Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 81.7%

Time: 25.9s

Alternatives: 15

Speedup: 1.9×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00086:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k\_m} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k\_m} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+89} \lor \neg \left(k\_m \leq 2.6 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00086)
   (/
    2.0
    (pow
     (*
      (* (cbrt (sin k_m)) (* t (pow (cbrt l) -2.0)))
      (* (cbrt k_m) (cbrt 2.0)))
     3.0))
   (if (<= k_m 1.65e+33)
     (*
      2.0
      (*
       (/ (pow l 2.0) (pow k_m 2.0))
       (/ (cos k_m) (* t (pow (sin k_m) 2.0)))))
     (if (or (<= k_m 5e+89) (not (<= k_m 2.6e+126)))
       (/
        2.0
        (pow
         (*
          (/ t (pow (cbrt l) 2.0))
          (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0))))))
         3.0))
       (/
        2.0
        (/
         (* (pow k_m 2.0) (* t (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
         (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00086) {
		tmp = 2.0 / pow(((cbrt(sin(k_m)) * (t * pow(cbrt(l), -2.0))) * (cbrt(k_m) * cbrt(2.0))), 3.0);
	} else if (k_m <= 1.65e+33) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0))));
	} else if ((k_m <= 5e+89) || !(k_m <= 2.6e+126)) {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (0.5 * cos((2.0 * k_m)))))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00086) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k_m)) * (t * Math.pow(Math.cbrt(l), -2.0))) * (Math.cbrt(k_m) * Math.cbrt(2.0))), 3.0);
	} else if (k_m <= 1.65e+33) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0))));
	} else if ((k_m <= 5e+89) || !(k_m <= 2.6e+126)) {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00086)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k_m)) * Float64(t * (cbrt(l) ^ -2.0))) * Float64(cbrt(k_m) * cbrt(2.0))) ^ 3.0));
	elseif (k_m <= 1.65e+33)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0)))));
	elseif ((k_m <= 5e+89) || !(k_m <= 2.6e+126))
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0)))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00086], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.65e+33], 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[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k$95$m, 5e+89], N[Not[LessEqual[k$95$m, 2.6e+126]], $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[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 8.59999999999999979e-4

   1. Initial program 59.0%

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

   3. Simplified 59.1%

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

      1. add-cube-cbrt 59.1%

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

      2. pow3 59.1%

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

      3. *-commutative 59.1%

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

      4. cbrt-prod 59.1%

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

      5. cbrt-div 60.5%

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

      6. rem-cbrt-cube 68.6%

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

      7. cbrt-prod 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. add-cube-cbrt 81.6%

