Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 83.5%

Time: 23.4s

Alternatives: 21

Speedup: 1.9×

• 26.5% 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: 55.1% 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: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 230000:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{\sin k\_m} \cdot \sqrt[3]{\tan k\_m \cdot t\_1}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 8.5 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{t\_1 \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k_m t) 2.0))))
   (if (<= k_m 230000.0)
     (/
      2.0
      (pow
       (*
        (* t (pow (cbrt l) -2.0))
        (* (cbrt (sin k_m)) (cbrt (* (tan k_m) t_1))))
       3.0))
     (if (<= k_m 8.5e+149)
       (*
        2.0
        (*
         (pow l 2.0)
         (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0)))))
       (/
        2.0
        (pow
         (* (/ t (pow (cbrt l) 2.0)) (cbrt (* t_1 (* (sin k_m) (tan k_m)))))
         3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 + pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 230000.0) {
		tmp = 2.0 / pow(((t * pow(cbrt(l), -2.0)) * (cbrt(sin(k_m)) * cbrt((tan(k_m) * t_1)))), 3.0);
	} else if (k_m <= 8.5e+149) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((t_1 * (sin(k_m) * tan(k_m))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 2.0 + Math.pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 230000.0) {
		tmp = 2.0 / Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt(Math.sin(k_m)) * Math.cbrt((Math.tan(k_m) * t_1)))), 3.0);
	} else if (k_m <= 8.5e+149) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((t_1 * (Math.sin(k_m) * Math.tan(k_m))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 + (Float64(k_m / t) ^ 2.0))
	tmp = 0.0
	if (k_m <= 230000.0)
		tmp = Float64(2.0 / (Float64(Float64(t * (cbrt(l) ^ -2.0)) * Float64(cbrt(sin(k_m)) * cbrt(Float64(tan(k_m) * t_1)))) ^ 3.0));
	elseif (k_m <= 8.5e+149)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(t_1 * Float64(sin(k_m) * tan(k_m))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 230000.0], N[(2.0 / N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.5e+149], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.3e5

   1. Initial program 59.9%

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

   3. Simplified 59.9%

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

      1. add-cube-cbrt 59.8%

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

      2. pow3 59.8%

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

      3. *-commutative 59.8%

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

      4. cbrt-prod 59.8%

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

      5. cbrt-div 60.7%

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

      6. rem-cbrt-cube 69.5%

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

      7. cbrt-prod 81.8%

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

      8. pow2 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

   8. Simplified 81.8%

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

      1. associate-*l/ 81.8%

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

      2. add-cbrt-cube 68.1%

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

      3. unpow3 68.1%

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

      4. cbrt-unprod 65.7%

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

      5. unpow2 65.7%

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

      6. cbrt-prod 60.8%

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

      7. cbrt-div 60.7%

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

      8. associate-*l/ 59.8%

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

      9. pow1 59.8%

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

   10. Applied egg-rr 81.8%

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

       1. unpow1 81.8%

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

       2. associate-*r* 81.8%

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

       3. *-commutative 81.8%

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

       4. *-commutative 81.8%

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

   12. Simplified 81.8%

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

       1. pow1/3 45.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{{\sin k}^{0.3333333333333333}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. add-cube-cbrt 45.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot {\color{blue}{\left(\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}\right)}}^{0.3333333333333333}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       3. unpow-prod-down 45.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       4. pow2 45.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\color{blue}{\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       5. pow1/3 80.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   14. Applied egg-rr 80.7%

       \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   15. Step-by-step derivation

       1. *-commutative 80.7%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot {\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. unpow1/3 81.7%

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

   16. Simplified 81.7%

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

       1. add-cube-cbrt 81.6%

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

       2. pow3 81.6%

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

   18. Applied egg-rr 87.6%

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

       1. associate-*l* 87.7%

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

       2. *-commutative 87.7%

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

   20. Simplified 87.7%

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

   if 2.3e5 < k < 8.49999999999999956e149

   1. Initial program 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 92.5%

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

      1. pow2 92.5%

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

      2. associate-/l* 92.4%

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

      3. pow2 92.4%

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

      4. associate-*r* 92.7%

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

   7. Applied egg-rr 92.7%

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

   if 8.49999999999999956e149 < k

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/r* 64.4%

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

      3. associate-+r+ 64.4%

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

      4. metadata-eval 64.4%

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

      5. associate-*l* 64.4%

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

      6. add-cube-cbrt 64.3%

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

      7. pow3 64.3%

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

   6. Applied egg-rr 82.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 230000:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := 2 + {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot t\_2\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k\_m}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 14:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right) \cdot \sqrt{t\_1}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{t\_2 \cdot t\_1}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))) (t_2 (+ 2.0 (pow (/ k_m t) 2.0))))
   (if (<= k_m 2e-147)
     (/
      2.0
      (*
       (* (tan k_m) t_2)
       (pow (* t (* (pow (cbrt l) -2.0) (cbrt (sin k_m)))) 3.0)))
     (if (<= k_m 14.0)
       (/
        2.0
        (pow
         (* (/ (pow t 1.5) l) (* (hypot 1.0 (hypot 1.0 (/ k_m t))) (sqrt t_1)))
         2.0))
       (if (<= k_m 1.7e+150)
         (*
          2.0
          (*
           (pow l 2.0)
           (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0)))))
         (/ 2.0 (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (* t_2 t_1))) 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 = 2.0 + pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((tan(k_m) * t_2) * pow((t * (pow(cbrt(l), -2.0) * cbrt(sin(k_m)))), 3.0));
	} else if (k_m <= 14.0) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t))) * sqrt(t_1))), 2.0);
	} else if (k_m <= 1.7e+150) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((t_2 * t_1))), 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 = 2.0 + Math.pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((Math.tan(k_m) * t_2) * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.sin(k_m)))), 3.0));
	} else if (k_m <= 14.0) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t))) * Math.sqrt(t_1))), 2.0);
	} else if (k_m <= 1.7e+150) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((t_2 * t_1))), 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(2.0 + (Float64(k_m / t) ^ 2.0))
	tmp = 0.0
	if (k_m <= 2e-147)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * t_2) * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(sin(k_m)))) ^ 3.0)));
	elseif (k_m <= 14.0)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t))) * sqrt(t_1))) ^ 2.0));
	elseif (k_m <= 1.7e+150)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(t_2 * t_1))) ^ 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[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2e-147], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 14.0], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.7e+150], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$2 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.9999999999999999e-147

