Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 95.4%

Time: 23.5s

Alternatives: 11

Speedup: 2.0×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.6% 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, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\cos k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.35 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k\_m}}{\sqrt{t\_2}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_2} \cdot {\sin k\_m}^{-2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ t_m (cos k_m))))
   (*
    t_s
    (if (<= k_m 1.35e-9)
      (* 2.0 (pow (/ (/ (/ l k_m) (sqrt t_2)) k_m) 2.0))
      (* 2.0 (* (/ (pow (/ l k_m) 2.0) t_2) (pow (sin k_m) -2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = t_m / cos(k_m);
	double tmp;
	if (k_m <= 1.35e-9) {
		tmp = 2.0 * pow((((l / k_m) / sqrt(t_2)) / k_m), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_2) * pow(sin(k_m), -2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m / cos(k_m)
    if (k_m <= 1.35d-9) then
        tmp = 2.0d0 * ((((l / k_m) / sqrt(t_2)) / k_m) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_2) * (sin(k_m) ** (-2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = t_m / Math.cos(k_m);
	double tmp;
	if (k_m <= 1.35e-9) {
		tmp = 2.0 * Math.pow((((l / k_m) / Math.sqrt(t_2)) / k_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_2) * Math.pow(Math.sin(k_m), -2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = t_m / math.cos(k_m)
	tmp = 0
	if k_m <= 1.35e-9:
		tmp = 2.0 * math.pow((((l / k_m) / math.sqrt(t_2)) / k_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_2) * math.pow(math.sin(k_m), -2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(t_m / cos(k_m))
	tmp = 0.0
	if (k_m <= 1.35e-9)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) / sqrt(t_2)) / k_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_2) * (sin(k_m) ^ -2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = t_m / cos(k_m);
	tmp = 0.0;
	if (k_m <= 1.35e-9)
		tmp = 2.0 * ((((l / k_m) / sqrt(t_2)) / k_m) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_2) * (sin(k_m) ^ -2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.35e-9], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3500000000000001e-9

   1. Initial program 30.6%

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

      1. associate-*l* 30.6%

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

      2. associate-/r* 30.5%

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

      3. sub-neg 30.5%

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

      4. distribute-rgt-in 23.2%

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

      5. unpow2 23.2%

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

      6. times-frac 14.9%

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

      7. sqr-neg 14.9%

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

      8. times-frac 23.2%

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

      9. unpow2 23.2%

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

      10. distribute-rgt-in 30.5%

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

      11. +-commutative 30.5%

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

      12. associate-+l+ 40.2%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 68.0%

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

      1. times-frac 67.9%

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

   7. Simplified 67.9%

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

      1. expm1-log1p-u 38.2%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 34.5%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 34.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 34.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 34.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 34.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 34.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 38.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 67.9%

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

       3. associate-/r* 67.9%

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

       4. associate-*r/ 67.9%

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

       5. associate-*l* 67.9%

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

       6. *-commutative 67.9%

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

       7. times-frac 69.0%

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

   11. Simplified 69.0%

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

   12. Taylor expanded in k around 0 63.3%

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

       1. expm1-log1p-u 36.4%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)\right)} \]

       2. expm1-udef 33.7%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)} - 1\right)} \]

   14. Applied egg-rr 28.0%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 32.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 32.7%

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

   16. Simplified 32.7%

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

   if 1.3500000000000001e-9 < k

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. associate-/r* 25.3%

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

      3. sub-neg 25.3%

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

      4. distribute-rgt-in 25.3%

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

      5. unpow2 25.3%

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

      6. times-frac 22.6%

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

      7. sqr-neg 22.6%

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

      8. times-frac 25.3%

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

      9. unpow2 25.3%

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

      10. distribute-rgt-in 25.3%

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

      11. +-commutative 25.3%

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

      12. associate-+l+ 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. times-frac 74.1%

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

   7. Simplified 74.1%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 52.0%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 52.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 52.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 52.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 52.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 52.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 63.1%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 74.1%