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

      2. pow3 81.6%

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

   8. Applied egg-rr 81.6%

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

      1. add-cube-cbrt 81.6%

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

      2. pow3 81.6%

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

   10. Applied egg-rr 87.6%

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

   11. Taylor expanded in k around 0 78.0%

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

   if 8.59999999999999979e-4 < k < 1.64999999999999988e33

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in t around 0 79.3%

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

      1. times-frac 79.3%

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

   7. Simplified 79.3%

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

   if 1.64999999999999988e33 < k < 4.99999999999999983e89 or 2.6e126 < k

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. add-cube-cbrt 49.4%

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

      2. pow3 49.4%

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

   6. Applied egg-rr 77.9%

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

   if 4.99999999999999983e89 < k < 2.6e126

   1. Initial program 21.5%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in t around 0 80.2%

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

      1. associate-/l* 61.9%

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

      2. associate-/l* 61.9%

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

   7. Simplified 61.9%

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

      1. unpow2 61.9%

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

      2. sin-mult 62.2%

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

   9. Applied egg-rr 62.2%

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

       1. div-sub 62.2%

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

       2. +-inverses 62.2%

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

       3. cos-0 62.2%

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

       4. metadata-eval 62.2%

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

       5. count-2 62.2%

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

   11. Simplified 62.2%

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

   12. Taylor expanded in k around inf 80.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00086:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+89} \lor \neg \left(k \leq 2.6 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{\sin k\_m}^{2}} \cdot \frac{\cos k\_m}{{k\_m}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k\_m} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.9e-132)
   (*
    2.0
    (/ (* (/ (pow l 2.0) (pow (sin k_m) 2.0)) (/ (cos k_m) (pow k_m 2.0))) t))
   (/
    2.0
    (pow
     (*
      (* (cbrt (sin k_m)) (* t (pow (cbrt l) -2.0)))
      (cbrt (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))))
     3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.9e-132) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(sin(k_m), 2.0)) * (cos(k_m) / pow(k_m, 2.0))) / t);
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k_m)) * (t * pow(cbrt(l), -2.0))) * cbrt((tan(k_m) * (2.0 + pow((k_m / t), 2.0))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.9e-132) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0)) * (Math.cos(k_m) / Math.pow(k_m, 2.0))) / t);
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k_m)) * (t * Math.pow(Math.cbrt(l), -2.0))) * Math.cbrt((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.9e-132)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (sin(k_m) ^ 2.0)) * Float64(cos(k_m) / (k_m ^ 2.0))) / t));
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k_m)) * Float64(t * (cbrt(l) ^ -2.0))) * cbrt(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.9e-132], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.89999999999999981e-132

   1. Initial program 51.8%

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

   3. Simplified 49.7%

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

      1. associate-*r* 57.9%

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

      2. *-un-lft-identity 57.9%

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

      3. times-frac 56.8%

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

      4. associate-*r* 59.1%

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

   6. Applied egg-rr 59.1%

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

      1. /-rgt-identity 59.1%

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

      2. *-commutative 59.1%

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

      3. associate-/r* 59.1%

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

   8. Simplified 59.1%

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

   9. Taylor expanded in t around 0 64.9%

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

       1. associate-/r* 65.2%

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

       2. *-commutative 65.2%

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

       3. associate-/r* 65.7%

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

       4. associate-/r* 63.4%

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

       5. *-commutative 63.4%

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

       6. times-frac 65.2%

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

   11. Simplified 65.2%

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

   if 4.89999999999999981e-132 < t

   1. Initial program 64.3%

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

   3. Simplified 64.3%

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

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.3%

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

      3. *-commutative 64.3%

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

      4. cbrt-prod 64.2%

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

      5. cbrt-div 65.4%

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

      6. rem-cbrt-cube 72.2%

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

      7. cbrt-prod 84.5%

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

      8. pow2 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. add-cube-cbrt 84.4%

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

      2. pow3 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. add-cube-cbrt 84.4%

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

      2. pow3 84.4%

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

   10. Applied egg-rr 95.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{{k}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00098:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k\_m} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k\_m} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00098)
   (/
    2.0
    (pow
     (*
      (* (cbrt (sin k_m)) (* t (pow (cbrt l) -2.0)))
      (* (cbrt k_m) (cbrt 2.0)))
     3.0))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00098) {
		tmp = 2.0 / pow(((cbrt(sin(k_m)) * (t * pow(cbrt(l), -2.0))) * (cbrt(k_m) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (0.5 * cos((2.0 * k_m)))))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00098) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k_m)) * (t * Math.pow(Math.cbrt(l), -2.0))) * (Math.cbrt(k_m) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00098)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k_m)) * Float64(t * (cbrt(l) ^ -2.0))) * Float64(cbrt(k_m) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00098], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.7999999999999997e-4

   1. Initial program 59.0%

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

   3. Simplified 59.1%

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

      1. add-cube-cbrt 59.1%

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

      2. pow3 59.1%

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

      3. *-commutative 59.1%

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

      4. cbrt-prod 59.1%

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

      5. cbrt-div 60.5%

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

      6. rem-cbrt-cube 68.6%

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

      7. cbrt-prod 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. add-cube-cbrt 81.6%

