Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 88.4%

Time: 34.3s

Alternatives: 25

Speedup: 35.0×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \sqrt[3]{t\_2}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := t\_4 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-290}:\\ \;\;\;\;\left(t\_4 \cdot \left(t \cdot {\left(t \cdot \left(t\_1 \cdot t\_3\right)\right)}^{-2}\right)\right) \cdot \frac{{\left(\sqrt{t\_5}\right)}^{2}}{t\_3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(t \cdot {\left(t\_3 \cdot \left(t \cdot t\_1\right)\right)}^{-2}\right)}{k\_m} \cdot \frac{t\_5}{t\_3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (* (sin k_m) (tan k_m)))
        (t_3 (cbrt t_2))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (* t_4 (pow (cbrt l) 2.0))))
   (if (<= (* l l) 1e-290)
     (*
      (* t_4 (* t (pow (* t (* t_1 t_3)) -2.0)))
      (/ (pow (sqrt t_5) 2.0) t_3))
     (if (<= (* l l) 2e+307)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_2)))
       (*
        (/ (* (sqrt 2.0) (* t (pow (* t_3 (* t t_1)) -2.0))) k_m)
        (/ t_5 t_3))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = cbrt(t_2);
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = t_4 * pow(cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 1e-290) {
		tmp = (t_4 * (t * pow((t * (t_1 * t_3)), -2.0))) * (pow(sqrt(t_5), 2.0) / t_3);
	} else if ((l * l) <= 2e+307) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = ((sqrt(2.0) * (t * pow((t_3 * (t * t_1)), -2.0))) / k_m) * (t_5 / t_3);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.cbrt(t_2);
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = t_4 * Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 1e-290) {
		tmp = (t_4 * (t * Math.pow((t * (t_1 * t_3)), -2.0))) * (Math.pow(Math.sqrt(t_5), 2.0) / t_3);
	} else if ((l * l) <= 2e+307) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = ((Math.sqrt(2.0) * (t * Math.pow((t_3 * (t * t_1)), -2.0))) / k_m) * (t_5 / t_3);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ -2.0
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = cbrt(t_2)
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(t_4 * (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (Float64(l * l) <= 1e-290)
		tmp = Float64(Float64(t_4 * Float64(t * (Float64(t * Float64(t_1 * t_3)) ^ -2.0))) * Float64((sqrt(t_5) ^ 2.0) / t_3));
	elseif (Float64(l * l) <= 2e+307)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_2)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(t * (Float64(t_3 * Float64(t * t_1)) ^ -2.0))) / k_m) * Float64(t_5 / t_3));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-290], N[(N[(t$95$4 * N[(t * N[Power[N[(t * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sqrt[t$95$5], $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+307], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[Power[N[(t$95$3 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(t$95$5 / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.0000000000000001e-290

   1. Initial program 24.2%

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

      1. *-commutative 24.2%

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

      2. associate-/r* 24.2%

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

   3. Simplified 31.4%

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

      1. add-sqr-sqrt 31.4%

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

      2. add-cube-cbrt 31.4%

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

      3. times-frac 31.4%

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

   6. Applied egg-rr 81.1%

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

      1. associate-/r/ 81.2%

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

      2. associate-/r* 81.2%

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

      3. associate-/r/ 81.2%

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

   8. Simplified 81.2%

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

      1. associate-*r/ 81.1%

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

   10. Applied egg-rr 81.2%

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

       1. associate-/l* 81.2%

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

       2. associate-*l* 81.2%

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

       3. associate-*l* 85.7%

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

       4. associate-/l* 87.1%

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

       5. *-inverses 87.1%

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

   12. Simplified 87.1%

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

       1. add-sqr-sqrt 49.5%

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

       2. pow2 49.5%

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

       3. *-rgt-identity 49.5%

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

   14. Applied egg-rr 49.5%

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

   if 1.0000000000000001e-290 < (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 41.6%

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

   3. Simplified 52.7%

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

      1. add-log-exp 43.4%

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

      2. *-commutative 43.4%

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

      3. exp-prod 37.4%

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

      4. pow2 37.4%

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

      5. associate-/r* 37.4%

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

      6. associate-*r* 37.4%

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

      7. *-commutative 37.4%

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

   6. Applied egg-rr 37.4%

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

   7. Taylor expanded in t around 0 87.0%

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

      1. associate-/r* 91.7%

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

   9. Simplified 91.7%

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

   if 1.99999999999999997e307 < (*.f64 l l)

   1. Initial program 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-/r* 29.1%

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

   3. Simplified 29.1%

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

      1. add-sqr-sqrt 29.1%

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

      2. add-cube-cbrt 29.1%

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

      3. times-frac 29.1%

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

   6. Applied egg-rr 87.5%

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

      1. associate-/r/ 87.5%

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

      2. associate-/r* 87.5%

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

      3. associate-/r/ 87.5%

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

   8. Simplified 87.5%

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

      1. associate-*r/ 87.5%

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

   10. Applied egg-rr 87.5%

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

       1. associate-/l* 87.6%

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

       2. associate-*l* 87.6%

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

       3. associate-*l* 87.6%

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

       4. associate-/l* 89.6%

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

       5. *-inverses 89.6%

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

   12. Simplified 89.6%

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

   13. Taylor expanded in k around 0 89.6%

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

       1. associate-*l/ 89.6%

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

   15. Applied egg-rr 89.6%

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

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \sqrt[3]{t\_2}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := \frac{t\_4 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_3}\\ \mathbf{if}\;\ell \leq 3.05 \cdot 10^{-161}:\\ \;\;\;\;\left(t\_4 \cdot \left(t \cdot {\left(t \cdot \left(t\_1 \cdot t\_3\right)\right)}^{-2}\right)\right) \cdot t\_5\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(t \cdot {\left(t\_3 \cdot \left(t \cdot t\_1\right)\right)}^{-2}\right)}{k\_m} \cdot t\_5\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (* (sin k_m) (tan k_m)))
        (t_3 (cbrt t_2))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (/ (* t_4 (pow (cbrt l) 2.0)) t_3)))
   (if (<= l 3.05e-161)
     (* (* t_4 (* t (pow (* t (* t_1 t_3)) -2.0))) t_5)
     (if (<= l 3.9e+153)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_2)))
       (* (/ (* (sqrt 2.0) (* t (pow (* t_3 (* t t_1)) -2.0))) k_m) t_5)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = cbrt(t_2);
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = (t_4 * pow(cbrt(l), 2.0)) / t_3;
	double tmp;
	if (l <= 3.05e-161) {
		tmp = (t_4 * (t * pow((t * (t_1 * t_3)), -2.0))) * t_5;
	} else if (l <= 3.9e+153) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = ((sqrt(2.0) * (t * pow((t_3 * (t * t_1)), -2.0))) / k_m) * t_5;
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.cbrt(t_2);
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = (t_4 * Math.pow(Math.cbrt(l), 2.0)) / t_3;
	double tmp;
	if (l <= 3.05e-161) {
		tmp = (t_4 * (t * Math.pow((t * (t_1 * t_3)), -2.0))) * t_5;
	} else if (l <= 3.9e+153) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = ((Math.sqrt(2.0) * (t * Math.pow((t_3 * (t * t_1)), -2.0))) / k_m) * t_5;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ -2.0
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = cbrt(t_2)
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(Float64(t_4 * (cbrt(l) ^ 2.0)) / t_3)
	tmp = 0.0
	if (l <= 3.05e-161)
		tmp = Float64(Float64(t_4 * Float64(t * (Float64(t * Float64(t_1 * t_3)) ^ -2.0))) * t_5);
	elseif (l <= 3.9e+153)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_2)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(t * (Float64(t_3 * Float64(t * t_1)) ^ -2.0))) / k_m) * t_5);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[l, 3.05e-161], N[(N[(t$95$4 * N[(t * N[Power[N[(t * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[l, 3.9e+153], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * N[Power[N[(t$95$3 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * t$95$5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.05000000000000008e-161