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

       3. associate-/r* 74.0%

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

       4. associate-*r/ 74.1%

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

       5. associate-*l* 74.0%

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

       6. *-commutative 74.0%

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

       7. times-frac 72.7%

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

   11. Simplified 72.7%

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

       1. associate-*l/ 72.7%

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

       2. associate-/l* 72.7%

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

   13. Applied egg-rr 72.7%

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

       1. div-inv 72.8%

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

       2. associate-*r/ 74.1%

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

       3. metadata-eval 74.1%

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

       4. pow-flip 74.1%

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

       5. div-inv 74.1%

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

       6. add-sqr-sqrt 74.0%

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

       7. pow2 74.0%

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

       8. sqrt-div 74.0%

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

       9. pow2 74.0%

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

       10. sqrt-prod 38.4%

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

       11. add-sqr-sqrt 78.2%

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

       12. unpow2 78.2%

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

       13. sqrt-prod 93.2%

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

       14. add-sqr-sqrt 93.3%

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

       15. pow-flip 93.3%

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

       16. metadata-eval 93.3%

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

   15. Applied egg-rr 93.3%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{t}{\cos k}} \cdot {\sin k}^{-2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-53}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k\_m}^{-2}\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m}}{\sqrt{\frac{t\_m}{\cos k\_m}} \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-53)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (* 2.0 (pow (/ (/ l k_m) (* (sqrt (/ t_m (cos k_m))) (sin k_m))) 2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-53) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * pow(((l / k_m) / (sqrt((t_m / cos(k_m))) * sin(k_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d-53) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 * (((l / k_m) / (sqrt((t_m / cos(k_m))) * sin(k_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-53) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * Math.pow(((l / k_m) / (Math.sqrt((t_m / Math.cos(k_m))) * Math.sin(k_m))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e-53:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 * math.pow(((l / k_m) / (math.sqrt((t_m / math.cos(k_m))) * math.sin(k_m))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e-53)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) / Float64(sqrt(Float64(t_m / cos(k_m))) * sin(k_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e-53)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 * (((l / k_m) / (sqrt((t_m / cos(k_m))) * sin(k_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-53], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[(N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5e-53

   1. Initial program 25.5%

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

      1. associate-*l* 25.5%

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

      2. associate-/r* 25.5%

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

      3. sub-neg 25.5%

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

      4. distribute-rgt-in 23.0%

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

      5. unpow2 23.0%

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

      6. times-frac 17.5%

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

      7. sqr-neg 17.5%

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

      8. times-frac 23.0%

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

      9. unpow2 23.0%

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

      10. distribute-rgt-in 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 57.4%

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

   7. Simplified 57.4%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   8. Step-by-step derivation

      1. div-inv 56.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

      2. pow-flip 56.6%

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

      3. metadata-eval 56.6%

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

   9. Applied egg-rr 56.6%

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

       1. associate-*l/ 59.0%

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

   11. Simplified 59.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 59.0%

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

       2. pow2 59.0%

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

       3. sqrt-prod 59.0%

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

       4. pow2 59.0%

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

       5. sqrt-prod 38.7%

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

       6. add-sqr-sqrt 74.7%

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

       7. sqrt-pow1 84.5%

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

       8. metadata-eval 84.5%

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

   13. Applied egg-rr 84.5%

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

   if 5e-53 < (*.f64 l l)

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-/r* 32.4%

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

      3. sub-neg 32.4%

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

      4. distribute-rgt-in 24.6%

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

      5. unpow2 24.6%

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

      6. times-frac 16.9%

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

      7. sqr-neg 16.9%

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

      8. times-frac 24.6%

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

      9. unpow2 24.6%

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

      10. distribute-rgt-in 32.4%

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

      11. +-commutative 32.4%

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

      12. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. times-frac 74.2%

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

   7. Simplified 74.2%

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

      1. expm1-log1p-u 35.3%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 27.4%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 27.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 35.3%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 74.2%

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

       3. associate-/r* 74.2%

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

       4. associate-*r/ 74.2%

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

       5. associate-*l* 74.2%

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

       6. *-commutative 74.2%

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

       7. times-frac 72.0%

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

   11. Simplified 72.0%

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

       1. associate-*l/ 73.4%

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

       2. associate-/l* 73.4%

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

   13. Applied egg-rr 73.4%

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

       1. expm1-log1p-u 34.4%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \frac{{k}^{-2}}{\frac{t}{\cos k}}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-udef 28.6%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \frac{{k}^{-2}}{\frac{t}{\cos k}}}{{\sin k}^{2}}\right)} - 1\right)} \]

   15. Applied egg-rr 32.3%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{\sin k}\right)}^{2}\right)} - 1\right)} \]
   16. Step-by-step derivation

       1. expm1-def 39.6%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{\sin k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 40.4%

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

       3. associate-/l/ 40.5%

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

   17. Simplified 40.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-53}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}} \cdot \sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-53}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k\_m}^{-2}\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k\_m}}{\sqrt{\frac{t\_m}{\cos k\_m}}}}{\sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-53)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (* 2.0 (pow (/ (/ (/ l k_m) (sqrt (/ t_m (cos k_m)))) (sin k_m)) 2.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-53) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * pow((((l / k_m) / sqrt((t_m / cos(k_m)))) / sin(k_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d-53) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / sin(k_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-53) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * Math.pow((((l / k_m) / Math.sqrt((t_m / Math.cos(k_m)))) / Math.sin(k_m)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e-53:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 * math.pow((((l / k_m) / math.sqrt((t_m / math.cos(k_m)))) / math.sin(k_m)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e-53)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) / sqrt(Float64(t_m / cos(k_m)))) / sin(k_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e-53)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / sin(k_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-53], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5e-53

