\right), k\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \ell\right), t\right), k\right)\right) \]

       6. /-lowering-/.f64 53.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), t\right), k\right)\right) \]

   14. Applied egg-rr 53.2%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{t}}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot t}}{k} \cdot \frac{2 + k \cdot \left(k \cdot -0.3333333333333333\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \frac{\ell}{k}}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 600000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot \left(k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 600000000000.0)
   (/ 2.0 (/ (/ (* t (* k_m (* k_m (* k_m k_m)))) l) l))
   (* -0.3333333333333333 (/ (/ l 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 <= 600000000000.0) {
		tmp = 2.0 / (((t * (k_m * (k_m * (k_m * k_m)))) / l) / l);
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0d0) then
        tmp = 2.0d0 / (((t * (k_m * (k_m * (k_m * k_m)))) / l) / l)
    else
        tmp = (-0.3333333333333333d0) * ((l / 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 <= 600000000000.0) {
		tmp = 2.0 / (((t * (k_m * (k_m * (k_m * k_m)))) / l) / l);
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0:
		tmp = 2.0 / (((t * (k_m * (k_m * (k_m * k_m)))) / l) / l)
	else:
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k_m * Float64(k_m * Float64(k_m * k_m)))) / l) / l));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / 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 <= 600000000000.0)
		tmp = 2.0 / (((t * (k_m * (k_m * (k_m * k_m)))) / l) / l);
	else
		tmp = -0.3333333333333333 * ((l / 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, 600000000000.0], N[(2.0 / N[(N[(N[(t * N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e11

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right), \ell\right), \ell\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)\right), \ell\right), \ell\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)\right), \ell\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot k\right)\right)\right), \ell\right), \ell\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right)\right) \]

      12. *-lowering-*.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right)\right) \]

   7. Applied egg-rr 72.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}{\ell}}{\ell}}} \]

   if 6e11 < k

   1. Initial program 22.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{k \cdot t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k} \cdot k}\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{t}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       9. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\ell}\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{k \cdot t}}{\ell}\right)\right)\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right) \]

       16. /-lowering-/.f64 50.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right) \]

   14. Applied egg-rr 50.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 600000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 600000000000.0)
   (/ 2.0 (/ (/ (* k_m (* k_m (* k_m k_m))) (/ l t)) l))
   (* -0.3333333333333333 (/ (/ l 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 <= 600000000000.0) {
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / (l / t)) / l);
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0d0) then
        tmp = 2.0d0 / (((k_m * (k_m * (k_m * k_m))) / (l / t)) / l)
    else
        tmp = (-0.3333333333333333d0) * ((l / 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 <= 600000000000.0) {
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / (l / t)) / l);
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0:
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / (l / t)) / l)
	else:
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * Float64(k_m * k_m))) / Float64(l / t)) / l));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / 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 <= 600000000000.0)
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / (l / t)) / l);
	else
		tmp = -0.3333333333333333 * ((l / 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, 600000000000.0], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e11

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right), \color{blue}{\ell}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{1}{\frac{\ell}{t}}\right), \ell\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\frac{\ell}{t}}\right), \ell\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot k\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\frac{\ell}{t}\right)\right), \ell\right)\right) \]

      13. /-lowering-/.f64 72.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(\ell, t\right)\right), \ell\right)\right) \]

   7. Applied egg-rr 72.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\frac{\ell}{t}}}{\ell}}} \]

   if 6e11 < k

   1. Initial program 22.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{k \cdot t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k} \cdot k}\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{t}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       9. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\ell}\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{k \cdot t}}{\ell}\right)\right)\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right) \]

       16. /-lowering-/.f64 50.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right) \]

