Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 95.4%

Time: 17.8s

Alternatives: 15

Speedup: 28.1×

• 39.1% 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: 35.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-134}:\\ \;\;\;\;{\left(\frac{k\_m}{\frac{\ell}{k\_m}}\right)}^{-1} \cdot {\left(\frac{\frac{k\_m}{\frac{\ell}{k\_m \cdot t}}}{2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot \frac{{\sin k\_m}^{2}}{\ell}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.8e-134)
   (* (pow (/ k_m (/ l k_m)) -1.0) (pow (/ (/ k_m (/ l (* k_m t))) 2.0) -1.0))
   (/
    2.0
    (/ (* (* k_m (* t (/ k_m l))) (/ (pow (sin k_m) 2.0) l)) (cos k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-134) {
		tmp = pow((k_m / (l / k_m)), -1.0) * pow(((k_m / (l / (k_m * t))) / 2.0), -1.0);
	} else {
		tmp = 2.0 / (((k_m * (t * (k_m / l))) * (pow(sin(k_m), 2.0) / l)) / cos(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 <= 5.8d-134) then
        tmp = ((k_m / (l / k_m)) ** (-1.0d0)) * (((k_m / (l / (k_m * t))) / 2.0d0) ** (-1.0d0))
    else
        tmp = 2.0d0 / (((k_m * (t * (k_m / l))) * ((sin(k_m) ** 2.0d0) / l)) / cos(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 <= 5.8e-134) {
		tmp = Math.pow((k_m / (l / k_m)), -1.0) * Math.pow(((k_m / (l / (k_m * t))) / 2.0), -1.0);
	} else {
		tmp = 2.0 / (((k_m * (t * (k_m / l))) * (Math.pow(Math.sin(k_m), 2.0) / l)) / Math.cos(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.8e-134:
		tmp = math.pow((k_m / (l / k_m)), -1.0) * math.pow(((k_m / (l / (k_m * t))) / 2.0), -1.0)
	else:
		tmp = 2.0 / (((k_m * (t * (k_m / l))) * (math.pow(math.sin(k_m), 2.0) / l)) / math.cos(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.8e-134)
		tmp = Float64((Float64(k_m / Float64(l / k_m)) ^ -1.0) * (Float64(Float64(k_m / Float64(l / Float64(k_m * t))) / 2.0) ^ -1.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(t * Float64(k_m / l))) * Float64((sin(k_m) ^ 2.0) / l)) / cos(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 <= 5.8e-134)
		tmp = ((k_m / (l / k_m)) ^ -1.0) * (((k_m / (l / (k_m * t))) / 2.0) ^ -1.0);
	else
		tmp = 2.0 / (((k_m * (t * (k_m / l))) * ((sin(k_m) ^ 2.0) / l)) / cos(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, 5.8e-134], N[(N[Power[N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m / N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.79999999999999986e-134

   1. Initial program 38.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 59.8%

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

   5. Simplified 59.8%

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

      1. 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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 54.1%

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

   7. Applied egg-rr 54.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 78.2%

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

   11. Applied egg-rr 80.9%

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

   if 5.79999999999999986e-134 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 79.0%

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

   5. Simplified 79.0%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 92.3%

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

   7. Applied egg-rr 92.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 95.4%

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

   9. Applied egg-rr 95.4%

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

Alternative 2: 95.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \left(k\_m \cdot \frac{t}{\ell \cdot \cos k\_m}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.7e-6)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/
    2.0
    (*
     (/ (* k_m (+ 0.5 (* (cos (* k_m 2.0)) -0.5))) l)
     (* k_m (/ t (* l (cos k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * (k_m * (t / (l * cos(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 <= 6.7d-6) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = 2.0d0 / (((k_m * (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0)))) / l) * (k_m * (t / (l * cos(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 <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * (0.5 + (Math.cos((k_m * 2.0)) * -0.5))) / l) * (k_m * (t / (l * Math.cos(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = 2.0 / (((k_m * (0.5 + (math.cos((k_m * 2.0)) * -0.5))) / l) * (k_m * (t / (l * math.cos(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5))) / l) * Float64(k_m * Float64(t / Float64(l * cos(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 <= 6.7e-6)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = 2.0 / (((k_m * (0.5 + (cos((k_m * 2.0)) * -0.5))) / l) * (k_m * (t / (l * cos(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, 6.7e-6], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * N[(t / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.7e-6