   1. Initial program 25.5%

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

      1. associate-*l* 25.5%

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

      2. associate-/r* 25.5%

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

      3. sub-neg 25.5%

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

      4. distribute-rgt-in 23.0%

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

      5. unpow2 23.0%

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

      6. times-frac 17.5%

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

      7. sqr-neg 17.5%

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

      8. times-frac 23.0%

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

      9. unpow2 23.0%

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

      10. distribute-rgt-in 25.5%

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

      11. +-commutative 25.5%

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

      12. associate-+l+ 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 57.4%

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

   7. Simplified 57.4%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   8. Step-by-step derivation

      1. div-inv 56.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

      2. pow-flip 56.6%

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

      3. metadata-eval 56.6%

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

   9. Applied egg-rr 56.6%

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

       1. associate-*l/ 59.0%

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

   11. Simplified 59.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 59.0%

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

       2. pow2 59.0%

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

       3. sqrt-prod 59.0%

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

       4. pow2 59.0%

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

       5. sqrt-prod 38.7%

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

       6. add-sqr-sqrt 74.7%

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

       7. sqrt-pow1 84.5%

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

       8. metadata-eval 84.5%

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

   13. Applied egg-rr 84.5%

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

   if 5e-53 < (*.f64 l l)

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-/r* 32.4%

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

      3. sub-neg 32.4%

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

      4. distribute-rgt-in 24.6%

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

      5. unpow2 24.6%

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

      6. times-frac 16.9%

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

      7. sqr-neg 16.9%

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

      8. times-frac 24.6%

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

      9. unpow2 24.6%

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

      10. distribute-rgt-in 32.4%

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

      11. +-commutative 32.4%

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

      12. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. times-frac 74.2%

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

   7. Simplified 74.2%

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

      1. expm1-log1p-u 35.3%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 27.4%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 27.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 27.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 35.3%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 74.2%

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

       3. associate-/r* 74.2%

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

       4. associate-*r/ 74.2%

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

       5. associate-*l* 74.2%

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

       6. *-commutative 74.2%

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

       7. times-frac 72.0%

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

   11. Simplified 72.0%

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

       1. associate-*l/ 73.4%

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

       2. associate-/l* 73.4%

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

   13. Applied egg-rr 73.4%

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

       1. add-sqr-sqrt 31.1%

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

       2. pow2 31.1%

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

   15. Applied egg-rr 40.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-53}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{\sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k\_m}}{\sqrt{\frac{t\_m}{\cos k\_m}}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.15e-7)
    (* 2.0 (pow (/ (/ (/ l k_m) (sqrt (/ t_m (cos k_m)))) k_m) 2.0))
    (*
     2.0
     (* (/ (pow (/ l k_m) 2.0) t_m) (/ (cos k_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-7) {
		tmp = 2.0 * pow((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * (cos(k_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.15d-7) then
        tmp = 2.0d0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * (cos(k_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-7) {
		tmp = 2.0 * Math.pow((((l / k_m) / Math.sqrt((t_m / Math.cos(k_m)))) / k_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * (Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.15e-7:
		tmp = 2.0 * math.pow((((l / k_m) / math.sqrt((t_m / math.cos(k_m)))) / k_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * (math.cos(k_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.15e-7)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) / sqrt(Float64(t_m / cos(k_m)))) / k_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(cos(k_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.15e-7)
		tmp = 2.0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * (cos(k_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.15e-7], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1500000000000001e-7

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. associate-/r* 30.4%

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

      3. sub-neg 30.4%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 14.8%

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

      7. sqr-neg 14.8%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 30.4%

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

      11. +-commutative 30.4%

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

      12. associate-+l+ 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 68.1%

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

      1. times-frac 68.1%

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

   7. Simplified 68.1%

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

      1. expm1-log1p-u 38.5%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 34.9%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 34.9%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 34.9%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 34.9%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 34.9%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 34.9%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 38.5%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 68.1%

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

       3. associate-/r* 68.1%

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

       4. associate-*r/ 68.1%

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

       5. associate-*l* 68.1%

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

       6. *-commutative 68.1%

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

       7. times-frac 69.1%

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

   11. Simplified 69.1%

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

   12. Taylor expanded in k around 0 63.5%

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

       1. expm1-log1p-u 36.7%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)\right)} \]

       2. expm1-udef 34.0%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)} - 1\right)} \]

   14. Applied egg-rr 28.3%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 32.5%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 33.1%

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

   16. Simplified 33.1%

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

   if 2.1500000000000001e-7 < k

   1. Initial program 25.6%

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

      1. associate-*l* 25.6%

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

      2. associate-/r* 25.6%

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

      3. sub-neg 25.6%

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

      4. distribute-rgt-in 25.6%

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

      5. unpow2 25.6%

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

      6. times-frac 22.9%

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

      7. sqr-neg 22.9%

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

      8. times-frac 25.6%

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

      9. unpow2 25.6%

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

      10. distribute-rgt-in 25.6%

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

      11. +-commutative 25.6%

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

      12. associate-+l+ 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in t around 0 72.5%

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

      1. times-frac 73.7%

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

   7. Simplified 73.7%

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

      1. expm1-log1p-u 62.7%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 51.5%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 51.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 51.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 51.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 51.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 51.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 62.7%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 73.7%