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.4%

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

      5. cbrt-div 59.9%

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

      6. rem-cbrt-cube 67.8%

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

      7. cbrt-prod 80.3%

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

      8. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

   8. Simplified 80.3%

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

      1. associate-*l/ 80.3%

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

      2. add-cbrt-cube 68.3%

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

      3. unpow3 68.4%

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

      4. cbrt-unprod 65.6%

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

      5. unpow2 65.6%

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

      6. cbrt-prod 59.9%

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

      7. cbrt-div 59.9%

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

      8. associate-*l/ 59.4%

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

      9. pow1 59.4%

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

   10. Applied egg-rr 80.3%

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

       1. unpow1 80.3%

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

       2. associate-*r* 80.3%

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

       3. *-commutative 80.3%

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

       4. *-commutative 80.3%

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

   12. Simplified 80.3%

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

       1. pow1/3 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{{\sin k}^{0.3333333333333333}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. add-cube-cbrt 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot {\color{blue}{\left(\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}\right)}}^{0.3333333333333333}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       3. unpow-prod-down 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       4. pow2 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\color{blue}{\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       5. pow1/3 79.2%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   14. Applied egg-rr 79.2%

       \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   15. Step-by-step derivation

       1. *-commutative 79.2%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot {\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. unpow1/3 80.2%

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

   16. Simplified 80.2%

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

       1. add-cube-cbrt 80.1%

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

       2. pow3 80.1%

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

   18. Applied egg-rr 86.5%

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

       1. *-commutative 86.5%

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

       2. cube-prod 80.2%

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

       3. rem-cube-cbrt 80.3%

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

       4. *-commutative 80.3%

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

       5. associate-*l* 80.3%

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

   20. Simplified 80.3%

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

   if 1.9999999999999999e-147 < k < 14

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   6. Applied egg-rr 32.1%

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

   if 14 < k < 1.69999999999999991e150

   1. Initial program 43.9%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. pow2 89.5%

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

      2. associate-/l* 89.4%

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

      3. pow2 89.4%

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

      4. associate-*r* 89.7%

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

   7. Applied egg-rr 89.7%

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

   if 1.69999999999999991e150 < k

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/r* 64.4%

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

      3. associate-+r+ 64.4%

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

      4. metadata-eval 64.4%

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

      5. associate-*l* 64.4%

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

      6. add-cube-cbrt 64.3%

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

      7. pow3 64.3%

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

   6. Applied egg-rr 82.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 14:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ t_2 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 1.55 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k\_m}\right)\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(t\_2 + 1\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 46:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right) \cdot \sqrt{t\_1}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 3.7 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + t\_2\right) \cdot t\_1}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))) (t_2 (pow (/ k_m t) 2.0)))
   (if (<= k_m 1.55e-159)
     (/
      2.0
      (*
       (pow (* (pow (cbrt l) -2.0) (* t (cbrt (sin k_m)))) 3.0)
       (* (tan k_m) (+ 1.0 (+ t_2 1.0)))))
     (if (<= k_m 46.0)
       (/
        2.0
        (pow
         (* (/ (pow t 1.5) l) (* (hypot 1.0 (hypot 1.0 (/ k_m t))) (sqrt t_1)))
         2.0))
       (if (<= k_m 3.7e+148)
         (*
          2.0
          (*
           (pow l 2.0)
           (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0)))))
         (/
          2.0
          (pow
           (* (/ t (pow (cbrt l) 2.0)) (cbrt (* (+ 2.0 t_2) t_1)))
           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 = pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 1.55e-159) {
		tmp = 2.0 / (pow((pow(cbrt(l), -2.0) * (t * cbrt(sin(k_m)))), 3.0) * (tan(k_m) * (1.0 + (t_2 + 1.0))));
	} else if (k_m <= 46.0) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t))) * sqrt(t_1))), 2.0);
	} else if (k_m <= 3.7e+148) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt(((2.0 + t_2) * t_1))), 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.pow((k_m / t), 2.0);
	double tmp;
	if (k_m <= 1.55e-159) {
		tmp = 2.0 / (Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt(Math.sin(k_m)))), 3.0) * (Math.tan(k_m) * (1.0 + (t_2 + 1.0))));
	} else if (k_m <= 46.0) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t))) * Math.sqrt(t_1))), 2.0);
	} else if (k_m <= 3.7e+148) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(((2.0 + t_2) * t_1))), 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(k_m / t) ^ 2.0
	tmp = 0.0
	if (k_m <= 1.55e-159)
		tmp = Float64(2.0 / Float64((Float64((cbrt(l) ^ -2.0) * Float64(t * cbrt(sin(k_m)))) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(t_2 + 1.0)))));
	elseif (k_m <= 46.0)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t))) * sqrt(t_1))) ^ 2.0));
	elseif (k_m <= 3.7e+148)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(Float64(2.0 + t_2) * t_1))) ^ 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[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 1.55e-159], N[(2.0 / N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 46.0], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.7e+148], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.55e-159