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 80.5%

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

       1. associate-*l/ N/A

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

       2. frac-2neg N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. sub0-neg N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-/r* N/A

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

       10. associate-/r* N/A

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

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

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

       13. remove-double-neg N/A

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

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

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

   11. Applied egg-rr 83.2%

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

   if 6.7e-6 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 72.1%

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

   5. Simplified 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. sqr-sin-a N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

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

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

   9. Applied egg-rr 93.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(0.5 + \cos \left(k \cdot 2\right) \cdot -0.5\right)}{\ell} \cdot \left(k \cdot \frac{t}{\ell \cdot \cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\frac{\ell}{t}} \cdot \left(k\_m \cdot \frac{0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5}{\ell \cdot \cos k\_m}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.7e-6)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/
    2.0
    (*
     (/ k_m (/ l t))
     (* k_m (/ (+ 0.5 (* (cos (* k_m 2.0)) -0.5)) (* l (cos k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / ((k_m / (l / t)) * (k_m * ((0.5 + (cos((k_m * 2.0)) * -0.5)) / (l * cos(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 <= 6.7d-6) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = 2.0d0 / ((k_m / (l / t)) * (k_m * ((0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0))) / (l * cos(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 <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / ((k_m / (l / t)) * (k_m * ((0.5 + (Math.cos((k_m * 2.0)) * -0.5)) / (l * Math.cos(k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = 2.0 / ((k_m / (l / t)) * (k_m * ((0.5 + (math.cos((k_m * 2.0)) * -0.5)) / (l * math.cos(k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(l / t)) * Float64(k_m * Float64(Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5)) / Float64(l * cos(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 <= 6.7e-6)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = 2.0 / ((k_m / (l / t)) * (k_m * ((0.5 + (cos((k_m * 2.0)) * -0.5)) / (l * cos(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, 6.7e-6], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.7e-6

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 80.5%

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

       1. associate-*l/ N/A

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

       2. frac-2neg N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. sub0-neg N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-/r* N/A

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

       10. associate-/r* N/A

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

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

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

       13. remove-double-neg N/A

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

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

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

   11. Applied egg-rr 83.2%

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

   if 6.7e-6 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 72.1%

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

   5. Simplified 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. associate-/l* N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

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

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

      15. associate-/l/ N/A

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

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

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

   9. Applied egg-rr 93.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \left(k \cdot \frac{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}{\ell \cdot \cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{\frac{\frac{k\_m}{\frac{\ell}{t}}}{\frac{\ell}{0.5 + \cos \left(k\_m \cdot 2\right) \cdot -0.5}}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.7e-6)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/
    2.0
    (*
     k_m
     (/
      (/ (/ k_m (/ l t)) (/ l (+ 0.5 (* (cos (* k_m 2.0)) -0.5))))
      (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (k_m * (((k_m / (l / t)) / (l / (0.5 + (cos((k_m * 2.0)) * -0.5)))) / cos(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 <= 6.7d-6) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = 2.0d0 / (k_m * (((k_m / (l / t)) / (l / (0.5d0 + (cos((k_m * 2.0d0)) * (-0.5d0))))) / cos(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 <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (k_m * (((k_m / (l / t)) / (l / (0.5 + (Math.cos((k_m * 2.0)) * -0.5)))) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = 2.0 / (k_m * (((k_m / (l / t)) / (l / (0.5 + (math.cos((k_m * 2.0)) * -0.5)))) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(Float64(k_m / Float64(l / t)) / Float64(l / Float64(0.5 + Float64(cos(Float64(k_m * 2.0)) * -0.5)))) / cos(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 <= 6.7e-6)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = 2.0 / (k_m * (((k_m / (l / t)) / (l / (0.5 + (cos((k_m * 2.0)) * -0.5)))) / cos(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, 6.7e-6], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(l / N[(0.5 + N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.7e-6

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 80.5%

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

       1. associate-*l/ N/A

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

       2. frac-2neg N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. sub0-neg N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-/r* N/A