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

       3. associate-/r* 73.7%

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

       4. associate-*r/ 73.7%

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

       5. associate-*l* 73.7%

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

       6. *-commutative 73.7%

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

       7. times-frac 72.4%

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

   11. Simplified 72.4%

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

       1. associate-*l/ 72.4%

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

       2. associate-/l* 72.4%

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

   13. Applied egg-rr 72.4%

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

   14. Taylor expanded in l around 0 72.5%

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

       1. associate-*r* 72.4%

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

       2. times-frac 72.4%

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

       3. associate-/r* 73.8%

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

       4. unpow2 73.8%

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

       5. unpow2 73.8%

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

       6. times-frac 93.2%

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

       7. unpow2 93.2%

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

   16. Simplified 93.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.92 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k\_m}^{-2}\right)}^{2}}{t\_m}\\ \mathbf{elif}\;k\_m \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k\_m}{{k\_m}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.92e-25)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (if (<= k_m 7.4e+77)
      (* 2.0 (* (/ (pow l 2.0) t_m) (/ (cos k_m) (pow k_m 4.0))))
      (* 2.0 (/ (/ (pow (/ l k_m) 2.0) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.92e-25) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else if (k_m <= 7.4e+77) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * (cos(k_m) / pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) / pow(sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.92d-25) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else if (k_m <= 7.4d+77) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (cos(k_m) / (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) / (sin(k_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.92e-25) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else if (k_m <= 7.4e+77) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k_m) / Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) / Math.pow(Math.sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.92e-25:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	elif k_m <= 7.4e+77:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * (math.cos(k_m) / math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) / math.pow(math.sin(k_m), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.92e-25)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	elseif (k_m <= 7.4e+77)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k_m) / (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.92e-25)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	elseif (k_m <= 7.4e+77)
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (cos(k_m) / (k_m ^ 4.0)));
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.92e-25], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.4e+77], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9200000000000001e-25

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.2%

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

      3. sub-neg 31.2%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 15.2%

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

      7. sqr-neg 15.2%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 31.2%

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

      11. +-commutative 31.2%

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

      12. associate-+l+ 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in k around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-/r* 56.0%

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

   7. Simplified 56.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   8. Step-by-step derivation

      1. div-inv 55.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

      2. pow-flip 55.4%

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

      3. metadata-eval 55.4%

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

   9. Applied egg-rr 55.4%

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

       1. associate-*l/ 57.1%

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

   11. Simplified 57.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.1%

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

       2. pow2 57.1%

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

       3. sqrt-prod 57.1%

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

       4. pow2 57.1%

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

       5. sqrt-prod 33.1%

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

       6. add-sqr-sqrt 68.6%

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

       7. sqrt-pow1 76.8%

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

       8. metadata-eval 76.8%

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

   13. Applied egg-rr 76.8%

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

   if 1.9200000000000001e-25 < k < 7.3999999999999999e77

   1. Initial program 20.7%

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

      1. associate-*l* 20.8%

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

      2. associate-/r* 20.7%

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

      3. sub-neg 20.7%

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

      4. distribute-rgt-in 20.7%

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

      5. unpow2 20.7%

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

      6. times-frac 20.7%

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

      7. sqr-neg 20.7%

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

      8. times-frac 20.7%

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

      9. unpow2 20.7%

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

      10. distribute-rgt-in 20.7%

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

      11. +-commutative 20.7%

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

      12. associate-+l+ 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. times-frac 86.7%

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

   7. Simplified 86.7%

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

      1. expm1-log1p-u 65.2%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 48.8%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 48.8%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 48.8%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 48.8%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 48.8%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 48.8%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 65.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 86.4%

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

       3. associate-/r* 86.4%

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

       4. associate-*r/ 86.4%

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

       5. associate-*l* 86.4%

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

       6. *-commutative 86.4%

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

       7. times-frac 92.9%

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

   11. Simplified 92.9%

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

   12. Taylor expanded in k around 0 67.3%

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

   13. Taylor expanded in l around 0 64.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
   14. Step-by-step derivation

       1. *-commutative 64.0%

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

       2. times-frac 70.8%

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

   15. Simplified 70.8%

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

   if 7.3999999999999999e77 < k

   1. Initial program 25.9%

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

      1. associate-*l* 25.9%

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

      2. associate-/r* 25.9%

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

      3. sub-neg 25.9%

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

      4. distribute-rgt-in 25.9%

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

      5. unpow2 25.9%

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

      6. times-frac 21.9%

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

      7. sqr-neg 21.9%

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

      8. times-frac 25.9%

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

      9. unpow2 25.9%

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

      10. distribute-rgt-in 25.9%

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

      11. +-commutative 25.9%

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

      12. associate-+l+ 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 59.6%

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

      1. times-frac 65.2%

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

   7. Simplified 65.2%

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

      1. expm1-log1p-u 59.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 50.0%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 50.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 59.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 65.3%