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   if 3.05000000000000008e-161 < l < 3.89999999999999983e153

   1. Initial program 43.2%

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

   3. Simplified 55.6%

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

      1. add-log-exp 45.4%

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

      2. *-commutative 45.4%

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

      3. exp-prod 40.3%

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

      4. pow2 40.3%

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

      5. associate-/r* 40.3%

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

      6. associate-*r* 40.3%

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

      7. *-commutative 40.3%

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

   6. Applied egg-rr 40.3%

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

   7. Taylor expanded in t around 0 89.8%

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

      1. associate-/r* 91.3%

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

   9. Simplified 91.3%

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

   if 3.89999999999999983e153 < l

   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. *-commutative 31.3%

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

      2. associate-/r* 31.3%

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

   3. Simplified 31.3%

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

      1. add-sqr-sqrt 31.3%

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

      2. add-cube-cbrt 31.3%

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

      3. times-frac 31.3%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 83.6%

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

   8. Simplified 83.6%

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

      1. associate-*r/ 83.6%

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

   10. Applied egg-rr 83.6%

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

       1. associate-/l* 83.6%

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

       2. associate-*l* 83.6%

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

       3. associate-*l* 83.7%

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

       4. associate-/l* 84.4%

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

       5. *-inverses 84.4%

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

   12. Simplified 84.4%

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

   13. Taylor expanded in k around 0 84.4%

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

       1. associate-*l/ 84.4%

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

   15. Applied egg-rr 84.4%

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

Alternative 3: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \sqrt[3]{t\_2}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := \frac{t\_4 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_3}\\ \mathbf{if}\;\ell \leq 3.05 \cdot 10^{-161}:\\ \;\;\;\;\left(t\_4 \cdot \left(t \cdot {\left(t \cdot \left(t\_1 \cdot t\_3\right)\right)}^{-2}\right)\right) \cdot t\_5\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(\sqrt{2} \cdot \frac{t \cdot {\left(t\_3 \cdot \left(t \cdot t\_1\right)\right)}^{-2}}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0))
        (t_2 (* (sin k_m) (tan k_m)))
        (t_3 (cbrt t_2))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (/ (* t_4 (pow (cbrt l) 2.0)) t_3)))
   (if (<= l 3.05e-161)
     (* (* t_4 (* t (pow (* t (* t_1 t_3)) -2.0))) t_5)
     (if (<= l 6.6e+153)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_2)))
       (* t_5 (* (sqrt 2.0) (/ (* t (pow (* t_3 (* t t_1)) -2.0)) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), -2.0);
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = cbrt(t_2);
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = (t_4 * pow(cbrt(l), 2.0)) / t_3;
	double tmp;
	if (l <= 3.05e-161) {
		tmp = (t_4 * (t * pow((t * (t_1 * t_3)), -2.0))) * t_5;
	} else if (l <= 6.6e+153) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = t_5 * (sqrt(2.0) * ((t * pow((t_3 * (t * t_1)), -2.0)) / k_m));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.cbrt(t_2);
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = (t_4 * Math.pow(Math.cbrt(l), 2.0)) / t_3;
	double tmp;
	if (l <= 3.05e-161) {
		tmp = (t_4 * (t * Math.pow((t * (t_1 * t_3)), -2.0))) * t_5;
	} else if (l <= 6.6e+153) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_2));
	} else {
		tmp = t_5 * (Math.sqrt(2.0) * ((t * Math.pow((t_3 * (t * t_1)), -2.0)) / k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ -2.0
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = cbrt(t_2)
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(Float64(t_4 * (cbrt(l) ^ 2.0)) / t_3)
	tmp = 0.0
	if (l <= 3.05e-161)
		tmp = Float64(Float64(t_4 * Float64(t * (Float64(t * Float64(t_1 * t_3)) ^ -2.0))) * t_5);
	elseif (l <= 6.6e+153)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_2)));
	else
		tmp = Float64(t_5 * Float64(sqrt(2.0) * Float64(Float64(t * (Float64(t_3 * Float64(t * t_1)) ^ -2.0)) / k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[l, 3.05e-161], N[(N[(t$95$4 * N[(t * N[Power[N[(t * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[l, 6.6e+153], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t * N[Power[N[(t$95$3 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.05000000000000008e-161

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   if 3.05000000000000008e-161 < l < 6.59999999999999989e153

   1. Initial program 43.2%

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

   3. Simplified 55.6%

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

      1. add-log-exp 45.4%

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

      2. *-commutative 45.4%

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

      3. exp-prod 40.3%

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

      4. pow2 40.3%

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

      5. associate-/r* 40.3%

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

      6. associate-*r* 40.3%

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

      7. *-commutative 40.3%

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

   6. Applied egg-rr 40.3%

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

   7. Taylor expanded in t around 0 89.8%

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

      1. associate-/r* 91.3%

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

   9. Simplified 91.3%

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

   if 6.59999999999999989e153 < l

   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. *-commutative 31.3%

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

      2. associate-/r* 31.3%

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

   3. Simplified 31.3%

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

      1. add-sqr-sqrt 31.3%

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

      2. add-cube-cbrt 31.3%

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

      3. times-frac 31.3%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 83.6%

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

   8. Simplified 83.6%

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

      1. associate-*r/ 83.6%

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

   10. Applied egg-rr 83.6%

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

       1. associate-/l* 83.6%

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

       2. associate-*l* 83.6%

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

       3. associate-*l* 83.7%

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

       4. associate-/l* 84.4%

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

       5. *-inverses 84.4%

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

   12. Simplified 84.4%

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

   13. Taylor expanded in k around 0 84.4%

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

       1. associate-*l/ 84.4%

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

   15. Applied egg-rr 84.4%

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

       1. associate-/l* 84.4%

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

   17. Simplified 84.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.05 \cdot 10^{-161}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\sqrt{2} \cdot \frac{t \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := \sqrt[3]{t\_1}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t_4 := \frac{t\_3 \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{t\_2}\\ t_5 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-156}:\\ \;\;\;\;t\_4 \cdot \left(t\_3 \cdot \left(t \cdot {\left(t \cdot \left(t\_5 \cdot \sqrt[3]{{k\_m}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \left(\sqrt{2} \cdot \frac{t \cdot {\left(t\_2 \cdot \left(t \cdot t\_5\right)\right)}^{-2}}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m)))
        (t_2 (cbrt t_1))
        (t_3 (/ (sqrt 2.0) k_m))
        (t_4 (/ (* t_3 (pow (cbrt l) 2.0)) t_2))
        (t_5 (pow (cbrt l) -2.0)))
   (if (<= l 2.5e-156)
     (* t_4 (* t_3 (* t (pow (* t (* t_5 (cbrt (pow k_m 2.0)))) -2.0))))
     (if (<= l 4.2e+153)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (* t_4 (* (sqrt 2.0) (/ (* t (pow (* t_2 (* t t_5)) -2.0)) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double t_2 = cbrt(t_1);
	double t_3 = sqrt(2.0) / k_m;
	double t_4 = (t_3 * pow(cbrt(l), 2.0)) / t_2;
	double t_5 = pow(cbrt(l), -2.0);
	double tmp;
	if (l <= 2.5e-156) {
		tmp = t_4 * (t_3 * (t * pow((t * (t_5 * cbrt(pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 4.2e+153) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = t_4 * (sqrt(2.0) * ((t * pow((t_2 * (t * t_5)), -2.0)) / k_m));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double t_2 = Math.cbrt(t_1);
	double t_3 = Math.sqrt(2.0) / k_m;
	double t_4 = (t_3 * Math.pow(Math.cbrt(l), 2.0)) / t_2;
	double t_5 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (l <= 2.5e-156) {
		tmp = t_4 * (t_3 * (t * Math.pow((t * (t_5 * Math.cbrt(Math.pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 4.2e+153) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = t_4 * (Math.sqrt(2.0) * ((t * Math.pow((t_2 * (t * t_5)), -2.0)) / k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	t_2 = cbrt(t_1)
	t_3 = Float64(sqrt(2.0) / k_m)
	t_4 = Float64(Float64(t_3 * (cbrt(l) ^ 2.0)) / t_2)
	t_5 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (l <= 2.5e-156)
		tmp = Float64(t_4 * Float64(t_3 * Float64(t * (Float64(t * Float64(t_5 * cbrt((k_m ^ 2.0)))) ^ -2.0))));
	elseif (l <= 4.2e+153)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(t_4 * Float64(sqrt(2.0) * Float64(Float64(t * (Float64(t_2 * Float64(t * t_5)) ^ -2.0)) / k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[l, 2.5e-156], N[(t$95$4 * N[(t$95$3 * N[(t * N[Power[N[(t * N[(t$95$5 * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+153], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t * N[Power[N[(t$95$2 * N[(t * t$95$5), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.50000000000000004e-156