   14. Applied egg-rr 50.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 600000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)}{\ell}}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 600000000000.0)
   (/ 2.0 (/ (/ (* k_m (* k_m (* k_m k_m))) l) (/ l t)))
   (* -0.3333333333333333 (/ (/ l 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 <= 600000000000.0) {
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / l) / (l / t));
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0d0) then
        tmp = 2.0d0 / (((k_m * (k_m * (k_m * k_m))) / l) / (l / t))
    else
        tmp = (-0.3333333333333333d0) * ((l / 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 <= 600000000000.0) {
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / l) / (l / t));
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0:
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / l) / (l / t))
	else:
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * Float64(k_m * k_m))) / l) / Float64(l / t)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / 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 <= 600000000000.0)
		tmp = 2.0 / (((k_m * (k_m * (k_m * k_m))) / l) / (l / t));
	else
		tmp = -0.3333333333333333 * ((l / 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, 600000000000.0], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e11

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}}{\color{blue}{\frac{\ell}{t}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell}\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right), \ell\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot k\right)\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right), \left(\frac{\ell}{t}\right)\right)\right) \]

      13. /-lowering-/.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right) \]

   7. Applied egg-rr 71.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}{\ell}}{\frac{\ell}{t}}}} \]

   if 6e11 < k

   1. Initial program 22.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{k \cdot t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k} \cdot k}\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{t}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       9. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\ell}\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{k \cdot t}}{\ell}\right)\right)\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right) \]

       16. /-lowering-/.f64 50.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right) \]

   14. Applied egg-rr 50.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 600000000000:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 600000000000.0)
   (* l (/ (/ 2.0 (* k_m (* k_m (* k_m k_m)))) (/ t l)))
   (* -0.3333333333333333 (/ (/ l 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 <= 600000000000.0) {
		tmp = l * ((2.0 / (k_m * (k_m * (k_m * k_m)))) / (t / l));
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0d0) then
        tmp = l * ((2.0d0 / (k_m * (k_m * (k_m * k_m)))) / (t / l))
    else
        tmp = (-0.3333333333333333d0) * ((l / 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 <= 600000000000.0) {
		tmp = l * ((2.0 / (k_m * (k_m * (k_m * k_m)))) / (t / l));
	} else {
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0:
		tmp = l * ((2.0 / (k_m * (k_m * (k_m * k_m)))) / (t / l))
	else:
		tmp = -0.3333333333333333 * ((l / 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 <= 600000000000.0)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k_m * Float64(k_m * Float64(k_m * k_m)))) / Float64(t / l)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / 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 <= 600000000000.0)
		tmp = l * ((2.0 / (k_m * (k_m * (k_m * k_m)))) / (t / l));
	else
		tmp = -0.3333333333333333 * ((l / 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, 600000000000.0], N[(l * N[(N[(2.0 / N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e11

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 63.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 71.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}} \cdot \ell} \]

   if 6e11 < k

   1. Initial program 22.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 7.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 50.4%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{k \cdot t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k} \cdot k}\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}}\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{t}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       9. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\ell}\right)}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{k \cdot t}}{\ell}\right)\right)\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right) \]

       16. /-lowering-/.f64 50.2%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right) \]

   14. Applied egg-rr 50.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 600000000000:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.0% accurate, 23.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\ell}} \cdot 0.3333333333333333}{0 - k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k\_m \cdot t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.2e-144)
   (/ (* (/ l (/ t l)) 0.3333333333333333) (- 0.0 (* k_m k_m)))
   (* -0.3333333333333333 (/ (* l (/ l (* k_m t))) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-144) {
		tmp = ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m));
	} else {
		tmp = -0.3333333333333333 * ((l * (l / (k_m * 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 <= 7.2d-144) then
        tmp = ((l / (t / l)) * 0.3333333333333333d0) / (0.0d0 - (k_m * k_m))
    else
        tmp = (-0.3333333333333333d0) * ((l * (l / (k_m * 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 <= 7.2e-144) {
		tmp = ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m));
	} else {
		tmp = -0.3333333333333333 * ((l * (l / (k_m * t))) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.2e-144:
		tmp = ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m))
	else:
		tmp = -0.3333333333333333 * ((l * (l / (k_m * t))) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-144)
		tmp = Float64(Float64(Float64(l / Float64(t / l)) * 0.3333333333333333) / Float64(0.0 - Float64(k_m * k_m)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l * Float64(l / Float64(k_m * 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 <= 7.2e-144)
		tmp = ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m));
	else
		tmp = -0.3333333333333333 * ((l * (l / (k_m * 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, 7.2e-144], N[(N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.0 - N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000001e-144