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

      2. pow3 81.6%

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

   8. Applied egg-rr 81.6%

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

      1. add-cube-cbrt 81.6%

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

      2. pow3 81.6%

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

   10. Applied egg-rr 87.6%

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

   11. Taylor expanded in k around 0 78.0%

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

   if 9.7999999999999997e-4 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00098:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot \ell}}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 0.0072:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.4e-173)
   (*
    (pow (/ (cbrt (* 2.0 l)) (* t (pow (cbrt k_m) 2.0))) 3.0)
    (/ l (+ 2.0 (pow (/ k_m t) 2.0))))
   (if (<= k_m 0.0072)
     (/
      2.0
      (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k_m 2.0))))) 3.0))
     (/
      2.0
      (/
       (* (pow k_m 2.0) (* t (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
       (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e-173) {
		tmp = pow((cbrt((2.0 * l)) / (t * pow(cbrt(k_m), 2.0))), 3.0) * (l / (2.0 + pow((k_m / t), 2.0)));
	} else if (k_m <= 0.0072) {
		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(k_m, 2.0) * (t * (0.5 - (0.5 * cos((2.0 * k_m)))))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e-173) {
		tmp = Math.pow((Math.cbrt((2.0 * l)) / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0) * (l / (2.0 + Math.pow((k_m / t), 2.0)));
	} else if (k_m <= 0.0072) {
		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(k_m, 2.0) * (t * (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.4e-173)
		tmp = Float64((Float64(cbrt(Float64(2.0 * l)) / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0) * Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0))));
	elseif (k_m <= 0.0072)
		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((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.4e-173], N[(N[Power[N[(N[Power[N[(2.0 * l), $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.0072], 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[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.3999999999999999e-173

   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 52.9%

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

      1. associate-*r* 62.6%

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

      2. *-un-lft-identity 62.6%

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

      3. times-frac 61.4%

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

      4. associate-*r* 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. /-rgt-identity 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-/r* 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in k around 0 59.5%

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

       1. add-cube-cbrt 59.4%

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

       2. pow3 59.4%

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

       3. associate-*r/ 59.4%

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

       4. cbrt-div 59.4%

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

       5. *-commutative 59.4%

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

       6. cbrt-prod 60.6%

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

       7. unpow3 60.6%

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

       8. add-cbrt-cube 66.1%

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

       9. unpow2 66.1%

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

       10. cbrt-prod 75.8%

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

       11. pow2 75.8%

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

   11. Applied egg-rr 75.8%

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

   if 5.3999999999999999e-173 < k < 0.0071999999999999998

   1. Initial program 73.6%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in k around 0 81.3%

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

      1. add-cube-cbrt 81.3%

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

      2. pow3 81.3%

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

      3. cbrt-prod 81.0%

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

      4. associate-/l/ 74.7%

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

      5. unpow2 74.7%

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

      6. cbrt-div 74.7%

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

      7. unpow3 74.6%

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

      8. add-cbrt-cube 81.0%

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

      9. unpow2 81.0%

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

      10. cbrt-prod 93.2%

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

      11. unpow2 93.2%

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

      12. div-inv 93.2%

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

      13. pow-flip 93.2%

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

      14. metadata-eval 93.2%

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

   7. Applied egg-rr 93.2%

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

      1. associate-*l* 93.2%

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

   9. Simplified 93.2%

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

   if 0.0071999999999999998 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot \ell}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 0.0072:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{elif}\;k\_m \leq 0.0045:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.4e-173)
   (*
    (/ l (+ 2.0 (pow (/ k_m t) 2.0)))
    (* l (/ 2.0 (pow (* t (pow (cbrt k_m) 2.0)) 3.0))))
   (if (<= k_m 0.0045)
     (/
      2.0
      (pow (* t (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k_m 2.0))))) 3.0))
     (/
      2.0
      (/
       (* (pow k_m 2.0) (* t (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
       (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e-173) {
		tmp = (l / (2.0 + pow((k_m / t), 2.0))) * (l * (2.0 / pow((t * pow(cbrt(k_m), 2.0)), 3.0)));
	} else if (k_m <= 0.0045) {
		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(k_m, 2.0) * (t * (0.5 - (0.5 * cos((2.0 * k_m)))))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.4e-173) {
		tmp = (l / (2.0 + Math.pow((k_m / t), 2.0))) * (l * (2.0 / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0)));
	} else if (k_m <= 0.0045) {
		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(k_m, 2.0) * (t * (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.4e-173)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(l * Float64(2.0 / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0))));
	elseif (k_m <= 0.0045)
		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((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.4e-173], N[(N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.0045], 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[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.3999999999999999e-173

   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 52.9%

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

      1. associate-*r* 62.6%

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

      2. *-un-lft-identity 62.6%

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

      3. times-frac 61.4%

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

      4. associate-*r* 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. /-rgt-identity 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-/r* 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in k around 0 59.5%