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 2.50000000000000004e-156 < l < 4.20000000000000033e153

   1. Initial program 43.2%

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

   3. Simplified 55.6%

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

      1. add-log-exp 45.4%

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

      2. *-commutative 45.4%

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

      3. exp-prod 40.3%

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

      4. pow2 40.3%

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

      5. associate-/r* 40.3%

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

      6. associate-*r* 40.3%

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

      7. *-commutative 40.3%

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

   6. Applied egg-rr 40.3%

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

   7. Taylor expanded in t around 0 89.8%

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

      1. associate-/r* 91.3%

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

   9. Simplified 91.3%

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

   if 4.20000000000000033e153 < l

   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. *-commutative 31.3%

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

      2. associate-/r* 31.3%

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

   3. Simplified 31.3%

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

      1. add-sqr-sqrt 31.3%

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

      2. add-cube-cbrt 31.3%

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

      3. times-frac 31.3%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.6%

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

      2. associate-/r* 83.6%

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

      3. associate-/r/ 83.6%

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

   8. Simplified 83.6%

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

      1. associate-*r/ 83.6%

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

   10. Applied egg-rr 83.6%

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

       1. associate-/l* 83.6%

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

       2. associate-*l* 83.6%

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

       3. associate-*l* 83.7%

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

       4. associate-/l* 84.4%

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

       5. *-inverses 84.4%

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

   12. Simplified 84.4%

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

   13. Taylor expanded in k around 0 84.4%

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

       1. associate-*l/ 84.4%

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

   15. Applied egg-rr 84.4%

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

       1. associate-/l* 84.4%

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

   17. Simplified 84.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\sqrt{2} \cdot \frac{t \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := \sqrt[3]{t\_1}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := t \cdot \frac{t\_2}{t\_4}\\ \mathbf{if}\;\ell \leq 10^{-160}:\\ \;\;\;\;\frac{t\_3 \cdot t\_4}{t\_2} \cdot \left(t\_3 \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k\_m}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(t\_3 \cdot t\right)}^{2}}{{t\_5}^{2}}}{t\_5}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m)))
        (t_2 (cbrt t_1))
        (t_3 (/ (sqrt 2.0) k_m))
        (t_4 (pow (cbrt l) 2.0))
        (t_5 (* t (/ t_2 t_4))))
   (if (<= l 1e-160)
     (*
      (/ (* t_3 t_4) t_2)
      (*
       t_3
       (* t (pow (* t (* (pow (cbrt l) -2.0) (cbrt (pow k_m 2.0)))) -2.0))))
     (if (<= l 4.5e+198)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (/ (/ (pow (* t_3 t) 2.0) (pow t_5 2.0)) t_5)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double t_2 = cbrt(t_1);
	double t_3 = sqrt(2.0) / k_m;
	double t_4 = pow(cbrt(l), 2.0);
	double t_5 = t * (t_2 / t_4);
	double tmp;
	if (l <= 1e-160) {
		tmp = ((t_3 * t_4) / t_2) * (t_3 * (t * pow((t * (pow(cbrt(l), -2.0) * cbrt(pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 4.5e+198) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (pow((t_3 * t), 2.0) / pow(t_5, 2.0)) / t_5;
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double t_2 = Math.cbrt(t_1);
	double t_3 = Math.sqrt(2.0) / k_m;
	double t_4 = Math.pow(Math.cbrt(l), 2.0);
	double t_5 = t * (t_2 / t_4);
	double tmp;
	if (l <= 1e-160) {
		tmp = ((t_3 * t_4) / t_2) * (t_3 * (t * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 4.5e+198) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (Math.pow((t_3 * t), 2.0) / Math.pow(t_5, 2.0)) / t_5;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	t_2 = cbrt(t_1)
	t_3 = Float64(sqrt(2.0) / k_m)
	t_4 = cbrt(l) ^ 2.0
	t_5 = Float64(t * Float64(t_2 / t_4))
	tmp = 0.0
	if (l <= 1e-160)
		tmp = Float64(Float64(Float64(t_3 * t_4) / t_2) * Float64(t_3 * Float64(t * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt((k_m ^ 2.0)))) ^ -2.0))));
	elseif (l <= 4.5e+198)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(Float64((Float64(t_3 * t) ^ 2.0) / (t_5 ^ 2.0)) / t_5);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t * N[(t$95$2 / t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1e-160], N[(N[(N[(t$95$3 * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$3 * N[(t * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+198], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t$95$3 * t), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.9999999999999999e-161

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 9.9999999999999999e-161 < l < 4.50000000000000001e198