   1. Initial program 59.2%

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

   3. Simplified 59.3%

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

      1. add-cube-cbrt 59.2%

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

      2. pow3 59.2%

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

      3. *-commutative 59.2%

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

      4. cbrt-prod 59.2%

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

      5. cbrt-div 59.7%

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

      6. rem-cbrt-cube 67.6%

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

      7. cbrt-prod 80.2%

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

      8. pow2 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

   8. Simplified 80.2%

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

      1. associate-*l/ 80.2%

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

      2. add-cbrt-cube 68.2%

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

      3. unpow3 68.2%

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

      4. cbrt-unprod 65.4%

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

      5. unpow2 65.4%

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

      6. cbrt-prod 59.7%

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

      7. cbrt-div 59.7%

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

      8. associate-*l/ 59.2%

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

      9. pow1 59.2%

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

   10. Applied egg-rr 80.2%

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

       1. unpow1 80.2%

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

       2. associate-*r* 80.2%

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

       3. *-commutative 80.2%

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

       4. *-commutative 80.2%

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

   12. Simplified 80.2%

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

   if 1.55e-159 < k < 46

   1. Initial program 66.2%

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

   3. Simplified 66.2%

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

   6. Applied egg-rr 34.7%

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

   if 46 < k < 3.7000000000000002e148

   1. Initial program 43.9%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. pow2 89.5%

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

      2. associate-/l* 89.4%

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

      3. pow2 89.4%

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

      4. associate-*r* 89.7%

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

   7. Applied egg-rr 89.7%

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

   if 3.7000000000000002e148 < k

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/r* 64.4%

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

      3. associate-+r+ 64.4%

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

      4. metadata-eval 64.4%

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

      5. associate-*l* 64.4%

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

      6. add-cube-cbrt 64.3%

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

      7. pow3 64.3%

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

   6. Applied egg-rr 82.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{elif}\;k \leq 46:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(1 + \left({\left(\frac{k\_m}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k\_m}\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 46:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right) \cdot \sqrt{\sin k\_m \cdot \tan k\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e-146)
   (/
    2.0
    (*
     (* (tan k_m) (+ 1.0 (+ (pow (/ k_m t) 2.0) 1.0)))
     (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt k_m)) 3.0)))
   (if (<= k_m 46.0)
     (/
      2.0
      (pow
       (*
        (/ (pow t 1.5) l)
        (* (hypot 1.0 (hypot 1.0 (/ k_m t))) (sqrt (* (sin k_m) (tan k_m)))))
       2.0))
     (*
      2.0
      (*
       (pow l 2.0)
       (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-146) {
		tmp = 2.0 / ((tan(k_m) * (1.0 + (pow((k_m / t), 2.0) + 1.0))) * pow(((t / pow(cbrt(l), 2.0)) * cbrt(k_m)), 3.0));
	} else if (k_m <= 46.0) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t))) * sqrt((sin(k_m) * tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-146) {
		tmp = 2.0 / ((Math.tan(k_m) * (1.0 + (Math.pow((k_m / t), 2.0) + 1.0))) * Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k_m)), 3.0));
	} else if (k_m <= 46.0) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t))) * Math.sqrt((Math.sin(k_m) * Math.tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e-146)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(1.0 + Float64((Float64(k_m / t) ^ 2.0) + 1.0))) * (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(k_m)) ^ 3.0)));
	elseif (k_m <= 46.0)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t))) * sqrt(Float64(sin(k_m) * tan(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e-146], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 46.0], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.29999999999999993e-146

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.4%

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

      5. cbrt-div 59.9%

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

      6. rem-cbrt-cube 67.8%

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

      7. cbrt-prod 80.3%

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

      8. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

   8. Simplified 80.3%

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

   9. Taylor expanded in k around 0 78.9%

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

   if 1.29999999999999993e-146 < k < 46

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   6. Applied egg-rr 32.1%

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

   if 46 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. pow2 80.9%

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

      2. associate-/l* 80.9%

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

      3. pow2 80.9%

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

      4. associate-*r* 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 46:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k\_m}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 13.8:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right) \cdot \sqrt{\sin k\_m \cdot \tan k\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-147)
   (/
    2.0
    (*
     (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
     (pow (* t (* (pow (cbrt l) -2.0) (cbrt (sin k_m)))) 3.0)))
   (if (<= k_m 13.8)
     (/
      2.0
      (pow
       (*
        (/ (pow t 1.5) l)
        (* (hypot 1.0 (hypot 1.0 (/ k_m t))) (sqrt (* (sin k_m) (tan k_m)))))
       2.0))
     (*
      2.0
      (*
       (pow l 2.0)
       (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * pow((t * (pow(cbrt(l), -2.0) * cbrt(sin(k_m)))), 3.0));
	} else if (k_m <= 13.8) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t))) * sqrt((sin(k_m) * tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.sin(k_m)))), 3.0));
	} else if (k_m <= 13.8) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t))) * Math.sqrt((Math.sin(k_m) * Math.tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-147)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(sin(k_m)))) ^ 3.0)));
	elseif (k_m <= 13.8)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t))) * sqrt(Float64(sin(k_m) * tan(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-147], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 13.8], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999999e-147