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

       3. associate-/r* 65.2%

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

       4. associate-*r/ 65.3%

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

       5. associate-*l* 65.3%

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

       6. *-commutative 65.3%

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

       7. times-frac 59.6%

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

   11. Simplified 59.6%

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

       1. associate-*l/ 59.6%

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

       2. associate-/l* 59.6%

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

   13. Applied egg-rr 59.6%

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

   14. Taylor expanded in k around 0 46.9%

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

       1. associate-/r* 47.7%

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

       2. unpow2 47.7%

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

       3. unpow2 47.7%

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

       4. times-frac 51.3%

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

       5. unpow2 51.3%

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

   16. Simplified 51.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.92 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k\_m \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k\_m}{{k\_m}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-15)
    (* 2.0 (pow (/ (/ l (* k_m (sqrt (/ t_m (cos k_m))))) k_m) 2.0))
    (if (<= k_m 7.4e+77)
      (* 2.0 (* (/ (pow l 2.0) t_m) (/ (cos k_m) (pow k_m 4.0))))
      (* 2.0 (/ (/ (pow (/ l k_m) 2.0) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-15) {
		tmp = 2.0 * pow(((l / (k_m * sqrt((t_m / cos(k_m))))) / k_m), 2.0);
	} else if (k_m <= 7.4e+77) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * (cos(k_m) / pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) / pow(sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-15) then
        tmp = 2.0d0 * (((l / (k_m * sqrt((t_m / cos(k_m))))) / k_m) ** 2.0d0)
    else if (k_m <= 7.4d+77) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (cos(k_m) / (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) / (sin(k_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-15) {
		tmp = 2.0 * Math.pow(((l / (k_m * Math.sqrt((t_m / Math.cos(k_m))))) / k_m), 2.0);
	} else if (k_m <= 7.4e+77) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k_m) / Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) / Math.pow(Math.sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-15:
		tmp = 2.0 * math.pow(((l / (k_m * math.sqrt((t_m / math.cos(k_m))))) / k_m), 2.0)
	elif k_m <= 7.4e+77:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * (math.cos(k_m) / math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) / math.pow(math.sin(k_m), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-15)
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(Float64(t_m / cos(k_m))))) / k_m) ^ 2.0));
	elseif (k_m <= 7.4e+77)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k_m) / (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-15)
		tmp = 2.0 * (((l / (k_m * sqrt((t_m / cos(k_m))))) / k_m) ^ 2.0);
	elseif (k_m <= 7.4e+77)
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (cos(k_m) / (k_m ^ 4.0)));
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-15], N[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 7.4e+77], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.5e-15

   1. Initial program 30.7%

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

      1. associate-*l* 30.7%

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

      2. associate-/r* 30.7%

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

      3. sub-neg 30.7%

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

      4. distribute-rgt-in 23.4%

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

      5. unpow2 23.4%

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

      6. times-frac 15.0%

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

      7. sqr-neg 15.0%

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

      8. times-frac 23.4%

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

      9. unpow2 23.4%

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

      10. distribute-rgt-in 30.7%

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

      11. +-commutative 30.7%

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

      12. associate-+l+ 40.4%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 67.8%

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

      1. times-frac 67.7%

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

   7. Simplified 67.7%

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

      1. expm1-log1p-u 37.9%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 34.7%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 37.9%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 67.8%

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

       3. associate-/r* 67.7%

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

       4. associate-*r/ 67.8%

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

       5. associate-*l* 67.7%

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

       6. *-commutative 67.7%

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

       7. times-frac 68.8%

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

   11. Simplified 68.8%

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

   12. Taylor expanded in k around 0 63.1%

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

       1. expm1-log1p-u 36.1%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)\right)} \]

       2. expm1-udef 33.9%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)} - 1\right)} \]

   14. Applied egg-rr 28.1%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 31.8%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 32.4%

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

       3. associate-/l/ 32.3%

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

   16. Simplified 32.3%

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

   if 2.5e-15 < k < 7.3999999999999999e77

   1. Initial program 23.1%

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

      1. associate-*l* 23.2%

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

      2. associate-/r* 23.1%

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

      3. sub-neg 23.1%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 23.1%

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

      7. sqr-neg 23.1%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 23.1%

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

      11. +-commutative 23.1%

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

      12. associate-+l+ 38.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. times-frac 92.5%

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

   7. Simplified 92.5%

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

      1. expm1-log1p-u 72.4%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 54.1%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 54.1%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 72.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 92.2%

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

       3. associate-/r* 92.2%

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

       4. associate-*r/ 92.3%

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

       5. associate-*l* 92.2%

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

       6. *-commutative 92.2%

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

       7. times-frac 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in k around 0 70.9%

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

   13. Taylor expanded in l around 0 67.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
   14. Step-by-step derivation

       1. *-commutative 67.1%

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

       2. times-frac 71.1%

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

   15. Simplified 71.1%

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

   if 7.3999999999999999e77 < k

   1. Initial program 25.9%

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

      1. associate-*l* 25.9%

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

      2. associate-/r* 25.9%

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

      3. sub-neg 25.9%

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

      4. distribute-rgt-in 25.9%

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

      5. unpow2 25.9%

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

      6. times-frac 21.9%

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

      7. sqr-neg 21.9%

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

      8. times-frac 25.9%

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

      9. unpow2 25.9%

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

      10. distribute-rgt-in 25.9%

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

      11. +-commutative 25.9%

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

      12. associate-+l+ 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 59.6%

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

      1. times-frac 65.2%

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

   7. Simplified 65.2%

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

      1. expm1-log1p-u 59.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 50.0%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 50.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 59.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 65.3%