   1. Initial program 40.7%

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

   3. Simplified 51.5%

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

      1. add-log-exp 42.6%

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

      2. *-commutative 42.6%

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

      3. exp-prod 38.1%

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

      4. pow2 38.1%

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

      5. associate-/r* 38.1%

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

      6. associate-*r* 38.1%

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

      7. *-commutative 38.1%

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

   6. Applied egg-rr 38.1%

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

   7. Taylor expanded in t around 0 88.3%

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

      1. associate-/r* 89.6%

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

   9. Simplified 89.6%

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

   if 4.50000000000000001e198 < l

   1. Initial program 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/r* 34.8%

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

   3. Simplified 34.8%

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

      1. add-sqr-sqrt 34.8%

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

      2. add-cube-cbrt 34.8%

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

      3. times-frac 34.8%

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

   6. Applied egg-rr 85.7%

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

      1. associate-*r/ 85.7%

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

      2. associate-*l/ 73.6%

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

      3. unpow2 73.6%

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

      4. associate-/r/ 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l/ 73.7%

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

      7. associate-/l* 73.6%

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

      8. associate-*l/ 73.6%

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

      9. associate-/l* 73.5%

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

   8. Simplified 73.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-160}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt{2}}{k} \cdot t\right)}^{2}}{{\left(t \cdot \frac{\sqrt[3]{\sin k \cdot \tan k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}}{t \cdot \frac{\sqrt[3]{\sin k \cdot \tan k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := \sqrt[3]{t\_1}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := t\_2 \cdot \frac{t}{t\_4}\\ \mathbf{if}\;\ell \leq 6.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_3 \cdot t\_4}{t\_2} \cdot \left(t\_3 \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k\_m}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_5}^{2}} \cdot \frac{{\left(\frac{k\_m}{t}\right)}^{-2}}{t\_5}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m)))
        (t_2 (cbrt t_1))
        (t_3 (/ (sqrt 2.0) k_m))
        (t_4 (pow (cbrt l) 2.0))
        (t_5 (* t_2 (/ t t_4))))
   (if (<= l 6.7e-158)
     (*
      (/ (* t_3 t_4) t_2)
      (*
       t_3
       (* t (pow (* t (* (pow (cbrt l) -2.0) (cbrt (pow k_m 2.0)))) -2.0))))
     (if (<= l 3.4e+198)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (* (/ 2.0 (pow t_5 2.0)) (/ (pow (/ k_m t) -2.0) t_5))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double t_2 = cbrt(t_1);
	double t_3 = sqrt(2.0) / k_m;
	double t_4 = pow(cbrt(l), 2.0);
	double t_5 = t_2 * (t / t_4);
	double tmp;
	if (l <= 6.7e-158) {
		tmp = ((t_3 * t_4) / t_2) * (t_3 * (t * pow((t * (pow(cbrt(l), -2.0) * cbrt(pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 3.4e+198) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 / pow(t_5, 2.0)) * (pow((k_m / t), -2.0) / t_5);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double t_2 = Math.cbrt(t_1);
	double t_3 = Math.sqrt(2.0) / k_m;
	double t_4 = Math.pow(Math.cbrt(l), 2.0);
	double t_5 = t_2 * (t / t_4);
	double tmp;
	if (l <= 6.7e-158) {
		tmp = ((t_3 * t_4) / t_2) * (t_3 * (t * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 3.4e+198) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 / Math.pow(t_5, 2.0)) * (Math.pow((k_m / t), -2.0) / t_5);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	t_2 = cbrt(t_1)
	t_3 = Float64(sqrt(2.0) / k_m)
	t_4 = cbrt(l) ^ 2.0
	t_5 = Float64(t_2 * Float64(t / t_4))
	tmp = 0.0
	if (l <= 6.7e-158)
		tmp = Float64(Float64(Float64(t_3 * t_4) / t_2) * Float64(t_3 * Float64(t * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt((k_m ^ 2.0)))) ^ -2.0))));
	elseif (l <= 3.4e+198)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(Float64(2.0 / (t_5 ^ 2.0)) * Float64((Float64(k_m / t) ^ -2.0) / t_5));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[(t / t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 6.7e-158], N[(N[(N[(t$95$3 * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$3 * N[(t * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.4e+198], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.7000000000000001e-158

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 6.7000000000000001e-158 < l < 3.4e198

   1. Initial program 40.7%

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

   3. Simplified 51.5%

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

      1. add-log-exp 42.6%

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

      2. *-commutative 42.6%

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

      3. exp-prod 38.1%

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

      4. pow2 38.1%

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

      5. associate-/r* 38.1%

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

      6. associate-*r* 38.1%

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

      7. *-commutative 38.1%

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

   6. Applied egg-rr 38.1%

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

   7. Taylor expanded in t around 0 88.3%

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

      1. associate-/r* 89.6%

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

   9. Simplified 89.6%

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

   if 3.4e198 < l

   1. Initial program 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/r* 34.8%

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

   3. Simplified 34.8%

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

      1. add-cube-cbrt 34.8%

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

      2. div-inv 34.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 34.8%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 73.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := \sqrt[3]{t\_1}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := \frac{t}{t\_3}\\ \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_4 \cdot t\_3}{t\_2} \cdot \left(t\_4 \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k\_m}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot t\_5\right)}^{2}} \cdot \frac{\frac{{\left(\frac{k\_m}{t}\right)}^{-2}}{t\_5}}{t\_2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m)))
        (t_2 (cbrt t_1))
        (t_3 (pow (cbrt l) 2.0))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (/ t t_3)))
   (if (<= l 7.2e-162)
     (*
      (/ (* t_4 t_3) t_2)
      (*
       t_4
       (* t (pow (* t (* (pow (cbrt l) -2.0) (cbrt (pow k_m 2.0)))) -2.0))))
     (if (<= l 7.5e+198)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (*
        (/ 2.0 (pow (* t_2 t_5) 2.0))
        (/ (/ (pow (/ k_m t) -2.0) t_5) t_2))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double t_2 = cbrt(t_1);
	double t_3 = pow(cbrt(l), 2.0);
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = t / t_3;
	double tmp;
	if (l <= 7.2e-162) {
		tmp = ((t_4 * t_3) / t_2) * (t_4 * (t * pow((t * (pow(cbrt(l), -2.0) * cbrt(pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 7.5e+198) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 / pow((t_2 * t_5), 2.0)) * ((pow((k_m / t), -2.0) / t_5) / t_2);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double t_2 = Math.cbrt(t_1);
	double t_3 = Math.pow(Math.cbrt(l), 2.0);
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = t / t_3;
	double tmp;
	if (l <= 7.2e-162) {
		tmp = ((t_4 * t_3) / t_2) * (t_4 * (t * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 7.5e+198) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 / Math.pow((t_2 * t_5), 2.0)) * ((Math.pow((k_m / t), -2.0) / t_5) / t_2);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	t_2 = cbrt(t_1)
	t_3 = cbrt(l) ^ 2.0
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(t / t_3)
	tmp = 0.0
	if (l <= 7.2e-162)
		tmp = Float64(Float64(Float64(t_4 * t_3) / t_2) * Float64(t_4 * Float64(t * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt((k_m ^ 2.0)))) ^ -2.0))));
	elseif (l <= 7.5e+198)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(Float64(2.0 / (Float64(t_2 * t_5) ^ 2.0)) * Float64(Float64((Float64(k_m / t) ^ -2.0) / t_5) / t_2));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(t / t$95$3), $MachinePrecision]}, If[LessEqual[l, 7.2e-162], N[(N[(N[(t$95$4 * t$95$3), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$4 * N[(t * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.5e+198], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(t$95$2 * t$95$5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.1999999999999996e-162

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 7.1999999999999996e-162 < l < 7.5000000000000002e198

   1. Initial program 40.7%

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

   3. Simplified 51.5%

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

      1. add-log-exp 42.6%

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

      2. *-commutative 42.6%

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

      3. exp-prod 38.1%

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

      4. pow2 38.1%

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

      5. associate-/r* 38.1%

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

      6. associate-*r* 38.1%

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

      7. *-commutative 38.1%

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

   6. Applied egg-rr 38.1%

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

   7. Taylor expanded in t around 0 88.3%

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

      1. associate-/r* 89.6%

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

   9. Simplified 89.6%

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

   if 7.5000000000000002e198 < l

   1. Initial program 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-/r* 34.8%

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

   3. Simplified 34.8%

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

      1. add-cube-cbrt 34.8%

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

      2. div-inv 34.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 34.8%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 73.6%