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.4%

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

      5. cbrt-div 59.9%

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

      6. rem-cbrt-cube 67.8%

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

      7. cbrt-prod 80.3%

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

      8. pow2 80.3%

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

   6. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

   8. Simplified 80.3%

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

      1. associate-*l/ 80.3%

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

      2. add-cbrt-cube 68.3%

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

      3. unpow3 68.4%

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

      4. cbrt-unprod 65.6%

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

      5. unpow2 65.6%

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

      6. cbrt-prod 59.9%

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

      7. cbrt-div 59.9%

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

      8. associate-*l/ 59.4%

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

      9. pow1 59.4%

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

   10. Applied egg-rr 80.3%

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

       1. unpow1 80.3%

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

       2. associate-*r* 80.3%

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

       3. *-commutative 80.3%

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

       4. *-commutative 80.3%

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

   12. Simplified 80.3%

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

       1. pow1/3 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{{\sin k}^{0.3333333333333333}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. add-cube-cbrt 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot {\color{blue}{\left(\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}\right)}}^{0.3333333333333333}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       3. unpow-prod-down 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       4. pow2 39.4%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\color{blue}{\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\sin k}\right)}^{0.3333333333333333}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       5. pow1/3 79.2%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   14. Applied egg-rr 79.2%

       \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left({\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   15. Step-by-step derivation

       1. *-commutative 79.2%

          \[\leadsto \frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot {\left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{0.3333333333333333}\right)}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

       2. unpow1/3 80.2%

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

   16. Simplified 80.2%

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

       1. add-cube-cbrt 80.1%

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

       2. pow3 80.1%

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

   18. Applied egg-rr 86.5%

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

       1. *-commutative 86.5%

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

       2. cube-prod 80.2%

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

       3. rem-cube-cbrt 80.3%

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

       4. *-commutative 80.3%

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

       5. associate-*l* 80.3%

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

   20. Simplified 80.3%

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

   if 1.9999999999999999e-147 < k < 13.800000000000001

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   6. Applied egg-rr 32.1%

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

   if 13.800000000000001 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. pow2 80.9%

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

      2. associate-/l* 80.9%

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

      3. pow2 80.9%

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

      4. associate-*r* 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 13.8:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(1 + \left({\left(\frac{k\_m}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k\_m}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-7)
   (/
    2.0
    (*
     (* (tan k_m) (+ 1.0 (+ (pow (/ k_m t) 2.0) 1.0)))
     (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt k_m)) 3.0)))
   (*
    2.0
    (*
     (pow l 2.0)
     (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-7) {
		tmp = 2.0 / ((tan(k_m) * (1.0 + (pow((k_m / t), 2.0) + 1.0))) * pow(((t / pow(cbrt(l), 2.0)) * cbrt(k_m)), 3.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-7) {
		tmp = 2.0 / ((Math.tan(k_m) * (1.0 + (Math.pow((k_m / t), 2.0) + 1.0))) * Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k_m)), 3.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-7)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(1.0 + Float64((Float64(k_m / t) ^ 2.0) + 1.0))) * (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(k_m)) ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e-7], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999959e-7

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. add-cube-cbrt 60.1%

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

      2. pow3 60.1%

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

      3. *-commutative 60.1%

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

      4. cbrt-prod 60.1%

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

      5. cbrt-div 61.0%

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

      6. rem-cbrt-cube 69.9%

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

      7. cbrt-prod 81.7%

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

      8. pow2 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. *-commutative 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in k around 0 80.5%

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

   if 8.99999999999999959e-7 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. pow2 80.9%

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

      2. associate-/l* 80.9%

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

      3. pow2 80.9%

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

      4. associate-*r* 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-6)
   (/
    2.0
    (*
     (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
     (pow (* (/ (pow t 1.5) l) (sqrt (sin k_m))) 2.0)))
   (*
    2.0
    (*
     (pow l 2.0)
     (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = 2.0 / ((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * pow(((pow(t, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-6) then
        tmp = 2.0d0 / ((tan(k_m) * (2.0d0 + ((k_m / t) ** 2.0d0))) * ((((t ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / ((t * (k_m ** 2.0d0)) * (sin(k_m) ** 2.0d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-6) {
		tmp = 2.0 / ((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-6:
		tmp = 2.0 / ((math.tan(k_m) * (2.0 + math.pow((k_m / t), 2.0))) * math.pow(((math.pow(t, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k_m) / ((t * math.pow(k_m, 2.0)) * math.pow(math.sin(k_m), 2.0))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-6)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * (Float64(Float64((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-6)
		tmp = 2.0 / ((tan(k_m) * (2.0 + ((k_m / t) ^ 2.0))) * ((((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / ((t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.4e-6], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999999e-6

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 29.1%

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

      2. pow2 29.1%

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

      3. *-commutative 29.1%

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

      4. sqrt-prod 15.1%

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

      5. sqrt-div 15.1%

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

      6. sqrt-pow1 17.6%

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

      7. metadata-eval 17.6%

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

      8. sqrt-prod 10.9%

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

      9. add-sqr-sqrt 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. *-commutative 20.2%

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

   12. Simplified 20.2%

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

   if 2.3999999999999999e-6 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. pow2 80.9%

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

      2. associate-/l* 80.9%

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

      3. pow2 80.9%

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

      4. associate-*r* 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-147)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (if (<= k_m 1.9e-5)
     (/
      2.0
      (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k_m 2.0))))) 3.0))
     (*
      2.0
      (*
       (pow l 2.0)
       (/ (cos k_m) (* (* t (pow k_m 2.0)) (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else if (k_m <= 1.9e-5) {
		tmp = 2.0 / pow((t * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k_m, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else if (k_m <= 1.9e-5) {
		tmp = 2.0 / Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k_m, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-147)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	elseif (k_m <= 1.9e-5)
		tmp = Float64(2.0 / (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k_m ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-147], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.9e-5], N[(2.0 / N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999999e-147

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

      1. distribute-lft-in 59.5%

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

      2. *-rgt-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. *-rgt-identity 59.5%