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

       3. associate-/r* 65.2%

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

       4. associate-*r/ 65.3%

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

       5. associate-*l* 65.3%

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

       6. *-commutative 65.3%

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

       7. times-frac 59.6%

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

   11. Simplified 59.6%

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

       1. associate-*l/ 59.6%

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

       2. associate-/l* 59.6%

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

   13. Applied egg-rr 59.6%

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

   14. Taylor expanded in k around 0 46.9%

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

       1. associate-/r* 47.7%

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

       2. unpow2 47.7%

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

       3. unpow2 47.7%

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

       4. times-frac 51.3%

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

       5. unpow2 51.3%

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

   16. Simplified 51.3%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k \cdot \sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k\_m}}{\sqrt{\frac{t\_m}{\cos k\_m}}}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 4.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k\_m}{{k\_m}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}}{{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.2e-15)
    (* 2.0 (pow (/ (/ (/ l k_m) (sqrt (/ t_m (cos k_m)))) k_m) 2.0))
    (if (<= k_m 4.4e+77)
      (* 2.0 (* (/ (pow l 2.0) t_m) (/ (cos k_m) (pow k_m 4.0))))
      (* 2.0 (/ (/ (pow (/ l k_m) 2.0) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-15) {
		tmp = 2.0 * pow((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m), 2.0);
	} else if (k_m <= 4.4e+77) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * (cos(k_m) / pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) / pow(sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-15) then
        tmp = 2.0d0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m) ** 2.0d0)
    else if (k_m <= 4.4d+77) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (cos(k_m) / (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) / (sin(k_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-15) {
		tmp = 2.0 * Math.pow((((l / k_m) / Math.sqrt((t_m / Math.cos(k_m)))) / k_m), 2.0);
	} else if (k_m <= 4.4e+77) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k_m) / Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) / Math.pow(Math.sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.2e-15:
		tmp = 2.0 * math.pow((((l / k_m) / math.sqrt((t_m / math.cos(k_m)))) / k_m), 2.0)
	elif k_m <= 4.4e+77:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * (math.cos(k_m) / math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) / math.pow(math.sin(k_m), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-15)
		tmp = Float64(2.0 * (Float64(Float64(Float64(l / k_m) / sqrt(Float64(t_m / cos(k_m)))) / k_m) ^ 2.0));
	elseif (k_m <= 4.4e+77)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k_m) / (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-15)
		tmp = 2.0 * ((((l / k_m) / sqrt((t_m / cos(k_m)))) / k_m) ^ 2.0);
	elseif (k_m <= 4.4e+77)
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (cos(k_m) / (k_m ^ 4.0)));
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) / (sin(k_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-15], N[(2.0 * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.4e+77], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.20000000000000009e-15

   1. Initial program 30.7%

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

      1. associate-*l* 30.7%

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

      2. associate-/r* 30.7%

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

      3. sub-neg 30.7%

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

      4. distribute-rgt-in 23.4%

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

      5. unpow2 23.4%

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

      6. times-frac 15.0%

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

      7. sqr-neg 15.0%

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

      8. times-frac 23.4%

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

      9. unpow2 23.4%

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

      10. distribute-rgt-in 30.7%

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

      11. +-commutative 30.7%

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

      12. associate-+l+ 40.4%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 67.8%

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

      1. times-frac 67.7%

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

   7. Simplified 67.7%

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

      1. expm1-log1p-u 37.9%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 34.7%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 34.7%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 34.7%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 37.9%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 67.8%

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

       3. associate-/r* 67.7%

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

       4. associate-*r/ 67.8%

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

       5. associate-*l* 67.7%

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

       6. *-commutative 67.7%

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

       7. times-frac 68.8%

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

   11. Simplified 68.8%

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

   12. Taylor expanded in k around 0 63.1%

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

       1. expm1-log1p-u 36.1%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)\right)} \]

       2. expm1-udef 33.9%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{{k}^{-2} \cdot \cos k}{t}\right)} - 1\right)} \]

   14. Applied egg-rr 28.1%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 31.8%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 32.4%

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

   16. Simplified 32.4%

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

   if 5.20000000000000009e-15 < k < 4.4000000000000001e77

   1. Initial program 23.1%

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

      1. associate-*l* 23.2%

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

      2. associate-/r* 23.1%

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

      3. sub-neg 23.1%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 23.1%

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

      7. sqr-neg 23.1%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 23.1%

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

      11. +-commutative 23.1%

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

      12. associate-+l+ 38.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around 0 99.7%

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

      1. times-frac 92.5%

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

   7. Simplified 92.5%

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

      1. expm1-log1p-u 72.4%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 54.1%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 54.1%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 54.1%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 72.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 92.2%