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

      1. associate-/r* 73.6%

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

   8. Simplified 73.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{{\left(\frac{k}{t}\right)}^{-2}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := \frac{\sqrt{2}}{k\_m}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_2 \cdot t\_3}{\sqrt[3]{t\_1}} \cdot \left(t\_2 \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k\_m}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+249}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k\_m}{t}\right)}^{-2}}{t\_1 \cdot {\left(\frac{t}{t\_3}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m)))
        (t_2 (/ (sqrt 2.0) k_m))
        (t_3 (pow (cbrt l) 2.0)))
   (if (<= l 1.2e-157)
     (*
      (/ (* t_2 t_3) (cbrt t_1))
      (*
       t_2
       (* t (pow (* t (* (pow (cbrt l) -2.0) (cbrt (pow k_m 2.0)))) -2.0))))
     (if (<= l 1.5e+249)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (/ (* 2.0 (pow (/ k_m t) -2.0)) (* t_1 (pow (/ t t_3) 3.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double t_2 = sqrt(2.0) / k_m;
	double t_3 = pow(cbrt(l), 2.0);
	double tmp;
	if (l <= 1.2e-157) {
		tmp = ((t_2 * t_3) / cbrt(t_1)) * (t_2 * (t * pow((t * (pow(cbrt(l), -2.0) * cbrt(pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 1.5e+249) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 * pow((k_m / t), -2.0)) / (t_1 * pow((t / t_3), 3.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double t_2 = Math.sqrt(2.0) / k_m;
	double t_3 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (l <= 1.2e-157) {
		tmp = ((t_2 * t_3) / Math.cbrt(t_1)) * (t_2 * (t * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.pow(k_m, 2.0)))), -2.0)));
	} else if (l <= 1.5e+249) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 * Math.pow((k_m / t), -2.0)) / (t_1 * Math.pow((t / t_3), 3.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	t_2 = Float64(sqrt(2.0) / k_m)
	t_3 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (l <= 1.2e-157)
		tmp = Float64(Float64(Float64(t_2 * t_3) / cbrt(t_1)) * Float64(t_2 * Float64(t * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt((k_m ^ 2.0)))) ^ -2.0))));
	elseif (l <= 1.5e+249)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(Float64(2.0 * (Float64(k_m / t) ^ -2.0)) / Float64(t_1 * (Float64(t / t_3) ^ 3.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 1.2e-157], N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] / N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Power[k$95$m, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e+249], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(t / t$95$3), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2e-157

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 29.8%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 36.1%

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

      2. add-cube-cbrt 36.1%

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

      3. times-frac 36.1%

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

   6. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.3%

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

      2. associate-/r* 79.3%

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

      3. associate-/r/ 79.3%

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

   8. Simplified 79.3%

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

      1. associate-*r/ 79.3%

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

   10. Applied egg-rr 79.4%

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

       1. associate-/l* 79.4%

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

       2. associate-*l* 79.4%

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

       3. associate-*l* 81.4%

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

       4. associate-/l* 84.6%

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

       5. *-inverses 84.6%

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

   12. Simplified 84.6%

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

   13. Taylor expanded in k around 0 84.6%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 1.2e-157 < l < 1.50000000000000008e249

   1. Initial program 41.0%

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

   3. Simplified 50.1%

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

      1. add-log-exp 42.7%

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

      2. *-commutative 42.7%

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

      3. exp-prod 39.0%

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

      4. pow2 39.0%

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

      5. associate-/r* 39.0%

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

      6. associate-*r* 39.0%

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

      7. *-commutative 39.0%

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

   6. Applied egg-rr 39.0%

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

   7. Taylor expanded in t around 0 85.7%

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

      1. associate-/r* 86.9%

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

   9. Simplified 86.9%

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

   if 1.50000000000000008e249 < l

   1. Initial program 22.2%

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

      1. *-commutative 22.2%

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

      2. associate-/r* 22.2%

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

   3. Simplified 22.2%

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

      1. add-cube-cbrt 22.2%

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

      2. div-inv 22.2%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 22.2%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 75.8%

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

      1. associate-*l/ 75.8%

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

      2. associate-/l* 75.8%

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

      3. associate-/l/ 75.8%

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

      4. unpow2 75.8%

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

      5. unpow3 75.7%

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

      6. *-commutative 75.7%

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

   8. Simplified 76.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)\right)}^{-2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+249}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k\_m}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{t\_1}\right)}\\ \mathbf{elif}\;k\_m \leq 2.7 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k\_m}{t}\right)}^{-2}}{t\_1 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= k_m 1.75e-163)
     (/
      (/ 2.0 (pow (/ k_m t) 2.0))
      (log (pow (exp (* (pow t 3.0) (pow l -2.0))) t_1)))
     (if (<= k_m 2.7e+135)
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))
       (/
        (* 2.0 (pow (/ k_m t) -2.0))
        (* t_1 (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (2.0 / pow((k_m / t), 2.0)) / log(pow(exp((pow(t, 3.0) * pow(l, -2.0))), t_1));
	} else if (k_m <= 2.7e+135) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 * pow((k_m / t), -2.0)) / (t_1 * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (2.0 / Math.pow((k_m / t), 2.0)) / Math.log(Math.pow(Math.exp((Math.pow(t, 3.0) * Math.pow(l, -2.0))), t_1));
	} else if (k_m <= 2.7e+135) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	} else {
		tmp = (2.0 * Math.pow((k_m / t), -2.0)) / (t_1 * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (k_m <= 1.75e-163)
		tmp = Float64(Float64(2.0 / (Float64(k_m / t) ^ 2.0)) / log((exp(Float64((t ^ 3.0) * (l ^ -2.0))) ^ t_1)));
	elseif (k_m <= 2.7e+135)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(Float64(2.0 * (Float64(k_m / t) ^ -2.0)) / Float64(t_1 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.75e-163], N[(N[(2.0 / N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[Power[N[Exp[N[(N[Power[t, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.7e+135], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.75000000000000014e-163

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-/r* 33.5%

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

   3. Simplified 37.9%

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

      1. add-log-exp 17.6%

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

      2. exp-prod 25.3%

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

      3. div-inv 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      4. pow2 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot \frac{1}{\color{blue}{{\ell}^{2}}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      5. pow-flip 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot \color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

      6. metadata-eval 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{\color{blue}{-2}}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)} \]

   6. Applied egg-rr 25.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}} \]

   if 1.75000000000000014e-163 < k < 2.69999999999999985e135

   1. Initial program 27.2%

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

   3. Simplified 37.7%

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

      1. add-log-exp 29.2%

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

      2. *-commutative 29.2%

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

      3. exp-prod 37.0%

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

      4. pow2 37.0%

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

      5. associate-/r* 37.0%

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

      6. associate-*r* 37.0%

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

      7. *-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in t around 0 82.8%

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

      1. associate-/r* 84.5%

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

   9. Simplified 84.5%

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

   if 2.69999999999999985e135 < k

   1. Initial program 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/r* 41.7%

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

   3. Simplified 51.4%

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

      1. add-cube-cbrt 51.4%

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

      2. div-inv 51.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 51.4%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 75.6%

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

      2. associate-/l* 75.6%

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

      3. associate-/l/ 70.6%

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

      4. unpow2 70.6%

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

      5. unpow3 70.6%

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

      6. *-commutative 70.6%

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

   8. Simplified 70.6%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\log \left({\left(e^{{t}^{3} \cdot {\ell}^{-2}}\right)}^{\left(\sin k \cdot \tan k\right)}\right)}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;\ell \leq 6 \cdot 10^{-169} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+249}\right):\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k\_m}{t}\right)}^{-2}}{t\_1 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (or (<= l 6e-169) (not (<= l 2.1e+249)))
     (/
      (* 2.0 (pow (/ k_m t) -2.0))
      (* t_1 (pow (/ t (pow (cbrt l) 2.0)) 3.0)))
     (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if ((l <= 6e-169) || !(l <= 2.1e+249)) {
		tmp = (2.0 * pow((k_m / t), -2.0)) / (t_1 * pow((t / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if ((l <= 6e-169) || !(l <= 2.1e+249)) {
		tmp = (2.0 * Math.pow((k_m / t), -2.0)) / (t_1 * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if ((l <= 6e-169) || !(l <= 2.1e+249))
		tmp = Float64(Float64(2.0 * (Float64(k_m / t) ^ -2.0)) / Float64(t_1 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[l, 6e-169], N[Not[LessEqual[l, 2.1e+249]], $MachinePrecision]], N[(N[(2.0 * N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9999999999999998e-169 or 2.0999999999999998e249 < l