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

       10. associate-/r* N/A

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

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

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

       13. remove-double-neg N/A

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

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

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

   11. Applied egg-rr 83.2%

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

   if 6.7e-6 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 72.1%

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

   5. Simplified 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\frac{\frac{k}{\frac{\ell}{t}}}{\frac{\ell}{0.5 + \cos \left(k \cdot 2\right) \cdot -0.5}}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot t}\\ \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - t\_1}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{t\_1}{k\_m}}{\frac{0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)}{\ell \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m t))))
   (if (<= k_m 6.7e-6)
     (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 t_1)))
     (*
      2.0
      (/ (/ t_1 k_m) (/ (- 0.5 (* 0.5 (cos (* k_m 2.0)))) (* l (cos k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * t);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	} else {
		tmp = 2.0 * ((t_1 / k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) / (l * cos(k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k_m * t)
    if (k_m <= 6.7d-6) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - t_1))
    else
        tmp = 2.0d0 * ((t_1 / k_m) / ((0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))) / (l * cos(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * t);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	} else {
		tmp = 2.0 * ((t_1 / k_m) / ((0.5 - (0.5 * Math.cos((k_m * 2.0)))) / (l * Math.cos(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * t)
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1))
	else:
		tmp = 2.0 * ((t_1 / k_m) / ((0.5 - (0.5 * math.cos((k_m * 2.0)))) / (l * math.cos(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * t))
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 / k_m) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))) / Float64(l * cos(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m * t);
	tmp = 0.0;
	if (k_m <= 6.7e-6)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	else
		tmp = 2.0 * ((t_1 / k_m) / ((0.5 - (0.5 * cos((k_m * 2.0)))) / (l * cos(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 6.7e-6], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 / k$95$m), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.7e-6

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 80.5%

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

       1. associate-*l/ N/A

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

       2. frac-2neg N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. sub0-neg N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-/r* N/A

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

       10. associate-/r* N/A

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

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

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

       13. remove-double-neg N/A

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

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

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

   11. Applied egg-rr 83.2%

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

   if 6.7e-6 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 72.1%

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

   5. Simplified 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 93.4%

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

   9. Applied egg-rr 93.4%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. clear-num N/A

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

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

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

   11. Applied egg-rr 90.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k \cdot t}}{k}}{\frac{0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)}{\ell \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot t}\\ \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot \frac{2}{k\_m \cdot \frac{0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)}{\ell \cdot t\_1}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m t))))
   (if (<= k_m 6.7e-6)
     (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 t_1)))
     (*
      (cos k_m)
      (/ 2.0 (* k_m (/ (- 0.5 (* 0.5 (cos (* k_m 2.0)))) (* l t_1))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * t);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	} else {
		tmp = cos(k_m) * (2.0 / (k_m * ((0.5 - (0.5 * cos((k_m * 2.0)))) / (l * t_1))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k_m * t)
    if (k_m <= 6.7d-6) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - t_1))
    else
        tmp = cos(k_m) * (2.0d0 / (k_m * ((0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))) / (l * t_1))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * t);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	} else {
		tmp = Math.cos(k_m) * (2.0 / (k_m * ((0.5 - (0.5 * Math.cos((k_m * 2.0)))) / (l * t_1))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l / (k_m * t)
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1))
	else:
		tmp = math.cos(k_m) * (2.0 / (k_m * ((0.5 - (0.5 * math.cos((k_m * 2.0)))) / (l * t_1))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * t))
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - t_1)));
	else
		tmp = Float64(cos(k_m) * Float64(2.0 / Float64(k_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0)))) / Float64(l * t_1)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l / (k_m * t);
	tmp = 0.0;
	if (k_m <= 6.7e-6)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - t_1));
	else
		tmp = cos(k_m) * (2.0 / (k_m * ((0.5 - (0.5 * cos((k_m * 2.0)))) / (l * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 6.7e-6], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 / N[(k$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.7e-6