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

       3. associate-/r* 92.2%

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

       4. associate-*r/ 92.3%

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

       5. associate-*l* 92.2%

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

       6. *-commutative 92.2%

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

       7. times-frac 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in k around 0 70.9%

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

   13. Taylor expanded in l around 0 67.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
   14. Step-by-step derivation

       1. *-commutative 67.1%

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

       2. times-frac 71.1%

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

   15. Simplified 71.1%

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

   if 4.4000000000000001e77 < k

   1. Initial program 25.9%

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

      1. associate-*l* 25.9%

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

      2. associate-/r* 25.9%

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

      3. sub-neg 25.9%

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

      4. distribute-rgt-in 25.9%

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

      5. unpow2 25.9%

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

      6. times-frac 21.9%

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

      7. sqr-neg 21.9%

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

      8. times-frac 25.9%

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

      9. unpow2 25.9%

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

      10. distribute-rgt-in 25.9%

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

      11. +-commutative 25.9%

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

      12. associate-+l+ 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 59.6%

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

      1. times-frac 65.2%

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

   7. Simplified 65.2%

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

      1. expm1-log1p-u 59.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 50.0%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 50.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 50.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 59.2%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 65.3%

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

       3. associate-/r* 65.2%

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

       4. associate-*r/ 65.3%

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

       5. associate-*l* 65.3%

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

       6. *-commutative 65.3%

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

       7. times-frac 59.6%

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

   11. Simplified 59.6%

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

       1. associate-*l/ 59.6%

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

       2. associate-/l* 59.6%

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

   13. Applied egg-rr 59.6%

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

   14. Taylor expanded in k around 0 46.9%

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

       1. associate-/r* 47.7%

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

       2. unpow2 47.7%

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

       3. unpow2 47.7%

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

       4. times-frac 51.3%

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

       5. unpow2 51.3%

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

   16. Simplified 51.3%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\frac{\ell}{k}}{\sqrt{\frac{t}{\cos k}}}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+77}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k\_m}^{-2}\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k\_m}{{k\_m}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.7e-27)
    (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))
    (* 2.0 (* (/ (pow l 2.0) t_m) (/ (cos k_m) (pow k_m 4.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 1.7e-27) {
		tmp = 2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * (cos(k_m) / pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.7d-27) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (cos(k_m) / (k_m ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 1.7e-27) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k_m) / Math.pow(k_m, 4.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 1.7e-27:
		tmp = 2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * (math.cos(k_m) / math.pow(k_m, 4.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 1.7e-27)
		tmp = Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k_m) / (k_m ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 1.7e-27)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (cos(k_m) / (k_m ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 1.7e-27], N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.69999999999999985e-27

   1. Initial program 29.2%

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

      1. associate-*l* 29.2%

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

      2. associate-/r* 29.2%

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

      3. sub-neg 29.2%

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

      4. distribute-rgt-in 24.0%

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

      5. unpow2 24.0%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 24.0%

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

      9. unpow2 24.0%

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

      10. distribute-rgt-in 29.2%

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

      11. +-commutative 29.2%

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

      12. associate-+l+ 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in k around 0 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-/r* 56.8%

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

   7. Simplified 56.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   8. Step-by-step derivation

      1. div-inv 56.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

      2. pow-flip 56.2%

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

      3. metadata-eval 56.2%

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

   9. Applied egg-rr 56.2%

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

       1. associate-*l/ 57.9%

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

   11. Simplified 57.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.8%

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

       2. pow2 57.8%

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

       3. sqrt-prod 57.9%

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

       4. pow2 57.9%

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

       5. sqrt-prod 25.3%

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

       6. add-sqr-sqrt 67.8%

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

       7. sqrt-pow1 74.8%

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

       8. metadata-eval 74.8%

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

   13. Applied egg-rr 74.8%

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

   if 1.69999999999999985e-27 < l

   1. Initial program 28.3%

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

      1. associate-*l* 28.3%

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

      2. associate-/r* 28.3%

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

      3. sub-neg 28.3%

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

      4. distribute-rgt-in 23.4%

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

      5. unpow2 23.4%

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

      6. times-frac 13.9%

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

      7. sqr-neg 13.9%

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

      8. times-frac 23.4%

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

      9. unpow2 23.4%

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

      10. distribute-rgt-in 28.3%

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

      11. +-commutative 28.3%

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

      12. associate-+l+ 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in t around 0 70.4%

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

      1. times-frac 71.8%

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

   7. Simplified 71.8%

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

      1. expm1-log1p-u 44.6%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 31.2%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]

      3. div-inv 31.2%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      4. pow-flip 31.2%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      5. metadata-eval 31.2%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-2}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right) \]

      6. associate-/r* 31.2%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)} - 1\right) \]

   9. Applied egg-rr 31.2%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 44.6%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-2}\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)} \]

       2. expm1-log1p 71.9%

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

       3. associate-/r* 71.8%

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

       4. associate-*r/ 71.8%

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

       5. associate-*l* 71.8%

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

       6. *-commutative 71.8%

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

       7. times-frac 67.4%

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

   11. Simplified 67.4%

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

   12. Taylor expanded in k around 0 54.2%

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

   13. Taylor expanded in l around 0 52.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t}} \]
   14. Step-by-step derivation