   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. *-commutative 29.2%

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

      2. associate-/r* 29.2%

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

   3. Simplified 35.2%

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

      1. add-cube-cbrt 35.1%

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

      2. div-inv 35.1%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 35.1%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 68.7%

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

      1. associate-*l/ 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/l/ 66.4%

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

      4. unpow2 66.4%

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

      5. unpow3 66.4%

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

      6. *-commutative 66.4%

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

   8. Simplified 58.6%

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

   if 5.9999999999999998e-169 < l < 2.0999999999999998e249

   1. Initial program 41.2%

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

   3. Simplified 50.1%

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

      1. add-log-exp 42.8%

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

      2. *-commutative 42.8%

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

      3. exp-prod 39.2%

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

      4. pow2 39.2%

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

      5. associate-/r* 39.2%

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

      6. associate-*r* 39.2%

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

      7. *-commutative 39.2%

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

   6. Applied egg-rr 39.2%

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

   7. Taylor expanded in t around 0 85.0%

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

      1. associate-/r* 86.1%

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

   9. Simplified 86.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-169} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+249}\right):\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sqrt{t\_1} \cdot \left(\frac{k\_m}{t} \cdot {t}^{1.5}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= (* l l) 0.0)
     (pow (* l (/ (sqrt 2.0) (* (sqrt t_1) (* (/ k_m t) (pow t 1.5))))) 2.0)
     (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = pow((l * (sqrt(2.0) / (sqrt(t_1) * ((k_m / t) * pow(t, 1.5))))), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * tan(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = (l * (sqrt(2.0d0) / (sqrt(t_1) * ((k_m / t) * (t ** 1.5d0))))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sqrt(t_1) * ((k_m / t) * Math.pow(t, 1.5))))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * math.tan(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = math.pow((l * (math.sqrt(2.0) / (math.sqrt(t_1) * ((k_m / t) * math.pow(t, 1.5))))), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / (t * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sqrt(t_1) * Float64(Float64(k_m / t) * (t ^ 1.5))))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * tan(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (l * (sqrt(2.0) / (sqrt(t_1) * ((k_m / t) * (t ^ 1.5))))) ^ 2.0;
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[(k$95$m / t), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.2%

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

   3. Simplified 31.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

   6. Applied egg-rr 27.4%

      \[\leadsto \color{blue}{{\left(\ell \cdot \frac{\sqrt{2}}{\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l* 34.6%

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

   8. Simplified 34.6%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 43.4%

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

      1. add-log-exp 37.8%

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

      2. *-commutative 37.8%

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

      3. exp-prod 35.0%

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

      4. pow2 35.0%

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

      5. associate-/r* 35.0%

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

      6. associate-*r* 35.0%

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

      7. *-commutative 35.0%

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

   6. Applied egg-rr 35.0%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. associate-/r* 79.3%

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

   9. Simplified 79.3%

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

Alternative 12: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k\_m}{t}\right)}^{-2}}{t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= (* l l) 0.0)
     (/ (* 2.0 (pow (/ k_m t) -2.0)) (* t_1 (pow (/ (pow t 1.5) l) 2.0)))
     (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 * pow((k_m / t), -2.0)) / (t_1 * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * tan(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 * ((k_m / t) ** (-2.0d0))) / (t_1 * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 * Math.pow((k_m / t), -2.0)) / (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * math.tan(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 * math.pow((k_m / t), -2.0)) / (t_1 * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / (t * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 * (Float64(k_m / t) ^ -2.0)) / Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * tan(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (2.0 * ((k_m / t) ^ -2.0)) / (t_1 * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 * N[Power[N[(k$95$m / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.2%

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

      1. *-commutative 24.2%

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

      2. associate-/r* 24.2%

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

   3. Simplified 31.8%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

      3. sqrt-div 13.6%

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

      4. sqrt-pow1 21.1%

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

      5. metadata-eval 21.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 16.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 28.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 28.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. clear-num 28.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{\left(\frac{k}{t}\right)}^{2} + 0}{2}}}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. +-rgt-identity 28.3%

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

      3. associate-/r/ 28.3%

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

      4. pow-flip 29.8%

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

      5. metadata-eval 29.8%

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

   8. Applied egg-rr 29.8%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 43.4%

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

      1. add-log-exp 37.8%

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

      2. *-commutative 37.8%

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

      3. exp-prod 35.0%

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

      4. pow2 35.0%

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

      5. associate-/r* 35.0%

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

      6. associate-*r* 35.0%

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

      7. *-commutative 35.0%

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

   6. Applied egg-rr 35.0%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. associate-/r* 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{1}{\frac{t}{k\_m} \cdot \frac{t}{k\_m}}}}{t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= (* l l) 0.0)
     (/
      (/ 2.0 (/ 1.0 (* (/ t k_m) (/ t k_m))))
      (* t_1 (pow (/ (pow t 1.5) l) 2.0)))
     (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / (1.0 / ((t / k_m) * (t / k_m)))) / (t_1 * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * tan(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 / (1.0d0 / ((t / k_m) * (t / k_m)))) / (t_1 * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / (1.0 / ((t / k_m) * (t / k_m)))) / (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * math.tan(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 / (1.0 / ((t / k_m) * (t / k_m)))) / (t_1 * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / (t * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 / Float64(Float64(t / k_m) * Float64(t / k_m)))) / Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * tan(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (2.0 / (1.0 / ((t / k_m) * (t / k_m)))) / (t_1 * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[(1.0 / N[(N[(t / k$95$m), $MachinePrecision] * N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.2%

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

      1. *-commutative 24.2%

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

      2. associate-/r* 24.2%

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

   3. Simplified 31.8%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

      3. sqrt-div 13.6%

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

      4. sqrt-pow1 21.1%

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

      5. metadata-eval 21.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 16.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 28.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 28.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. unpow2 28.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. clear-num 28.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. clear-num 28.4%

         \[\leadsto \frac{\frac{2}{\frac{1}{\frac{t}{k}} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. frac-times 28.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{1 \cdot 1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      5. metadata-eval 28.3%

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{1}}{\frac{t}{k} \cdot \frac{t}{k}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 28.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 43.4%

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

      1. add-log-exp 37.8%

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

      2. *-commutative 37.8%

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

      3. exp-prod 35.0%

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

      4. pow2 35.0%

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

      5. associate-/r* 35.0%

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

      6. associate-*r* 35.0%

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

      7. *-commutative 35.0%

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

   6. Applied egg-rr 35.0%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. associate-/r* 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;t \leq 1.95 \cdot 10^{-109}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k\_m}{t}\right)}^{2}}}{\frac{t\_1 \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= t 1.95e-109)
     (*
      (* l l)
      (/ 2.0 (* (* k_m k_m) (/ (* t (pow (sin k_m) 2.0)) (cos k_m)))))
     (if (<= t 1.45e+73)
       (/ (/ 2.0 (pow (/ k_m t) 2.0)) (/ (* t_1 (/ (pow t 3.0) l)) l))
       (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if (t <= 1.95e-109) {
		tmp = (l * l) * (2.0 / ((k_m * k_m) * ((t * pow(sin(k_m), 2.0)) / cos(k_m))));
	} else if (t <= 1.45e+73) {
		tmp = (2.0 / pow((k_m / t), 2.0)) / ((t_1 * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * tan(k_m)
    if (t <= 1.95d-109) then
        tmp = (l * l) * (2.0d0 / ((k_m * k_m) * ((t * (sin(k_m) ** 2.0d0)) / cos(k_m))))
    else if (t <= 1.45d+73) then
        tmp = (2.0d0 / ((k_m / t) ** 2.0d0)) / ((t_1 * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if (t <= 1.95e-109) {
		tmp = (l * l) * (2.0 / ((k_m * k_m) * ((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m))));
	} else if (t <= 1.45e+73) {
		tmp = (2.0 / Math.pow((k_m / t), 2.0)) / ((t_1 * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * math.tan(k_m)
	tmp = 0
	if t <= 1.95e-109:
		tmp = (l * l) * (2.0 / ((k_m * k_m) * ((t * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m))))
	elif t <= 1.45e+73:
		tmp = (2.0 / math.pow((k_m / t), 2.0)) / ((t_1 * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / (t * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (t <= 1.95e-109)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)))));
	elseif (t <= 1.45e+73)
		tmp = Float64(Float64(2.0 / (Float64(k_m / t) ^ 2.0)) / Float64(Float64(t_1 * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * tan(k_m);
	tmp = 0.0;
	if (t <= 1.95e-109)
		tmp = (l * l) * (2.0 / ((k_m * k_m) * ((t * (sin(k_m) ^ 2.0)) / cos(k_m))));
	elseif (t <= 1.45e+73)
		tmp = (2.0 / ((k_m / t) ^ 2.0)) / ((t_1 * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.95e-109], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+73], N[(N[(2.0 / N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.95000000000000011e-109