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 60.5%

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

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 80.5%

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

       1. associate-*l/ N/A

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

       2. frac-2neg N/A

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

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

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

       4. metadata-eval N/A

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

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

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

       6. sub0-neg N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-/r* N/A

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

       10. associate-/r* N/A

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

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

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

       13. remove-double-neg N/A

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

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

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

   11. Applied egg-rr 83.2%

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

   if 6.7e-6 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 72.1%

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

   5. Simplified 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 93.4%

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

   9. Applied egg-rr 93.4%

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

       1. associate-/r/ N/A

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

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

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

   11. Applied egg-rr 87.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{k \cdot \frac{0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)}{\ell \cdot \frac{\ell}{k \cdot t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{k\_m \cdot k\_m}{\ell}\right) \cdot \frac{t}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k\_m}{\frac{\ell}{k\_m}}\right)}^{-1} \cdot {\left(\frac{\frac{k\_m}{\frac{\ell}{k\_m \cdot t}}}{2}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 7.6e-66)
   (/ 2.0 (/ (* (* (* k_m k_m) (/ (* k_m k_m) l)) (/ t l)) (cos k_m)))
   (*
    (pow (/ k_m (/ l k_m)) -1.0)
    (pow (/ (/ k_m (/ l (* k_m t))) 2.0) -1.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7.6e-66) {
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m));
	} else {
		tmp = pow((k_m / (l / k_m)), -1.0) * pow(((k_m / (l / (k_m * t))) / 2.0), -1.0);
	}
	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 (t <= 7.6d-66) then
        tmp = 2.0d0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m))
    else
        tmp = ((k_m / (l / k_m)) ** (-1.0d0)) * (((k_m / (l / (k_m * t))) / 2.0d0) ** (-1.0d0))
    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 (t <= 7.6e-66) {
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / Math.cos(k_m));
	} else {
		tmp = Math.pow((k_m / (l / k_m)), -1.0) * Math.pow(((k_m / (l / (k_m * t))) / 2.0), -1.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 7.6e-66:
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / math.cos(k_m))
	else:
		tmp = math.pow((k_m / (l / k_m)), -1.0) * math.pow(((k_m / (l / (k_m * t))) / 2.0), -1.0)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 7.6e-66)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * Float64(Float64(k_m * k_m) / l)) * Float64(t / l)) / cos(k_m)));
	else
		tmp = Float64((Float64(k_m / Float64(l / k_m)) ^ -1.0) * (Float64(Float64(k_m / Float64(l / Float64(k_m * t))) / 2.0) ^ -1.0));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 7.6e-66)
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m));
	else
		tmp = ((k_m / (l / k_m)) ^ -1.0) * (((k_m / (l / (k_m * t))) / 2.0) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 7.6e-66], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m / N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999995e-66

   1. Initial program 37.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 80.7%

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

   5. Simplified 80.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

      14. /-lowering-/.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 69.5%

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

   10. Simplified 69.5%

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

   if 7.5999999999999995e-66 < t

   1. Initial program 31.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 64.6%

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

   5. Simplified 64.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      15. neg-sub0 N/A

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

      16. --lowering--.f64 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

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

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

   9. Applied egg-rr 82.2%

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

   11. Applied egg-rr 82.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{\frac{\ell}{k}}\right)}^{-1} \cdot {\left(\frac{\frac{k}{\frac{\ell}{k \cdot t}}}{2}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{k\_m \cdot k\_m}{\ell}\right) \cdot \frac{t}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6e-66)
   (/ 2.0 (/ (* (* (* k_m k_m) (/ (* k_m k_m) l)) (/ t l)) (cos k_m)))
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6e-66) {
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m));
	} else {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * 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 (t <= 6d-66) then
        tmp = 2.0d0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m))
    else
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * 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 (t <= 6e-66) {
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / Math.cos(k_m));
	} else {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 6e-66:
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / math.cos(k_m))
	else:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6e-66)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * Float64(Float64(k_m * k_m) / l)) * Float64(t / l)) / cos(k_m)));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 6e-66)
		tmp = 2.0 / ((((k_m * k_m) * ((k_m * k_m) / l)) * (t / l)) / cos(k_m));
	else
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6e-66], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000004e-66

   1. Initial program 37.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. cos-lowering-cos.f64 80.7%

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

   5. Simplified 80.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

      14. /-lowering-/.f64 70.5%

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

   7. Applied egg-rr 70.5%

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

   8. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 69.5%

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

   10. Simplified 69.5%

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

   if 6.0000000000000004e-66 < t

   1. Initial program 31.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 64.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 66.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

   9. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       6. sub0-neg N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot -2}{\color{blue}{k} \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)} \]

       9. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot -2}{k}}{\color{blue}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       10. associate-/r* N/A

           \[\leadsto \frac{\frac{\frac{\ell \cdot -2}{k}}{k}}{\color{blue}{0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)} \]

       13. remove-double-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)\right), \color{blue}{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right) \]