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

       1. add-cube-cbrt 59.4%

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

       2. pow3 59.4%

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

       3. *-commutative 59.4%

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

       4. cbrt-prod 59.4%

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

       5. unpow3 59.4%

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

       6. add-cbrt-cube 63.7%

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

       7. unpow2 63.7%

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

       8. cbrt-prod 70.5%

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

       9. pow2 70.5%

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

   11. Applied egg-rr 70.5%

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

   if 5.3999999999999999e-173 < k < 0.00449999999999999966

   1. Initial program 73.6%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in k around 0 81.3%

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

      1. add-cube-cbrt 81.3%

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

      2. pow3 81.3%

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

      3. cbrt-prod 81.0%

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

      4. associate-/l/ 74.7%

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

      5. unpow2 74.7%

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

      6. cbrt-div 74.7%

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

      7. unpow3 74.6%

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

      8. add-cbrt-cube 81.0%

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

      9. unpow2 81.0%

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

      10. cbrt-prod 93.2%

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

      11. unpow2 93.2%

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

      12. div-inv 93.2%

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

      13. pow-flip 93.2%

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

      14. metadata-eval 93.2%

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

   7. Applied egg-rr 93.2%

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

      1. associate-*l* 93.2%

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

   9. Simplified 93.2%

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

   if 0.00449999999999999966 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{elif}\;k \leq 0.0045:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{if}\;t \leq 4.3 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\_m\right)}{2}}{\cos k\_m \cdot \left(\ell \cdot \ell\right)}\right)}\\ \mathbf{elif}\;t \leq 7.7 \cdot 10^{+86}:\\ \;\;\;\;t\_1 \cdot \frac{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k\_m}}{\tan k\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (+ 2.0 (pow (/ k_m t) 2.0)))))
   (if (<= t 4.3e-103)
     (/
      2.0
      (*
       (pow k_m 2.0)
       (* t (/ (- 0.5 (/ (cos (* 2.0 k_m)) 2.0)) (* (cos k_m) (* l l))))))
     (if (<= t 7.7e+86)
       (* t_1 (/ (* l (/ (/ 2.0 (pow t 3.0)) (sin k_m))) (tan k_m)))
       (* t_1 (* l (/ 2.0 (pow (* t (pow (cbrt k_m) 2.0)) 3.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (2.0 + pow((k_m / t), 2.0));
	double tmp;
	if (t <= 4.3e-103) {
		tmp = 2.0 / (pow(k_m, 2.0) * (t * ((0.5 - (cos((2.0 * k_m)) / 2.0)) / (cos(k_m) * (l * l)))));
	} else if (t <= 7.7e+86) {
		tmp = t_1 * ((l * ((2.0 / pow(t, 3.0)) / sin(k_m))) / tan(k_m));
	} else {
		tmp = t_1 * (l * (2.0 / pow((t * pow(cbrt(k_m), 2.0)), 3.0)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / (2.0 + Math.pow((k_m / t), 2.0));
	double tmp;
	if (t <= 4.3e-103) {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * (t * ((0.5 - (Math.cos((2.0 * k_m)) / 2.0)) / (Math.cos(k_m) * (l * l)))));
	} else if (t <= 7.7e+86) {
		tmp = t_1 * ((l * ((2.0 / Math.pow(t, 3.0)) / Math.sin(k_m))) / Math.tan(k_m));
	} else {
		tmp = t_1 * (l * (2.0 / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0)))
	tmp = 0.0
	if (t <= 4.3e-103)
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t * Float64(Float64(0.5 - Float64(cos(Float64(2.0 * k_m)) / 2.0)) / Float64(cos(k_m) * Float64(l * l))))));
	elseif (t <= 7.7e+86)
		tmp = Float64(t_1 * Float64(Float64(l * Float64(Float64(2.0 / (t ^ 3.0)) / sin(k_m))) / tan(k_m)));
	else
		tmp = Float64(t_1 * Float64(l * Float64(2.0 / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 4.3e-103], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(N[(0.5 - N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.7e+86], N[(t$95$1 * N[(N[(l * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l * N[(2.0 / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.30000000000000023e-103