   1. Initial program 35.4%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in t around 0 70.7%

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

      1. associate-/l* 70.7%

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

   7. Simplified 70.7%

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

      1. unpow2 70.7%

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

   9. Applied egg-rr 70.7%

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

   if 1.95000000000000011e-109 < t < 1.4500000000000001e73

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-/r* 57.5%

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

   3. Simplified 62.7%

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

      1. associate-/r* 78.5%

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

      2. associate-*l/ 83.8%

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

   6. Applied egg-rr 83.8%

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

   if 1.4500000000000001e73 < t

   1. Initial program 8.2%

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

   3. Simplified 22.4%

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

      1. add-log-exp 22.4%

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

      2. *-commutative 22.4%

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

      3. exp-prod 31.6%

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

      4. pow2 31.6%

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

      5. associate-/r* 31.6%

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

      6. associate-*r* 31.6%

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

      7. *-commutative 31.6%

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

   6. Applied egg-rr 31.6%

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

   7. Taylor expanded in t around 0 72.0%

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

      1. associate-/r* 75.9%

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

   9. Simplified 75.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-109}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{k\_m}{t} \cdot \frac{k\_m}{t}}}{t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= (* l l) 0.0)
     (/ (/ 2.0 (* (/ k_m t) (/ k_m t))) (* t_1 (pow (/ (pow t 1.5) l) 2.0)))
     (* 2.0 (/ (/ (pow l 2.0) (pow k_m 2.0)) (* t t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / ((k_m / t) * (k_m / t))) / (t_1 * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * tan(k_m)
    if ((l * l) <= 0.0d0) then
        tmp = (2.0d0 / ((k_m / t) * (k_m / t))) / (t_1 * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * Math.tan(k_m);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (2.0 / ((k_m / t) * (k_m / t))) / (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * math.tan(k_m)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (2.0 / ((k_m / t) * (k_m / t))) / (t_1 * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / (t * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) * Float64(k_m / t))) / Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * tan(k_m);
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = (2.0 / ((k_m / t) * (k_m / t))) / (t_1 * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.2%

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

      1. *-commutative 24.2%

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

      2. associate-/r* 24.2%

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

   3. Simplified 31.8%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

      3. sqrt-div 13.6%

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

      4. sqrt-pow1 21.1%

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

      5. metadata-eval 21.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 16.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 28.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 28.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. unpow2 28.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 28.3%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 43.4%

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

      1. add-log-exp 37.8%

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

      2. *-commutative 37.8%

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

      3. exp-prod 35.0%

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

      4. pow2 35.0%

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

      5. associate-/r* 35.0%

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

      6. associate-*r* 35.0%

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

      7. *-commutative 35.0%

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

   6. Applied egg-rr 35.0%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. associate-/r* 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) / (t * (sin(k_m) * tan(k_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

   1. add-log-exp 36.2%

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

   2. *-commutative 36.2%

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

   3. exp-prod 39.5%

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

   4. pow2 39.5%

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

   5. associate-/r* 39.5%

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

   6. associate-*r* 39.5%

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

   7. *-commutative 39.5%

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

6. Applied egg-rr 39.5%

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

7. Taylor expanded in t around 0 70.1%

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

   1. associate-/r* 72.0%

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

9. Simplified 72.0%

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

Alternative 17: 46.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 32000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{e^{\log t + 4 \cdot \log k\_m}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 32000.0)
   (* (* l l) (/ 2.0 (exp (+ (log t) (* 4.0 (log k_m))))))
   (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow k_m 2.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 32000.0) {
		tmp = (l * l) * (2.0 / exp((log(t) + (4.0 * log(k_m)))));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 32000.0d0) then
        tmp = (l * l) * (2.0d0 / exp((log(t) + (4.0d0 * log(k_m)))))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (k_m ** 2.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 32000.0) {
		tmp = (l * l) * (2.0 / Math.exp((Math.log(t) + (4.0 * Math.log(k_m)))));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 32000.0:
		tmp = (l * l) * (2.0 / math.exp((math.log(t) + (4.0 * math.log(k_m)))))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 32000.0)
		tmp = Float64(Float64(l * l) * Float64(2.0 / exp(Float64(log(t) + Float64(4.0 * log(k_m))))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 32000.0)
		tmp = (l * l) * (2.0 / exp((log(t) + (4.0 * log(k_m)))));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (k_m ^ 2.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 32000.0], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Exp[N[(N[Log[t], $MachinePrecision] + N[(4.0 * N[Log[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 32000

   1. Initial program 32.0%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in k around 0 59.8%

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

      1. add-log-exp 30.9%

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

      2. *-commutative 30.9%

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

      3. exp-prod 28.4%

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

   7. Applied egg-rr 28.4%

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

      1. pow-exp 30.9%

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

      2. rem-log-exp 59.8%

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

      3. add-exp-log 28.9%

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

      4. pow-to-exp 8.5%

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

      5. prod-exp 9.4%

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

      6. rem-log-exp 8.5%

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

      7. pow-to-exp 28.9%

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

      8. log-pow 9.4%

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

   9. Applied egg-rr 9.4%

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

   if 32000 < k

   1. Initial program 37.4%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 20.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 20.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2} \cdot {\ell}^{2}}{t}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{4}} \]

      2. associate-/l* 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{{k}^{2} \cdot \frac{{\ell}^{2}}{t}}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      3. associate-*r/ 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}\right)}{{k}^{4}} \]

      4. *-commutative 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \frac{\color{blue}{{\ell}^{2} \cdot 2}}{t}\right)}{{k}^{4}} \]

      5. associate-/l* 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{{\ell}^{2} \cdot \frac{2}{t}}\right)}{{k}^{4}} \]

   7. Simplified 23.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, {\ell}^{2} \cdot \frac{2}{t}\right)}{{k}^{4}}} \]

   8. Taylor expanded in k around inf 61.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/r* 61.1%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

   10. Simplified 61.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 32000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{e^{\log t + 4 \cdot \log k}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) * (2.0d0 / ((k_m * k_m) * ((t * (sin(k_m) ** 2.0d0)) / cos(k_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around 0 70.1%

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

   1. associate-/l* 70.1%

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

7. Simplified 70.1%

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

   1. unpow2 70.1%

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

9. Applied egg-rr 70.1%

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

10. Final simplification 70.1%

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

Alternative 19: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \frac{t \cdot {k\_m}^{2}}{\cos k\_m}} \end{array} \]

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * ((t * (k_m ** 2.0d0)) / cos(k_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