       1. *-commutative 52.4%

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

       2. times-frac 52.3%

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

   15. Simplified 52.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot {k\_m}^{-4}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (/ (pow l 2.0) t_m) (pow k_m -4.0)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((pow(l, 2.0) / t_m) * pow(k_m, -4.0)));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l ** 2.0d0) / t_m) * (k_m ** (-4.0d0))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) * Math.pow(k_m, -4.0)));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow(l, 2.0) / t_m) * math.pow(k_m, -4.0)))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * (k_m ^ -4.0))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l ^ 2.0) / t_m) * (k_m ^ -4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

   1. associate-*l* 29.0%

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

   2. associate-/r* 29.0%

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

   3. sub-neg 29.0%

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

   4. distribute-rgt-in 23.8%

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

   5. unpow2 23.8%

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

   6. times-frac 17.2%

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

   7. sqr-neg 17.2%

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

   8. times-frac 23.8%

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

   9. unpow2 23.8%

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

   10. distribute-rgt-in 29.0%

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

   11. +-commutative 29.0%

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

   12. associate-+l+ 40.1%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-/r* 54.0%

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

7. Simplified 54.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
8. Step-by-step derivation

   1. div-inv 53.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

   2. pow-flip 53.6%

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

   3. metadata-eval 53.6%

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

9. Applied egg-rr 53.6%

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

10. Final simplification 53.6%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right) \]
11. Add Preprocessing

Alternative 10: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2} \cdot {k\_m}^{-4}}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow l 2.0) (pow k_m -4.0)) t_m))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((pow(l, 2.0) * pow(k_m, -4.0)) / t_m));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l ** 2.0d0) * (k_m ** (-4.0d0))) / t_m))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) * Math.pow(k_m, -4.0)) / t_m));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow(l, 2.0) * math.pow(k_m, -4.0)) / t_m))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) * (k_m ^ -4.0)) / t_m)))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l ^ 2.0) * (k_m ^ -4.0)) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

   1. associate-*l* 29.0%

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

   2. associate-/r* 29.0%

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

   3. sub-neg 29.0%

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

   4. distribute-rgt-in 23.8%

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

   5. unpow2 23.8%

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

   6. times-frac 17.2%

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

   7. sqr-neg 17.2%

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

   8. times-frac 23.8%

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

   9. unpow2 23.8%

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

   10. distribute-rgt-in 29.0%

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

   11. +-commutative 29.0%

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

   12. associate-+l+ 40.1%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-/r* 54.0%

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

7. Simplified 54.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
8. Step-by-step derivation

   1. div-inv 53.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

   2. pow-flip 53.6%

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

   3. metadata-eval 53.6%

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

9. Applied egg-rr 53.6%

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

    1. associate-*l/ 54.5%

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

11. Simplified 54.5%

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

12. Final simplification 54.5%

    \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \]
13. Add Preprocessing

Alternative 11: 70.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\left(\ell \cdot {k\_m}^{-2}\right)}^{2}}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t_m))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (pow((l * pow(k_m, -2.0)), 2.0) / t_m));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l * (k_m ** (-2.0d0))) ** 2.0d0) / t_m))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (Math.pow((l * Math.pow(k_m, -2.0)), 2.0) / t_m));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (math.pow((l * math.pow(k_m, -2.0)), 2.0) / t_m))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64((Float64(l * (k_m ^ -2.0)) ^ 2.0) / t_m)))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l * (k_m ^ -2.0)) ^ 2.0) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

   1. associate-*l* 29.0%

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

   2. associate-/r* 29.0%

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

   3. sub-neg 29.0%

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

   4. distribute-rgt-in 23.8%

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

   5. unpow2 23.8%

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

   6. times-frac 17.2%

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

   7. sqr-neg 17.2%

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

   8. times-frac 23.8%

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

   9. unpow2 23.8%

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

   10. distribute-rgt-in 29.0%

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

   11. +-commutative 29.0%

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

   12. associate-+l+ 40.1%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 54.5%

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

   1. *-commutative 54.5%

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

   2. associate-/r* 54.0%

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

7. Simplified 54.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
8. Step-by-step derivation

   1. div-inv 53.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}\right)} \]

   2. pow-flip 53.6%

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

   3. metadata-eval 53.6%

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

9. Applied egg-rr 53.6%

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

    1. associate-*l/ 54.5%

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

11. Simplified 54.5%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot {k}^{-4}}{t}} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 54.5%

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

    2. pow2 54.5%

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

    3. sqrt-prod 54.5%

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

    4. pow2 54.5%

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

    5. sqrt-prod 30.1%

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

    6. add-sqr-sqrt 62.2%

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

    7. sqrt-pow1 67.8%

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

    8. metadata-eval 67.8%

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

13. Applied egg-rr 67.8%

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

14. Final simplification 67.8%

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

Reproduce

herbie shell --seed 2024027 
(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))))