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 31.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 18.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 18.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 20.1%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \frac{\ell}{k \cdot t}\right)}{\color{blue}{k}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{k \cdot t}}{k} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k}}{k} \]

       4. associate-/l* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{k}}{k} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{k}}{k} \]

       6. clear-num N/A

          \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}}{k}}{k} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}}{\color{blue}{k \cdot k}} \]

       8. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}\right)}{\color{blue}{\mathsf{neg}\left(k \cdot k\right)}} \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}\right)\right), \color{blue}{\left(\mathsf{neg}\left(k \cdot k\right)\right)}\right) \]

   14. Applied egg-rr 50.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{t}{\ell}} \cdot 0.3333333333333333}{0 - k \cdot k}} \]

   if 7.2000000000000001e-144 < k

   1. Initial program 27.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   7. Simplified 15.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 32.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

       2. associate-/l* N/A

          \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

       8. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

       12. *-lowering-*.f64 35.1%

           \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

   12. Applied egg-rr 35.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
   13. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\mathsf{neg}\left(\ell\right)}{\mathsf{neg}\left(k \cdot t\right)}\right)\right), k\right)\right) \]

       2. distribute-frac-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(\frac{\ell}{\mathsf{neg}\left(k \cdot t\right)}\right)\right)\right), k\right)\right) \]

       3. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\left(\frac{\ell}{\mathsf{neg}\left(k \cdot t\right)}\right)\right)\right), k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\ell, \left(\mathsf{neg}\left(k \cdot t\right)\right)\right)\right)\right), k\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\ell, \left(0 - k \cdot t\right)\right)\right)\right), k\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{\_.f64}\left(0, \left(k \cdot t\right)\right)\right)\right)\right), k\right)\right) \]

       7. *-lowering-*.f64 36.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, t\right)\right)\right)\right)\right), k\right)\right) \]

   14. Applied egg-rr 36.5%

       \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \color{blue}{\left(-\frac{\ell}{0 - k \cdot t}\right)}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\ell}} \cdot 0.3333333333333333}{0 - k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.3% accurate, 32.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell}{\frac{t}{\ell}} \cdot 0.3333333333333333}{0 - k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (/ l (/ t l)) 0.3333333333333333) (- 0.0 (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / (t / l)) * 0.3333333333333333) / (0.0 - (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 / (t / l)) * 0.3333333333333333d0) / (0.0d0 - (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 / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / Float64(t / l)) * 0.3333333333333333) / Float64(0.0 - Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / (t / l)) * 0.3333333333333333) / (0.0 - (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.0 - N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 27.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   12. *-lowering-*.f64 22.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

    2. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

    12. *-lowering-*.f64 24.4%

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

12. Applied egg-rr 24.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \frac{\ell}{k \cdot t}\right)}{\color{blue}{k}} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{k \cdot t}}{k} \]

    3. associate-/l/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k}}{k} \]

    4. associate-/l* N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{k}}{k} \]

    5. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{k}}{k} \]

    6. clear-num N/A

       \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}}{k}}{k} \]

    7. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}}{\color{blue}{k \cdot k}} \]

    8. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}\right)}{\color{blue}{\mathsf{neg}\left(k \cdot k\right)}} \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\frac{t}{\ell \cdot \ell}} \cdot \frac{-1}{3}\right)\right), \color{blue}{\left(\mathsf{neg}\left(k \cdot k\right)\right)}\right) \]

14. Applied egg-rr 45.6%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{t}{\ell}} \cdot 0.3333333333333333}{0 - k \cdot k}} \]
15. Add Preprocessing