   11. Applied egg-rr 82.7%

       \[\leadsto \color{blue}{\frac{-\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{\frac{\ell}{k \cdot t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 17.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 21000000000000:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \left(\ell \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell \cdot \ell}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 21000000000000.0)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/ 2.0 (/ (* (* k_m k_m) (* l (/ (* t (* k_m k_m)) (* l l)))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 21000000000000.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (l * ((t * (k_m * k_m)) / (l * l)))) / 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 <= 21000000000000.0d0) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = 2.0d0 / (((k_m * k_m) * (l * ((t * (k_m * k_m)) / (l * l)))) / 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 <= 21000000000000.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (l * ((t * (k_m * k_m)) / (l * l)))) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 21000000000000.0:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = 2.0 / (((k_m * k_m) * (l * ((t * (k_m * k_m)) / (l * l)))) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 21000000000000.0)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(l * Float64(Float64(t * Float64(k_m * k_m)) / Float64(l * l)))) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 21000000000000.0)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = 2.0 / (((k_m * k_m) * (l * ((t * (k_m * k_m)) / (l * l)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 21000000000000.0], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(l * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e13

   1. Initial program 38.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 60.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-*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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 57.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 57.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

   9. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       6. sub0-neg N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot -2}{\color{blue}{k} \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)} \]

       9. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot -2}{k}}{\color{blue}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       10. associate-/r* N/A

           \[\leadsto \frac{\frac{\frac{\ell \cdot -2}{k}}{k}}{\color{blue}{0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)} \]

       13. remove-double-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)\right), \color{blue}{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right) \]

   11. Applied egg-rr 82.5%

       \[\leadsto \color{blue}{\frac{-\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{\frac{\ell}{k \cdot t}}}} \]

   if 2.1e13 < k

   1. Initial program 25.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 44.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 44.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-*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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 50.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\mathsf{neg}\left(\ell\right)}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{0 - \ell}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      3. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\frac{0 \cdot 0 - \ell \cdot \ell}{0 + \ell}}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\frac{0 - \ell \cdot \ell}{0 + \ell}}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\frac{\mathsf{neg}\left(\ell \cdot \ell\right)}{0 + \ell}}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      6. +-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\frac{\mathsf{neg}\left(\ell \cdot \ell\right)}{\ell}}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\mathsf{neg}\left(\ell \cdot \ell\right)} \cdot \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)}{\mathsf{neg}\left(\ell \cdot \ell\right)}\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(k \cdot k\right) \cdot t\right)\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(k \cdot k\right) \cdot t\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(k \cdot k\right) \cdot t\right)\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot \left(k \cdot k\right)\right)\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\mathsf{neg}\left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \left(0 - \ell \cdot \ell\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{\_.f64}\left(0, \left(\ell \cdot \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      17. *-lowering-*.f64 49.5%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   9. Applied egg-rr 49.5%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{0 - t \cdot \left(k \cdot k\right)}{0 - \ell \cdot \ell} \cdot \ell}\right)}{0 - \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 21000000000000:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\ell \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1950:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot \frac{k\_m}{\frac{\ell}{t}}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1950.0)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/ 2.0 (/ (* (* k_m k_m) (* k_m (/ k_m (/ l t)))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1950.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (k_m * (k_m / (l / t)))) / l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1950.0d0) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = 2.0d0 / (((k_m * k_m) * (k_m * (k_m / (l / t)))) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1950.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (k_m * (k_m / (l / t)))) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1950.0:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = 2.0 / (((k_m * k_m) * (k_m * (k_m / (l / t)))) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1950.0)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(k_m * Float64(k_m / Float64(l / t)))) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1950.0)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = 2.0 / (((k_m * k_m) * (k_m * (k_m / (l / t)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1950.0], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1950

   1. Initial program 39.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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 60.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 60.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. 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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 58.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 58.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

   9. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       6. sub0-neg N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot -2}{\color{blue}{k} \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)} \]

       9. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot -2}{k}}{\color{blue}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       10. associate-/r* N/A

           \[\leadsto \frac{\frac{\frac{\ell \cdot -2}{k}}{k}}{\color{blue}{0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)} \]

       13. remove-double-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)\right), \color{blue}{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right) \]