   1. Initial program 51.2%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in t around 0 65.3%

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

      1. associate-/l* 65.7%

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

      2. associate-/l* 66.2%

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

   7. Simplified 66.2%

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

      1. unpow2 66.2%

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

      2. sin-mult 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. div-sub 61.0%

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

       2. +-inverses 61.0%

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

       3. cos-0 61.0%

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

       4. metadata-eval 61.0%

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

       5. count-2 61.0%

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

   11. Simplified 61.0%

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

       1. unpow2 59.4%

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

   13. Applied egg-rr 61.0%

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

   if 4.30000000000000023e-103 < t < 7.70000000000000053e86

   1. Initial program 75.0%

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

   3. Simplified 73.4%

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

      1. associate-*r* 80.2%

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

      2. *-un-lft-identity 80.2%

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

      3. times-frac 84.6%

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

      4. associate-*r* 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. /-rgt-identity 87.0%

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

      2. *-commutative 87.0%

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

      3. associate-/r* 87.1%

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

   8. Simplified 87.1%

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

      1. associate-*r/ 91.8%

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

      2. associate-/r* 91.8%

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

   10. Applied egg-rr 91.8%

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

   if 7.70000000000000053e86 < t

   1. Initial program 54.2%

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

   3. Simplified 50.1%

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

      1. associate-*r* 57.6%

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

      2. *-un-lft-identity 57.6%

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

      3. times-frac 57.6%

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

      4. associate-*r* 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. /-rgt-identity 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-/r* 61.8%

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

   8. Simplified 61.8%

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

   9. Taylor expanded in k around 0 57.6%

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

       1. add-cube-cbrt 57.6%

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

       2. pow3 57.6%

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

       3. *-commutative 57.6%

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

       4. cbrt-prod 57.6%

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

       5. unpow3 57.6%

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

       6. add-cbrt-cube 61.5%

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

       7. unpow2 61.5%

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

       8. cbrt-prod 80.5%

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

       9. pow2 80.5%

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

   11. Applied egg-rr 80.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}\\ \mathbf{elif}\;t \leq 7.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00092:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00092)
   (*
    (/ l (+ 2.0 (pow (/ k_m t) 2.0)))
    (* l (/ 2.0 (pow (* t (pow (cbrt k_m) 2.0)) 3.0))))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00092) {
		tmp = (l / (2.0 + pow((k_m / t), 2.0))) * (l * (2.0 / pow((t * pow(cbrt(k_m), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (0.5 * cos((2.0 * k_m)))))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00092) {
		tmp = (l / (2.0 + Math.pow((k_m / t), 2.0))) * (l * (2.0 / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (0.5 * Math.cos((2.0 * k_m)))))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00092)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(l * Float64(2.0 / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00092], N[(N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000003e-4

   1. Initial program 59.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 64.6%

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

      4. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-/r* 68.0%

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

   8. Simplified 68.0%

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

   9. Taylor expanded in k around 0 63.0%

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

       1. add-cube-cbrt 62.9%

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

       2. pow3 62.9%

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

       3. *-commutative 62.9%

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

       4. cbrt-prod 62.9%

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

       5. unpow3 62.9%

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

       6. add-cbrt-cube 67.6%

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

       7. unpow2 67.6%

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

       8. cbrt-prod 73.7%

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

       9. pow2 73.7%

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

   11. Applied egg-rr 73.7%

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

   if 9.2000000000000003e-4 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00092:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00047:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\_m\right)}{2}}{\cos k\_m \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00047)
   (*
    (/ l (+ 2.0 (pow (/ k_m t) 2.0)))
    (* l (/ 2.0 (pow (* t (pow (cbrt k_m) 2.0)) 3.0))))
   (/
    2.0
    (*
     (pow k_m 2.0)
     (* t (/ (- 0.5 (/ (cos (* 2.0 k_m)) 2.0)) (* (cos k_m) (* l l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00047) {
		tmp = (l / (2.0 + pow((k_m / t), 2.0))) * (l * (2.0 / pow((t * pow(cbrt(k_m), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * (t * ((0.5 - (cos((2.0 * k_m)) / 2.0)) / (cos(k_m) * (l * l)))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00047) {
		tmp = (l / (2.0 + Math.pow((k_m / t), 2.0))) * (l * (2.0 / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * (t * ((0.5 - (Math.cos((2.0 * k_m)) / 2.0)) / (Math.cos(k_m) * (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00047)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(l * Float64(2.0 / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t * Float64(Float64(0.5 - Float64(cos(Float64(2.0 * k_m)) / 2.0)) / Float64(cos(k_m) * Float64(l * l))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00047], N[(N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(N[(0.5 - N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.69999999999999986e-4