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

      2. distribute-lft-out 59.5%

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

      3. associate-+r+ 59.5%

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

      4. metadata-eval 59.5%

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

   8. Simplified 59.5%

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

      1. add-sqr-sqrt 27.2%

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

      2. pow2 27.2%

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

      3. sqrt-div 27.2%

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

      4. sqrt-pow1 30.2%

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

      5. metadata-eval 30.2%

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

      6. sqrt-prod 18.6%

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

      7. add-sqr-sqrt 34.8%

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

   10. Applied egg-rr 34.8%

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

   11. Taylor expanded in k around 0 32.3%

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

       1. *-commutative 32.3%

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

   13. Simplified 32.3%

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

   if 1.9999999999999999e-147 < k < 1.9000000000000001e-5

   1. Initial program 64.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in k around 0 65.3%

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

      1. add-cube-cbrt 65.3%

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

      2. pow3 65.2%

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

      3. cbrt-prod 65.2%

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

      4. associate-/r* 64.8%

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

      5. cbrt-div 68.5%

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

      6. unpow3 68.5%

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

      7. add-cbrt-cube 87.7%

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

      8. cbrt-prod 95.1%

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

      9. unpow2 95.1%

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

      10. div-inv 95.1%

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

      11. pow-flip 95.1%

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

      12. metadata-eval 95.1%

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

   7. Applied egg-rr 95.1%

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

      1. associate-*l* 95.0%

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

   9. Simplified 95.0%

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

   if 1.9000000000000001e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. pow2 80.9%

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

      2. associate-/l* 80.9%

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

      3. pow2 80.9%

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

      4. associate-*r* 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-147)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (if (<= k_m 5.8e-5)
     (/
      2.0
      (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k_m 2.0))))) 3.0))
     (*
      2.0
      (/
       (* (pow l 2.0) (cos k_m))
       (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else if (k_m <= 5.8e-5) {
		tmp = 2.0 / pow((t * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k_m, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-147) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else if (k_m <= 5.8e-5) {
		tmp = 2.0 / Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k_m, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-147)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	elseif (k_m <= 5.8e-5)
		tmp = Float64(2.0 / (Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k_m ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-147], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.8e-5], N[(2.0 / N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999999e-147

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

      1. distribute-lft-in 59.5%

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

      2. *-rgt-identity 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. *-rgt-identity 59.5%

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

      2. distribute-lft-out 59.5%

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

      3. associate-+r+ 59.5%

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

      4. metadata-eval 59.5%

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

   8. Simplified 59.5%

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

      1. add-sqr-sqrt 27.2%

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

      2. pow2 27.2%

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

      3. sqrt-div 27.2%

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

      4. sqrt-pow1 30.2%

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

      5. metadata-eval 30.2%

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

      6. sqrt-prod 18.6%

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

      7. add-sqr-sqrt 34.8%

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

   10. Applied egg-rr 34.8%

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

   11. Taylor expanded in k around 0 32.3%

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

       1. *-commutative 32.3%

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

   13. Simplified 32.3%

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

   if 1.9999999999999999e-147 < k < 5.8e-5

   1. Initial program 64.8%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in k around 0 65.3%

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

      1. add-cube-cbrt 65.3%

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

      2. pow3 65.2%

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

      3. cbrt-prod 65.2%

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

      4. associate-/r* 64.8%

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

      5. cbrt-div 68.5%

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

      6. unpow3 68.5%

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

      7. add-cbrt-cube 87.7%

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

      8. cbrt-prod 95.1%

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

      9. unpow2 95.1%

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

      10. div-inv 95.1%

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

      11. pow-flip 95.1%

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

      12. metadata-eval 95.1%

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

   7. Applied egg-rr 95.1%

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

      1. associate-*l* 95.0%

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

   9. Simplified 95.0%

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

   if 5.8e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. unpow2 80.9%

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

      2. sin-mult 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. div-sub 80.9%

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

      2. +-inverses 80.9%

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

      3. cos-0 80.9%

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

      4. metadata-eval 80.9%

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

      5. count-2 80.9%

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

      6. *-commutative 80.9%

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

   9. Simplified 80.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 360000:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \cdot \left(\sin k\_m \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 360000.0)
   (/
    2.0
    (*
     (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
     (* (sin k_m) (* (/ (pow t 2.0) l) (/ t l)))))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k_m))
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 360000.0) {
		tmp = 2.0 / ((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * (sin(k_m) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 360000.0d0) then
        tmp = 2.0d0 / ((tan(k_m) * (2.0d0 + ((k_m / t) ** 2.0d0))) * (sin(k_m) * (((t ** 2.0d0) / l) * (t / l))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k_m)) / ((k_m ** 2.0d0) * (t * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 360000.0) {
		tmp = 2.0 / ((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * (Math.sin(k_m) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 360000.0:
		tmp = 2.0 / ((math.tan(k_m) * (2.0 + math.pow((k_m / t), 2.0))) * (math.sin(k_m) * ((math.pow(t, 2.0) / l) * (t / l))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k_m)) / (math.pow(k_m, 2.0) * (t * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 360000.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(sin(k_m) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 360000.0)
		tmp = 2.0 / ((tan(k_m) * (2.0 + ((k_m / t) ^ 2.0))) * (sin(k_m) * (((t ^ 2.0) / l) * (t / l))));
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k_m)) / ((k_m ^ 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 360000.0], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.6e5

   1. Initial program 59.9%

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

   3. Simplified 59.9%

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

      1. distribute-lft-in 59.9%

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

      2. *-rgt-identity 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. *-rgt-identity 59.9%