   11. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\frac{-\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{\frac{\ell}{k \cdot t}}}} \]

   if 1950 < k

   1. Initial program 24.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 43.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 43.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 48.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 48.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot \left(k \cdot t\right)}{\ell}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(k \cdot \frac{k \cdot t}{\ell}\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{k \cdot t}{\ell} \cdot k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\frac{k \cdot t}{\ell}\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(k \cdot \frac{t}{\ell}\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(k \cdot \frac{1}{\frac{\ell}{t}}\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\frac{k}{\frac{\ell}{t}}\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

      9. /-lowering-/.f64 50.3%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right), k\right)\right)\right), \mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

   9. Applied egg-rr 50.3%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{k}{\frac{\ell}{t}} \cdot k}\right)}{0 - \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1950:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot \frac{k}{\frac{\ell}{t}}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4000000000:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -2}{k\_m \cdot \left(\frac{k\_m \cdot k\_m}{\frac{\ell}{t}} \cdot \left(0 - k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4000000000.0)
   (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t)))))
   (/ (* l -2.0) (* k_m (* (/ (* k_m k_m) (/ l t)) (- 0.0 k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4000000000.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = (l * -2.0) / (k_m * (((k_m * k_m) / (l / t)) * (0.0 - 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 <= 4000000000.0d0) then
        tmp = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
    else
        tmp = (l * (-2.0d0)) / (k_m * (((k_m * k_m) / (l / t)) * (0.0d0 - 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 <= 4000000000.0) {
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	} else {
		tmp = (l * -2.0) / (k_m * (((k_m * k_m) / (l / t)) * (0.0 - k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4000000000.0:
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))
	else:
		tmp = (l * -2.0) / (k_m * (((k_m * k_m) / (l / t)) * (0.0 - k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4000000000.0)
		tmp = Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))));
	else
		tmp = Float64(Float64(l * -2.0) / Float64(k_m * Float64(Float64(Float64(k_m * k_m) / Float64(l / t)) * Float64(0.0 - 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 <= 4000000000.0)
		tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
	else
		tmp = (l * -2.0) / (k_m * (((k_m * k_m) / (l / t)) * (0.0 - 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, 4000000000.0], N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * -2.0), $MachinePrecision] / N[(k$95$m * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(0.0 - k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4e9

   1. Initial program 39.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 58.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

      3. distribute-frac-neg N/A

         \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

      4. associate-/r/ N/A

         \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

   9. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)} \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       6. sub0-neg N/A

          \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot -2}{\color{blue}{k} \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)} \]

       9. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot -2}{k}}{\color{blue}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

       10. associate-/r* N/A

           \[\leadsto \frac{\frac{\frac{\ell \cdot -2}{k}}{k}}{\color{blue}{0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

       11. frac-2neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

       12. sub0-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)} \]

       13. remove-double-neg N/A

           \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)\right), \color{blue}{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right) \]

   11. Applied egg-rr 82.9%

       \[\leadsto \color{blue}{\frac{-\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{\frac{\ell}{k \cdot t}}}} \]

   if 4e9 < k

   1. Initial program 24.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 43.6%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 43.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

      16. --lowering--.f64 49.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

   7. Applied egg-rr 49.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)} \cdot \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2 \cdot \left(\mathsf{neg}\left(\ell\right)\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \ell\right)\right), \color{blue}{\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\ell \cdot 2\right)\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\mathsf{neg}\left(2\right)\right)\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(\left(k \cdot \color{blue}{k}\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \left(k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}\right)\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \ell\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 49.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \ell\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot -2}{k \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t}{\ell}}\right)\right)\right)\right)\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \left(\left(k \cdot k\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right)\right)\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \left(\frac{k \cdot k}{\color{blue}{\frac{\ell}{t}}}\right)\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{\ell}{t}\right)}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{\ell}}{t}\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 50.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, -2\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 50.9%

       \[\leadsto \frac{\ell \cdot -2}{k \cdot \left(k \cdot \left(0 - \color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4000000000:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -2}{k \cdot \left(\frac{k \cdot k}{\frac{\ell}{t}} \cdot \left(0 - k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 24.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\frac{-2}{\frac{k\_m}{\ell}}}{k\_m}}{\frac{k\_m}{0 - \frac{\ell}{k\_m \cdot t}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (/ (/ -2.0 (/ k_m l)) k_m) (/ k_m (- 0.0 (/ l (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((-2.0d0) / (k_m / l)) / k_m) / (k_m / (0.0d0 - (l / (k_m * t))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(-2.0 / Float64(k_m / l)) / k_m) / Float64(k_m / Float64(0.0 - Float64(l / Float64(k_m * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((-2.0 / (k_m / l)) / k_m) / (k_m / (0.0 - (l / (k_m * t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(-2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m / N[(0.0 - N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 56.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 56.4%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
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-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