   1. Initial program 59.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 64.6%

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

      4. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-/r* 68.0%

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

   8. Simplified 68.0%

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

   9. Taylor expanded in k around 0 63.0%

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

       1. add-cube-cbrt 62.9%

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

       2. pow3 62.9%

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

       3. *-commutative 62.9%

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

       4. cbrt-prod 62.9%

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

       5. unpow3 62.9%

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

       6. add-cbrt-cube 67.6%

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

       7. unpow2 67.6%

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

       8. cbrt-prod 73.7%

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

       9. pow2 73.7%

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

   11. Applied egg-rr 73.7%

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

   if 4.69999999999999986e-4 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

       1. unpow2 61.6%

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

   13. Applied egg-rr 70.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00047:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0054:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k\_m \cdot {t}^{3}}}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\_m\right)}{2}}{\cos k\_m \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0054)
   (* (* l (/ (/ 2.0 (* (sin k_m) (pow t 3.0))) (tan k_m))) (* l 0.5))
   (/
    2.0
    (*
     (pow k_m 2.0)
     (* t (/ (- 0.5 (/ (cos (* 2.0 k_m)) 2.0)) (* (cos k_m) (* l l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0054) {
		tmp = (l * ((2.0 / (sin(k_m) * pow(t, 3.0))) / tan(k_m))) * (l * 0.5);
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * (t * ((0.5 - (cos((2.0 * k_m)) / 2.0)) / (cos(k_m) * (l * l)))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.0054d0) then
        tmp = (l * ((2.0d0 / (sin(k_m) * (t ** 3.0d0))) / tan(k_m))) * (l * 0.5d0)
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * (t * ((0.5d0 - (cos((2.0d0 * k_m)) / 2.0d0)) / (cos(k_m) * (l * l)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0054) {
		tmp = (l * ((2.0 / (Math.sin(k_m) * Math.pow(t, 3.0))) / Math.tan(k_m))) * (l * 0.5);
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * (t * ((0.5 - (Math.cos((2.0 * k_m)) / 2.0)) / (Math.cos(k_m) * (l * l)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.0054:
		tmp = (l * ((2.0 / (math.sin(k_m) * math.pow(t, 3.0))) / math.tan(k_m))) * (l * 0.5)
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * (t * ((0.5 - (math.cos((2.0 * k_m)) / 2.0)) / (math.cos(k_m) * (l * l)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0054)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(sin(k_m) * (t ^ 3.0))) / tan(k_m))) * Float64(l * 0.5));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t * Float64(Float64(0.5 - Float64(cos(Float64(2.0 * k_m)) / 2.0)) / Float64(cos(k_m) * Float64(l * l))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.0054)
		tmp = (l * ((2.0 / (sin(k_m) * (t ^ 3.0))) / tan(k_m))) * (l * 0.5);
	else
		tmp = 2.0 / ((k_m ^ 2.0) * (t * ((0.5 - (cos((2.0 * k_m)) / 2.0)) / (cos(k_m) * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0054], N[(N[(l * N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(N[(0.5 - N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0054000000000000003

   1. Initial program 59.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 64.6%

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

      4. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-/r* 68.0%

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

   8. Simplified 68.0%

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

   9. Taylor expanded in k around 0 66.1%

      \[\leadsto \left(\ell \cdot \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.6%

          \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.1%

       \[\leadsto \left(\ell \cdot \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 0.0054000000000000003 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