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

      2. distribute-lft-out 59.9%

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

      3. associate-+r+ 59.9%

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

      4. metadata-eval 59.9%

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

   8. Simplified 59.9%

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

      1. unpow3 59.9%

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

      2. times-frac 72.7%

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

      3. pow2 72.7%

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

   10. Applied egg-rr 72.7%

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

   if 3.6e5 < k

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in t around 0 82.2%

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

      1. unpow2 82.2%

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

      2. sin-mult 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. div-sub 82.1%

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

      2. +-inverses 82.1%

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

      3. cos-0 82.1%

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

      4. metadata-eval 82.1%

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

      5. count-2 82.1%

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

      6. *-commutative 82.1%

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

   9. Simplified 82.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 360000:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.6e-7)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (*
    2.0
    (*
     (/ (pow l 2.0) (pow k_m 2.0))
     (/ (cos k_m) (* t (- 0.5 (* 0.5 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-7) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t * (0.5 - (0.5 * cos((k_m * 2.0)))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.6d-7) then
        tmp = 2.0d0 / ((sin(k_m) * (((t ** 1.5d0) / l) ** 2.0d0)) * (k_m * 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) * (cos(k_m) / (t * (0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.6e-7) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * (0.5 - (0.5 * Math.cos((k_m * 2.0)))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.6e-7:
		tmp = 2.0 / ((math.sin(k_m) * math.pow((math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) * (math.cos(k_m) / (t * (0.5 - (0.5 * math.cos((k_m * 2.0)))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.6e-7)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0))))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.6e-7)
		tmp = 2.0 / ((sin(k_m) * (((t ^ 1.5) / l) ^ 2.0)) * (k_m * 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) * (cos(k_m) / (t * (0.5 - (0.5 * cos((k_m * 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.6e-7], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 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[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999999e-7

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

      3. sqrt-div 26.8%

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

      4. sqrt-pow1 29.4%

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

      5. metadata-eval 29.4%

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

      6. sqrt-prod 17.8%

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

      7. add-sqr-sqrt 33.5%

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

   10. Applied egg-rr 33.5%

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

   11. Taylor expanded in k around 0 31.4%

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

       1. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 2.59999999999999999e-7 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. unpow2 80.9%

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

      2. sin-mult 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. div-sub 80.9%

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

      2. +-inverses 80.9%

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

      3. cos-0 80.9%

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

      4. metadata-eval 80.9%

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

      5. count-2 80.9%

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

      6. *-commutative 80.9%

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

   9. Simplified 80.9%

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

       1. times-frac 76.5%

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

       2. div-inv 76.5%

          \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot \left(0.5 - \color{blue}{\cos \left(k \cdot 2\right) \cdot \frac{1}{2}}\right)}\right) \]

       3. metadata-eval 76.5%

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

   11. Applied egg-rr 76.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{\left(t \cdot {k\_m}^{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.8e-5)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (*
    2.0
    (*
     (pow l 2.0)
     (/
      (cos k_m)
      (* (* t (pow k_m 2.0)) (- 0.5 (* 0.5 (cos (* k_m 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-5) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / ((t * pow(k_m, 2.0)) * (0.5 - (0.5 * cos((k_m * 2.0)))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-5) then
        tmp = 2.0d0 / ((sin(k_m) * (((t ** 1.5d0) / l) ** 2.0d0)) * (k_m * 2.0d0))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / ((t * (k_m ** 2.0d0)) * (0.5d0 - (0.5d0 * cos((k_m * 2.0d0)))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-5) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / ((t * Math.pow(k_m, 2.0)) * (0.5 - (0.5 * Math.cos((k_m * 2.0)))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.8e-5:
		tmp = 2.0 / ((math.sin(k_m) * math.pow((math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k_m) / ((t * math.pow(k_m, 2.0)) * (0.5 - (0.5 * math.cos((k_m * 2.0)))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-5)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(Float64(t * (k_m ^ 2.0)) * Float64(0.5 - Float64(0.5 * cos(Float64(k_m * 2.0))))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-5)
		tmp = 2.0 / ((sin(k_m) * (((t ^ 1.5) / l) ^ 2.0)) * (k_m * 2.0));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / ((t * (k_m ^ 2.0)) * (0.5 - (0.5 * cos((k_m * 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.8e-5], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000005e-5

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

      3. sqrt-div 26.8%

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

      4. sqrt-pow1 29.4%

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

      5. metadata-eval 29.4%

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

      6. sqrt-prod 17.8%

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

      7. add-sqr-sqrt 33.5%

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

   10. Applied egg-rr 33.5%

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

   11. Taylor expanded in k around 0 31.4%

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

       1. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 1.80000000000000005e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. unpow2 80.9%

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

      2. sin-mult 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. div-sub 80.9%

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

      2. +-inverses 80.9%

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

      3. cos-0 80.9%

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

      4. metadata-eval 80.9%

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

      5. count-2 80.9%

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

      6. *-commutative 80.9%

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

   9. Simplified 80.9%

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

       1. associate-/l* 80.8%

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

       2. associate-*r* 80.9%

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

       3. div-inv 80.9%

          \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left({k}^{2} \cdot t\right) \cdot \left(0.5 - \color{blue}{\cos \left(k \cdot 2\right) \cdot \frac{1}{2}}\right)}\right) \]

       4. metadata-eval 80.9%

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

   11. Applied egg-rr 80.9%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-6)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k_m))
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-6) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9d-6) then
        tmp = 2.0d0 / ((sin(k_m) * (((t ** 1.5d0) / l) ** 2.0d0)) * (k_m * 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k_m)) / ((k_m ** 2.0d0) * (t * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-6) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9e-6:
		tmp = 2.0 / ((math.sin(k_m) * math.pow((math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k_m)) / (math.pow(k_m, 2.0) * (t * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e-6)
		tmp = 2.0 / ((sin(k_m) * (((t ^ 1.5) / l) ^ 2.0)) * (k_m * 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k_m)) / ((k_m ^ 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e-6], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000023e-6