   16. --lowering--.f64 56.0%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 56.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

   3. distribute-frac-neg N/A

      \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

9. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

    2. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \ell\right)}{\color{blue}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

    3. distribute-lft-neg-in N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

    4. metadata-eval N/A

       \[\leadsto \frac{-2 \cdot \ell}{\mathsf{neg}\left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)} \]

    5. distribute-rgt-neg-in N/A

       \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

    6. sub0-neg N/A

       \[\leadsto \frac{-2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

    7. associate-*r* N/A

       \[\leadsto \frac{-2 \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\ell \cdot -2}{\color{blue}{k} \cdot \left(k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)} \]

    9. associate-/r* N/A

       \[\leadsto \frac{\frac{\ell \cdot -2}{k}}{\color{blue}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

    10. associate-/r* N/A

        \[\leadsto \frac{\frac{\frac{\ell \cdot -2}{k}}{k}}{\color{blue}{0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

    11. frac-2neg N/A

        \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\color{blue}{\mathsf{neg}\left(\left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

    12. sub0-neg N/A

        \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)} \]

    13. remove-double-neg N/A

        \[\leadsto \frac{\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)}{\frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell}}} \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{\ell \cdot -2}{k}}{k}\right)\right), \color{blue}{\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right) \]

11. Applied egg-rr 74.1%

    \[\leadsto \color{blue}{\frac{-\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{\frac{\ell}{k \cdot t}}}} \]

12. Final simplification 74.1%

    \[\leadsto \frac{\frac{\frac{-2}{\frac{k}{\ell}}}{k}}{\frac{k}{0 - \frac{\ell}{k \cdot t}}} \]
13. Add Preprocessing

Alternative 13: 73.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m} \cdot \frac{2}{\frac{k\_m \cdot t}{\frac{\ell}{k\_m \cdot k\_m}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l k_m) (/ 2.0 (/ (* k_m t) (/ l (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / k_m) * (2.0 / ((k_m * t) / (l / (k_m * k_m))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l / k_m) * (2.0d0 / ((k_m * t) / (l / (k_m * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / k_m) * (2.0 / ((k_m * t) / (l / (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / k_m) * (2.0 / ((k_m * t) / (l / (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / k_m) * Float64(2.0 / Float64(Float64(k_m * t) / Float64(l / Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / k_m) * (2.0 / ((k_m * t) / (l / (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / k$95$m), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] / N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 56.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 56.4%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
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-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

   16. --lowering--.f64 56.0%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 56.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

   3. distribute-frac-neg N/A

      \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

9. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\ell \cdot 2}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

    3. associate-*l* N/A

       \[\leadsto \frac{\ell \cdot 2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

    4. times-frac N/A

       \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{2}{k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

    5. frac-2neg N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{-2}{\mathsf{neg}\left(\color{blue}{k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

    7. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{-2}{k \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}} \]

    8. sub0-neg N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{-2}{k \cdot \left(0 - \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\right)} \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\frac{-2}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right)}\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\color{blue}{-2}}{k \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{k} \cdot \left(0 - \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right) \]

    12. sub0-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\mathsf{neg}\left(2\right)}{k \cdot \left(\mathsf{neg}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)}\right)\right) \]

    13. distribute-rgt-neg-out N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{\mathsf{neg}\left(2\right)}{\mathsf{neg}\left(k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right) \]

    14. frac-2neg N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\frac{2}{\color{blue}{k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}}\right)\right) \]

    15. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \color{blue}{\left(k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)}\right)\right) \]

    16. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(k \cdot \left(t \cdot \color{blue}{\frac{k \cdot k}{\ell}}\right)\right)\right)\right) \]

    17. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(\left(k \cdot t\right) \cdot \color{blue}{\frac{k \cdot k}{\ell}}\right)\right)\right) \]

    18. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{/.f64}\left(2, \left(\left(k \cdot t\right) \cdot \frac{1}{\color{blue}{\frac{\ell}{k \cdot k}}}\right)\right)\right) \]