       1. unpow2 61.6%

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

   13. Applied egg-rr 70.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0054:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k \cdot {t}^{3}}}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00095:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k\_m \cdot {t}^{3}}}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00095)
   (* (* l (/ (/ 2.0 (* (sin k_m) (pow t 3.0))) (tan k_m))) (* l 0.5))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00095) {
		tmp = (l * ((2.0 / (sin(k_m) * pow(t, 3.0))) / tan(k_m))) * (l * 0.5);
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00095d0) then
        tmp = (l * ((2.0d0 / (sin(k_m) * (t ** 3.0d0))) / tan(k_m))) * (l * 0.5d0)
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00095) {
		tmp = (l * ((2.0 / (Math.sin(k_m) * Math.pow(t, 3.0))) / Math.tan(k_m))) * (l * 0.5);
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.00095:
		tmp = (l * ((2.0 / (math.sin(k_m) * math.pow(t, 3.0))) / math.tan(k_m))) * (l * 0.5)
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00095)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(sin(k_m) * (t ^ 3.0))) / tan(k_m))) * Float64(l * 0.5));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00095)
		tmp = (l * ((2.0 / (sin(k_m) * (t ^ 3.0))) / tan(k_m))) * (l * 0.5);
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00095], N[(N[(l * N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.49999999999999998e-4

   1. Initial program 59.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 64.6%

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

      4. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-/r* 68.0%

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

   8. Simplified 68.0%

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

   9. Taylor expanded in k around 0 66.1%

      \[\leadsto \left(\ell \cdot \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.6%

          \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 66.1%

       \[\leadsto \left(\ell \cdot \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 9.49999999999999998e-4 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

   13. Taylor expanded in k around 0 64.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00095:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k \cdot {t}^{3}}}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0024:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0024)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (pow (/ t (cbrt l)) 3.0) l)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0024) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * (pow((t / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0024) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * (Math.pow((t / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0024)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0024], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00239999999999999979

   1. Initial program 59.0%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in k around 0 64.7%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

      3. cbrt-div 64.6%

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

      4. rem-cbrt-cube 67.3%

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

   7. Applied egg-rr 67.3%

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

   if 0.00239999999999999979 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

   13. Taylor expanded in k around 0 64.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0024:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0053:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0053)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t 2.0) (/ t l)) l)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0053) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 2.0) * (t / l)) / l));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.0053d0) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 2.0d0) * (t / l)) / l))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0053) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 2.0) * (t / l)) / l));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.0053:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 2.0) * (t / l)) / l))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0053)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 2.0) * Float64(t / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.0053)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 2.0) * (t / l)) / l));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0053], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00530000000000000002

   1. Initial program 59.0%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in k around 0 64.7%

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

      1. unpow3 64.7%

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

      2. *-un-lft-identity 64.7%

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

      3. times-frac 66.3%

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

      4. pow2 66.3%

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

   7. Applied egg-rr 66.3%

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

   if 0.00530000000000000002 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. sin-mult 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 70.6%

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

       2. +-inverses 70.6%

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

       3. cos-0 70.6%

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

       4. metadata-eval 70.6%

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

       5. count-2 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in k around inf 73.3%

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

   13. Taylor expanded in k around 0 64.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0053:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0013:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({k\_m}^{4} \cdot {\ell}^{-2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0013)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t 2.0) (/ t l)) l)))
   (/ 2.0 (* t (* (pow k_m 4.0) (pow l -2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0013) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 2.0) * (t / l)) / l));
	} else {
		tmp = 2.0 / (t * (pow(k_m, 4.0) * pow(l, -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 <= 0.0013d0) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 2.0d0) * (t / l)) / l))
    else
        tmp = 2.0d0 / (t * ((k_m ** 4.0d0) * (l ** (-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 <= 0.0013) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 2.0) * (t / l)) / l));
	} else {
		tmp = 2.0 / (t * (Math.pow(k_m, 4.0) * Math.pow(l, -2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.0013:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 2.0) * (t / l)) / l))
	else:
		tmp = 2.0 / (t * (math.pow(k_m, 4.0) * math.pow(l, -2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0013)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 2.0) * Float64(t / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(t * Float64((k_m ^ 4.0) * (l ^ -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 <= 0.0013)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 2.0) * (t / l)) / l));
	else
		tmp = 2.0 / (t * ((k_m ^ 4.0) * (l ^ -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, 0.0013], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0012999999999999999

   1. Initial program 59.0%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in k around 0 64.7%

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

      1. unpow3 64.7%

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

      2. *-un-lft-identity 64.7%

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

      3. times-frac 66.3%

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

      4. pow2 66.3%

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

   7. Applied egg-rr 66.3%

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

   if 0.0012999999999999999 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in k around 0 61.8%