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

      3. sqrt-div 26.8%

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

      4. sqrt-pow1 29.4%

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

      5. metadata-eval 29.4%

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

      6. sqrt-prod 17.8%

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

      7. add-sqr-sqrt 33.5%

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

   10. Applied egg-rr 33.5%

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

   11. Taylor expanded in k around 0 31.4%

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

       1. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 9.00000000000000023e-6 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. unpow2 80.9%

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

      2. sin-mult 80.9%

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

   7. Applied egg-rr 80.9%

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

      1. div-sub 80.9%

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

      2. +-inverses 80.9%

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

      3. cos-0 80.9%

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

      4. metadata-eval 80.9%

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

      5. count-2 80.9%

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

      6. *-commutative 80.9%

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

   9. Simplified 80.9%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.5e-9)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (*
    2.0
    (* (/ (pow l 2.0) (pow k_m 2.0)) (/ (/ (cos k_m) (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 <= 2.5e-9) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * ((cos(k_m) / 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 <= 2.5d-9) then
        tmp = 2.0d0 / ((sin(k_m) * (((t ** 1.5d0) / l) ** 2.0d0)) * (k_m * 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) * ((cos(k_m) / (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 <= 2.5e-9) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * ((Math.cos(k_m) / 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 <= 2.5e-9:
		tmp = 2.0 / ((math.sin(k_m) * math.pow((math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 2.0)) * ((math.cos(k_m) / 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 <= 2.5e-9)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(Float64(cos(k_m) / (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 <= 2.5e-9)
		tmp = 2.0 / ((sin(k_m) * (((t ^ 1.5) / l) ^ 2.0)) * (k_m * 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) * ((cos(k_m) / (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, 2.5e-9], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 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[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e-9

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

      3. sqrt-div 26.8%

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

      4. sqrt-pow1 29.4%

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

      5. metadata-eval 29.4%

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

      6. sqrt-prod 17.8%

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

      7. add-sqr-sqrt 33.5%

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

   10. Applied egg-rr 33.5%

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

   11. Taylor expanded in k around 0 31.4%

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

       1. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 2.5000000000000001e-9 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

   6. Taylor expanded in k around 0 60.3%

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

      1. times-frac 60.3%

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

      2. *-commutative 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-/r* 60.3%

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

   10. Simplified 60.3%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{\cos k}{{k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k\_m \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{t \cdot {k\_m}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.7e-160)
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t 3.0) (* l l)))))
   (if (<= k_m 1.6e-9)
     (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t 3.0) l)) l))
     (* 2.0 (* (pow l 2.0) (/ (cos k_m) (* t (pow k_m 4.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.7e-160) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t, 3.0) / (l * l))));
	} else if (k_m <= 1.6e-9) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / (t * pow(k_m, 4.0))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.7d-160) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t ** 3.0d0) / (l * l))))
    else if (k_m <= 1.6d-9) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / (t * (k_m ** 4.0d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.7e-160) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))));
	} else if (k_m <= 1.6e-9) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / (t * Math.pow(k_m, 4.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.7e-160:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l))))
	elif k_m <= 1.6e-9:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k_m) / (t * math.pow(k_m, 4.0))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.7e-160)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))));
	elseif (k_m <= 1.6e-9)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(t * (k_m ^ 4.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.7e-160)
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t ^ 3.0) / (l * l))));
	elseif (k_m <= 1.6e-9)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / (t * (k_m ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.7e-160], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6e-9], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-160

   1. Initial program 59.2%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in k around 0 59.6%

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

   if 2.7000000000000001e-160 < k < 1.60000000000000006e-9

   1. Initial program 66.2%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in k around 0 66.6%

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

      1. associate-*l/ 70.4%

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

   7. Applied egg-rr 70.4%

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

   if 1.60000000000000006e-9 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

   6. Taylor expanded in k around 0 60.3%

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

      1. associate-/l* 60.3%

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

      2. associate-*r* 60.3%

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

      3. pow-prod-up 60.3%

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

      4. metadata-eval 60.3%

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

   8. Applied egg-rr 60.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{t \cdot {k\_m}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-8)
   (/ 2.0 (* (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)) (* k_m 2.0)))
   (* 2.0 (* (pow l 2.0) (/ (cos k_m) (* t (pow k_m 4.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-8) {
		tmp = 2.0 / ((sin(k_m) * pow((pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k_m) / (t * pow(k_m, 4.0))));
	}
	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 <= 8.5d-8) then
        tmp = 2.0d0 / ((sin(k_m) * (((t ** 1.5d0) / l) ** 2.0d0)) * (k_m * 2.0d0))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k_m) / (t * (k_m ** 4.0d0))))
    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 <= 8.5e-8) {
		tmp = 2.0 / ((Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k_m) / (t * Math.pow(k_m, 4.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-8:
		tmp = 2.0 / ((math.sin(k_m) * math.pow((math.pow(t, 1.5) / l), 2.0)) * (k_m * 2.0))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k_m) / (t * math.pow(k_m, 4.0))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-8)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(t * (k_m ^ 4.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-8)
		tmp = 2.0 / ((sin(k_m) * (((t ^ 1.5) / l) ^ 2.0)) * (k_m * 2.0));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k_m) / (t * (k_m ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-8], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.49999999999999935e-8