11. Applied egg-rr 73.6%

    \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{2}{\frac{k \cdot t}{\frac{\ell}{k \cdot k}}}} \]
12. Add Preprocessing

Alternative 14: 72.4% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot \left(k\_m \cdot \frac{k\_m}{\ell}\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ 2.0 (* (* k_m k_m) (* t (* k_m (/ k_m l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (2.0 / ((k_m * k_m) * (t * (k_m * (k_m / l)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (2.0d0 / ((k_m * k_m) * (t * (k_m * (k_m / l)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (2.0 / ((k_m * k_m) * (t * (k_m * (k_m / l)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (2.0 / ((k_m * k_m) * (t * (k_m * (k_m / l)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(t * Float64(k_m * Float64(k_m / l))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (2.0 / ((k_m * k_m) * (t * (k_m * (k_m / l)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 56.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 56.4%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
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-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

   16. --lowering--.f64 56.0%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 56.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

   3. distribute-frac-neg N/A

      \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

9. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\left(t \cdot k\right) \cdot k}{\ell}\right)\right)\right), \ell\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\left(k \cdot t\right) \cdot k}{\ell}\right)\right)\right), \ell\right) \]

    3. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{k \cdot t}{\ell} \cdot k\right)\right)\right), \ell\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{t \cdot k}{\ell} \cdot k\right)\right)\right), \ell\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\left(t \cdot \frac{k}{\ell}\right) \cdot k\right)\right)\right), \ell\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)\right)\right), \ell\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \left(\frac{k}{\ell} \cdot k\right)\right)\right)\right), \ell\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{k}{\ell}\right), k\right)\right)\right)\right), \ell\right) \]

    9. /-lowering-/.f64 73.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), k\right)\right)\right)\right), \ell\right) \]

11. Applied egg-rr 73.6%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)}} \cdot \ell \]

12. Final simplification 73.6%

    \[\leadsto \ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \]
13. Add Preprocessing

Alternative 15: 70.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{2}{t \cdot \left(\frac{k\_m}{\frac{\ell}{k\_m}} \cdot \left(k\_m \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ 2.0 (* t (* (/ k_m (/ l k_m)) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (2.0 / (t * ((k_m / (l / k_m)) * (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 * (2.0d0 / (t * ((k_m / (l / k_m)) * (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 * (2.0 / (t * ((k_m / (l / k_m)) * (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (2.0 / (t * ((k_m / (l / k_m)) * (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(2.0 / Float64(t * Float64(Float64(k_m / Float64(l / k_m)) * Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (2.0 / (t * ((k_m / (l / k_m)) * (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(2.0 / N[(t * N[(N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 56.4%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 56.4%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
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-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\frac{\left(k \cdot k\right) \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right) \]

   5. frac-2neg N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \frac{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\mathsf{neg}\left(\ell\right)}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\color{blue}{\ell}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \left(\frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(\mathsf{neg}\left(\ell\right)\right)\right)\right) \]

   15. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \left(0 - \color{blue}{\ell}\right)\right)\right) \]

   16. --lowering--.f64 56.0%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\ell}\right)\right)\right) \]

7. Applied egg-rr 56.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{0 - \ell}}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\mathsf{neg}\left(\ell\right)}} \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\frac{\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)}{\ell}\right)} \]

   3. distribute-frac-neg N/A

      \[\leadsto \frac{2}{\frac{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}{\color{blue}{\ell}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)} \cdot \color{blue}{\ell} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\mathsf{neg}\left(\left(k \cdot k\right) \cdot \left(0 - \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right), \color{blue}{\ell}\right) \]

9. Applied egg-rr 72.1%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \ell} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot k\right)\right)\right), \ell\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\frac{k \cdot k}{\ell} \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{k \cdot k}{\ell}\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot \frac{k}{\ell}\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    7. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot \frac{1}{\frac{\ell}{k}}\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    8. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{k}{\frac{\ell}{k}}\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\ell}{k}\right)\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), \left(k \cdot k\right)\right)\right)\right), \ell\right) \]

    11. *-lowering-*.f64 71.4%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right) \]

11. Applied egg-rr 71.4%

    \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

12. Final simplification 71.4%

    \[\leadsto \ell \cdot \frac{2}{t \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \left(k \cdot k\right)\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))