Alternative 15: 31.3% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m}}{\frac{k\_m}{\frac{\ell}{t}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* -0.3333333333333333 (/ (/ l k_m) (/ k_m (/ l t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.3333333333333333 * ((l / k_m) / (k_m / (l / 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 = (-0.3333333333333333d0) * ((l / k_m) / (k_m / (l / t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.3333333333333333 * ((l / k_m) / (k_m / (l / t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.3333333333333333 * ((l / k_m) / (k_m / (l / t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.3333333333333333 * Float64(Float64(l / k_m) / Float64(k_m / Float64(l / t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.3333333333333333 * ((l / k_m) / (k_m / (l / t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] / N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 27.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   12. *-lowering-*.f64 22.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

    2. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

    12. *-lowering-*.f64 24.4%

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

12. Applied egg-rr 24.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{k \cdot t}}{k}\right)\right) \]

    2. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k}\right)\right) \]

    3. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot k}}\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k} \cdot k}\right)\right) \]

    5. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{k}}\right)\right) \]

    6. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{t \cdot k}}\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{t}}\right)\right) \]

    8. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\ell}{k} \cdot \frac{1}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

    9. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\ell}{k}}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{k \cdot t}{\ell}\right)}\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{k \cdot t}}{\ell}\right)\right)\right) \]

    12. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right) \]

    13. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

    14. un-div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right) \]

    15. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right) \]

    16. /-lowering-/.f64 24.5%

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right) \]

14. Applied egg-rr 24.5%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{t}}}} \]
15. Add Preprocessing

Alternative 16: 31.3% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.3333333333333333 \cdot \frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{t}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* -0.3333333333333333 (/ (* (/ l k_m) (/ l t)) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.3333333333333333 * (((l / k_m) * (l / t)) / 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 = (-0.3333333333333333d0) * (((l / k_m) * (l / t)) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.3333333333333333 * (((l / k_m) * (l / t)) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.3333333333333333 * (((l / k_m) * (l / t)) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k_m) * Float64(l / t)) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.3333333333333333 * (((l / k_m) * (l / t)) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.3333333333333333 * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 27.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   12. *-lowering-*.f64 22.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

    2. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

    12. *-lowering-*.f64 24.4%

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

12. Applied egg-rr 24.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{k \cdot t}\right), k\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot k}\right), k\right)\right) \]

    3. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right), k\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{\ell}{k}\right)\right), k\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{k}\right)\right), k\right)\right) \]

    6. /-lowering-/.f64 24.5%

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, k\right)\right), k\right)\right) \]

14. Applied egg-rr 24.5%

    \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{k}}}{k} \]

15. Final simplification 24.5%

    \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k} \]
16. Add Preprocessing

Alternative 17: 31.2% accurate, 38.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k\_m \cdot t}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* -0.3333333333333333 (/ (* l (/ l (* k_m t))) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.3333333333333333 * ((l * (l / (k_m * t))) / 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 = (-0.3333333333333333d0) * ((l * (l / (k_m * t))) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.3333333333333333 * ((l * (l / (k_m * t))) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.3333333333333333 * ((l * (l / (k_m * t))) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.3333333333333333 * Float64(Float64(l * Float64(l / Float64(k_m * t))) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.3333333333333333 * ((l * (l / (k_m * t))) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.3333333333333333 * N[(N[(l * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{2}{t}}{\frac{\frac{t \cdot t}{\ell}}{\ell}}}{\sin k}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-1}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}\right), \left({k}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 27.2%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{k \cdot k}}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   12. *-lowering-*.f64 22.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}{\color{blue}{k} \cdot k} \]

    2. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)}\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{\color{blue}{k}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k}\right), \color{blue}{k}\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \frac{\ell}{t}}{k}\right), k\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\frac{\ell}{t}}{k}\right), k\right)\right) \]

    8. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot k}\right), k\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\ell}{t \cdot k}\right)\right), k\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right)\right), k\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right)\right), k\right)\right) \]

    12. *-lowering-*.f64 24.4%

        \[\leadsto \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right) \]

12. Applied egg-rr 24.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))