5. Taylor expanded in t around 0 70.1%

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

   1. associate-/l* 70.1%

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

7. Simplified 70.1%

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

8. Taylor expanded in k around 0 62.3%

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

9. Final simplification 62.3%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{\cos k}} \]
10. Add Preprocessing

Alternative 20: 64.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k\_m}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e+14)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k_m 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k_m 4.0)))
   (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow k_m 2.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e+14) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k_m, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d+14) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k_m ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k_m ** 4.0d0))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (k_m ** 2.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e+14) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k_m, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5e+14:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k_m, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k_m, 4.0))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e+14)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k_m ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k_m ^ 4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e+14)
		tmp = (l * l) * (((-0.3333333333333333 * ((k_m ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k_m ^ 4.0));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (k_m ^ 2.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e+14], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5e14

   1. Initial program 32.0%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 54.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5e14 < k

   1. Initial program 37.6%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in k around 0 20.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 20.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2} \cdot {\ell}^{2}}{t}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{4}} \]

      2. associate-/l* 23.8%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{{k}^{2} \cdot \frac{{\ell}^{2}}{t}}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      3. associate-*r/ 23.8%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}\right)}{{k}^{4}} \]

      4. *-commutative 23.8%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \frac{\color{blue}{{\ell}^{2} \cdot 2}}{t}\right)}{{k}^{4}} \]

      5. associate-/l* 23.8%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{{\ell}^{2} \cdot \frac{2}{t}}\right)}{{k}^{4}} \]

   7. Simplified 23.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, {\ell}^{2} \cdot \frac{2}{t}\right)}{{k}^{4}}} \]

   8. Taylor expanded in k around inf 60.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/r* 60.5%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

   10. Simplified 60.5%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 32000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 32000.0)
   (* (* l l) (/ (/ 2.0 t) (pow k_m 4.0)))
   (* -0.3333333333333333 (/ (/ (pow l 2.0) (pow k_m 2.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 32000.0) {
		tmp = (l * l) * ((2.0 / t) / pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((pow(l, 2.0) / pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 32000.0d0) then
        tmp = (l * l) * ((2.0d0 / t) / (k_m ** 4.0d0))
    else
        tmp = (-0.3333333333333333d0) * (((l ** 2.0d0) / (k_m ** 2.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 32000.0) {
		tmp = (l * l) * ((2.0 / t) / Math.pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 32000.0:
		tmp = (l * l) * ((2.0 / t) / math.pow(k_m, 4.0))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 32000.0)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / t) / (k_m ^ 4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 32000.0)
		tmp = (l * l) * ((2.0 / t) / (k_m ^ 4.0));
	else
		tmp = -0.3333333333333333 * (((l ^ 2.0) / (k_m ^ 2.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 32000.0], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 32000

   1. Initial program 32.0%

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

   3. Simplified 37.2%

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

      1. unpow2 37.2%

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

      2. clear-num 37.2%

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

      3. frac-times 30.9%

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

      4. *-un-lft-identity 30.9%

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

   6. Applied egg-rr 30.9%

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

   7. Taylor expanded in k around 0 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-/r* 59.8%

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

   9. Simplified 59.8%

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

   if 32000 < k

   1. Initial program 37.4%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 20.1%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 20.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2} \cdot {\ell}^{2}}{t}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{4}} \]

      2. associate-/l* 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{{k}^{2} \cdot \frac{{\ell}^{2}}{t}}, 2 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      3. associate-*r/ 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}\right)}{{k}^{4}} \]

      4. *-commutative 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \frac{\color{blue}{{\ell}^{2} \cdot 2}}{t}\right)}{{k}^{4}} \]

      5. associate-/l* 23.0%

         \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, \color{blue}{{\ell}^{2} \cdot \frac{2}{t}}\right)}{{k}^{4}} \]

   7. Simplified 23.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, {k}^{2} \cdot \frac{{\ell}^{2}}{t}, {\ell}^{2} \cdot \frac{2}{t}\right)}{{k}^{4}}} \]

   8. Taylor expanded in k around inf 61.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/r* 61.1%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

   10. Simplified 61.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 32000:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 63.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{k\_m}^{4}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* l l) (/ (/ 2.0 t) (pow k_m 4.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * ((2.0 / t) / pow(k_m, 4.0));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) * ((2.0d0 / t) / (k_m ** 4.0d0))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * l) * ((2.0 / t) / Math.pow(k_m, 4.0));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * l) * ((2.0 / t) / math.pow(k_m, 4.0))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * l) * Float64(Float64(2.0 / t) / (k_m ^ 4.0)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * l) * ((2.0 / t) / (k_m ^ 4.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

   1. unpow2 40.4%

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

   2. clear-num 40.4%

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

   3. frac-times 33.7%

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

   4. *-un-lft-identity 33.7%

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

6. Applied egg-rr 33.7%

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

7. Taylor expanded in k around 0 59.2%

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

   1. *-commutative 59.2%

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

   2. associate-/r* 59.2%

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

9. Simplified 59.2%

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

10. Final simplification 59.2%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{k}^{4}} \]
11. Add Preprocessing

Alternative 23: 63.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k\_m}^{4}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* l l) (/ 2.0 (* t (pow k_m 4.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * (2.0 / (t * pow(k_m, 4.0)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) * (2.0d0 / (t * (k_m ** 4.0d0)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * l) * (2.0 / (t * Math.pow(k_m, 4.0)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * l) * (2.0 / (t * math.pow(k_m, 4.0)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k_m ^ 4.0))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * l) * (2.0 / (t * (k_m ^ 4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

5. Taylor expanded in k around 0 59.2%

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

6. Final simplification 59.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 24: 41.9% accurate, 35.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-5) (* (* l l) (/ 2.0 0.0)) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-5) {
		tmp = (l * l) * (2.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.15d-5) then
        tmp = (l * l) * (2.0d0 / 0.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-5) {
		tmp = (l * l) * (2.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-5:
		tmp = (l * l) * (2.0 / 0.0)
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-5)
		tmp = Float64(Float64(l * l) * Float64(2.0 / 0.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.15e-5)
		tmp = (l * l) * (2.0 / 0.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-5], N[(N[(l * l), $MachinePrecision] * N[(2.0 / 0.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-5

   1. Initial program 32.0%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in k around 0 59.8%

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

      1. add-log-exp 30.9%

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

      2. *-commutative 30.9%

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

      3. exp-prod 28.4%

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

   7. Applied egg-rr 28.4%

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

   8. Taylor expanded in t around 0 20.3%

      \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]

   if 1.15e-5 < k

   1. Initial program 37.4%

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

   3. Simplified 49.2%

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

      1. add-log-exp 45.0%

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

      2. *-commutative 45.0%

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

      3. exp-prod 57.6%

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

      4. pow2 57.6%

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

      5. associate-/r* 57.6%

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

      6. associate-*r* 57.6%

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

      7. *-commutative 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in l around 0 55.2%

      \[\leadsto \log \color{blue}{1} \]
   8. Step-by-step derivation

      1. metadata-eval 55.2%

         \[\leadsto \color{blue}{0} \]

   9. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 29.7% accurate, 421.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 0 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 0.0)

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 0.0;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 0.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 0.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 0.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return 0.0
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 0.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := 0.0

Derivation

1. Initial program 33.4%

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

3. Simplified 40.4%

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

   1. add-log-exp 36.2%

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

   2. *-commutative 36.2%

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

   3. exp-prod 39.5%

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

   4. pow2 39.5%

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

   5. associate-/r* 39.5%

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

   6. associate-*r* 39.5%

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

   7. *-commutative 39.5%

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

6. Applied egg-rr 39.5%

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

7. Taylor expanded in l around 0 26.6%

   \[\leadsto \log \color{blue}{1} \]
8. Step-by-step derivation

   1. metadata-eval 26.6%

      \[\leadsto \color{blue}{0} \]

9. Applied egg-rr 26.6%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))