   1. Initial program 60.2%

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

   3. Simplified 60.2%

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

      1. distribute-lft-in 60.2%

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

      2. *-rgt-identity 60.2%

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

   6. Applied egg-rr 60.2%

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

      1. *-rgt-identity 60.2%

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

      2. distribute-lft-out 60.2%

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

      3. associate-+r+ 60.2%

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

      4. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

      3. sqrt-div 26.8%

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

      4. sqrt-pow1 29.4%

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

      5. metadata-eval 29.4%

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

      6. sqrt-prod 17.8%

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

      7. add-sqr-sqrt 33.5%

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

   10. Applied egg-rr 33.5%

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

   11. Taylor expanded in k around 0 31.4%

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

       1. *-commutative 31.4%

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

   13. Simplified 31.4%

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

   if 8.49999999999999935e-8 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

   6. Taylor expanded in k around 0 60.3%

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

      1. associate-/l* 60.3%

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

      2. associate-*r* 60.3%

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

      3. pow-prod-up 60.3%

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

      4. metadata-eval 60.3%

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

   8. Applied egg-rr 60.3%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-35}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.2e-35)
   (* 2.0 (/ (/ (pow l 2.0) t) (pow k_m 4.0)))
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t 3.0) (* l l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.2e-35) {
		tmp = 2.0 * ((pow(l, 2.0) / t) / pow(k_m, 4.0));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 5.2d-35) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t) / (k_m ** 4.0d0))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t ** 3.0d0) / (l * l))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.2e-35) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k_m, 4.0));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5.2e-35:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k_m, 4.0))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.2e-35)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k_m ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 5.2e-35)
		tmp = 2.0 * (((l ^ 2.0) / t) / (k_m ^ 4.0));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t ^ 3.0) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.2e-35], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.20000000000000009e-35

   1. Initial program 56.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 65.8%

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

   6. Taylor expanded in k around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. associate-/r* 55.5%

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

   8. Simplified 55.5%

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

   if 5.20000000000000009e-35 < t

   1. Initial program 65.1%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in k around 0 63.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-35}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k\_m}^{-4}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e-5)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (/ (pow t 3.0) l) l)))
   (* 2.0 (* (/ (pow l 2.0) t) (pow k_m -4.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-5) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 3.0) / l) / l));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t) * pow(k_m, -4.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.2d-5) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 3.0d0) / l) / l))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t) * (k_m ** (-4.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-5) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 3.0) / l) / l));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) * Math.pow(k_m, -4.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.2e-5:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 3.0) / l) / l))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) * math.pow(k_m, -4.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e-5)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) * (k_m ^ -4.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.2e-5)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 3.0) / l) / l));
	else
		tmp = 2.0 * (((l ^ 2.0) / t) * (k_m ^ -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e-5], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999999e-5

   1. Initial program 60.2%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in k around 0 57.3%

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

   if 2.1999999999999999e-5 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

   6. Taylor expanded in k around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/r* 56.8%

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

   8. Simplified 56.8%

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

      1. *-un-lft-identity 56.8%

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

      2. div-inv 56.8%

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

      3. pow-flip 56.8%

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

      4. metadata-eval 56.8%

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

   10. Applied egg-rr 56.8%

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 56.8%

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

   12. Simplified 56.8%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k\_m}^{-4}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-7)
   (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t 3.0) l)) l))
   (* 2.0 (* (/ (pow l 2.0) t) (pow k_m -4.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-7) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / t) * pow(k_m, -4.0));
	}
	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 <= 8.5d-7) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / t) * (k_m ** (-4.0d0)))
    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 <= 8.5e-7) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t) * Math.pow(k_m, -4.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-7:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / t) * math.pow(k_m, -4.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t) * (k_m ^ -4.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-7)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 * (((l ^ 2.0) / t) * (k_m ^ -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-7], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.50000000000000014e-7

   1. Initial program 60.2%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in k around 0 57.3%

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

      1. associate-*l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

   if 8.50000000000000014e-7 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in t around 0 80.9%

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

   6. Taylor expanded in k around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/r* 56.8%

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

   8. Simplified 56.8%

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

      1. *-un-lft-identity 56.8%

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

      2. div-inv 56.8%

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

      3. pow-flip 56.8%

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

      4. metadata-eval 56.8%

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

   10. Applied egg-rr 56.8%

       \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 56.8%

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

   12. Simplified 56.8%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k\_m}^{-4}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (* (/ (pow l 2.0) t) (pow k_m -4.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * ((pow(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 = 2.0d0 * (((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 2.0 * ((Math.pow(l, 2.0) / t) * Math.pow(k_m, -4.0));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * ((math.pow(l, 2.0) / t) * math.pow(k_m, -4.0))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / t) * (k_m ^ -4.0)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * (((l ^ 2.0) / t) * (k_m ^ -4.0));
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] / t), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

3. Simplified 52.4%

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

5. Taylor expanded in t around 0 62.2%

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

6. Taylor expanded in k around 0 52.6%

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

   1. *-commutative 52.6%

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

   2. associate-/r* 52.9%

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

8. Simplified 52.9%

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

   1. *-un-lft-identity 52.9%

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

   2. div-inv 52.9%

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

   3. pow-flip 52.9%

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

   4. metadata-eval 52.9%

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

10. Applied egg-rr 52.9%

    \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]
11. Step-by-step derivation

    1. *-lft-identity 52.9%

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

12. Simplified 52.9%

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

13. Final simplification 52.9%

    \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right) \]
14. Add Preprocessing

Alternative 21: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k\_m}^{4}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (/ (/ (pow l 2.0) t) (pow k_m 4.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * ((pow(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 = 2.0d0 * (((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 2.0 * ((Math.pow(l, 2.0) / t) / Math.pow(k_m, 4.0));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * ((math.pow(l, 2.0) / t) / math.pow(k_m, 4.0))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / t) / (k_m ^ 4.0)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * (((l ^ 2.0) / t) / (k_m ^ 4.0));
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] / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.0%

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

3. Simplified 52.4%

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

5. Taylor expanded in t around 0 62.2%

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

6. Taylor expanded in k around 0 52.6%

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

   1. *-commutative 52.6%

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

   2. associate-/r* 52.9%

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

   \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))