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

      1. associate-/l* 61.6%

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

   10. Simplified 61.6%

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

       1. pow1 61.6%

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

       2. div-inv 61.6%

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

       3. pow-flip 61.6%

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

       4. metadata-eval 61.6%

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

   12. Applied egg-rr 61.6%

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

       1. unpow1 61.6%

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

       2. associate-*r* 61.8%

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

       3. *-commutative 61.8%

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

       4. associate-*l* 61.8%

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

   14. Simplified 61.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0013:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0056:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({k\_m}^{4} \cdot {\ell}^{-2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0056)
   (* (* l 0.5) (* 2.0 (/ (/ l (pow t 3.0)) (pow k_m 2.0))))
   (/ 2.0 (* t (* (pow k_m 4.0) (pow l -2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0056) {
		tmp = (l * 0.5) * (2.0 * ((l / pow(t, 3.0)) / pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / (t * (pow(k_m, 4.0) * pow(l, -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 <= 0.0056d0) then
        tmp = (l * 0.5d0) * (2.0d0 * ((l / (t ** 3.0d0)) / (k_m ** 2.0d0)))
    else
        tmp = 2.0d0 / (t * ((k_m ** 4.0d0) * (l ** (-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 <= 0.0056) {
		tmp = (l * 0.5) * (2.0 * ((l / Math.pow(t, 3.0)) / Math.pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / (t * (Math.pow(k_m, 4.0) * Math.pow(l, -2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.0056:
		tmp = (l * 0.5) * (2.0 * ((l / math.pow(t, 3.0)) / math.pow(k_m, 2.0)))
	else:
		tmp = 2.0 / (t * (math.pow(k_m, 4.0) * math.pow(l, -2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0056)
		tmp = Float64(Float64(l * 0.5) * Float64(2.0 * Float64(Float64(l / (t ^ 3.0)) / (k_m ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(t * Float64((k_m ^ 4.0) * (l ^ -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 <= 0.0056)
		tmp = (l * 0.5) * (2.0 * ((l / (t ^ 3.0)) / (k_m ^ 2.0)));
	else
		tmp = 2.0 / (t * ((k_m ^ 4.0) * (l ^ -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, 0.0056], N[(N[(l * 0.5), $MachinePrecision] * N[(2.0 * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00559999999999999994

   1. Initial program 59.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. times-frac 64.6%

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

      4. associate-*r* 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. *-commutative 68.0%

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

      3. associate-/r* 68.0%

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

   8. Simplified 68.0%

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

   9. Taylor expanded in k around 0 63.0%

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

       1. *-commutative 63.0%

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

       2. associate-/r* 63.8%

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

   11. Simplified 63.8%

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

   12. Taylor expanded in k around 0 65.6%

       \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right) \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   13. Step-by-step derivation

       1. *-commutative 65.6%

          \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   14. Simplified 65.6%

       \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right) \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 0.00559999999999999994 < k

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-/l* 70.5%

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

      2. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in k around 0 61.8%

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

      1. associate-/l* 61.6%

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

   10. Simplified 61.6%

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

       1. pow1 61.6%

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

       2. div-inv 61.6%

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

       3. pow-flip 61.6%

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

       4. metadata-eval 61.6%

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

   12. Applied egg-rr 61.6%

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

       1. unpow1 61.6%

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

       2. associate-*r* 61.8%

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

       3. *-commutative 61.8%

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

       4. associate-*l* 61.8%

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

   14. Simplified 61.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0056:\\ \;\;\;\;\left(\ell \cdot 0.5\right) \cdot \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{{k\_m}^{4} \cdot \frac{t}{\ell \cdot \ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (/ 2.0 (* (pow k_m 4.0) (/ t (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (pow(k_m, 4.0) * (t / (l * l)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m ** 4.0d0) * (t / (l * l)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (Math.pow(k_m, 4.0) * (t / (l * l)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (math.pow(k_m, 4.0) * (t / (l * l)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t / Float64(l * l))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m ^ 4.0) * (t / (l * l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Simplified 55.4%

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

5. Taylor expanded in t around 0 61.7%

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

   1. associate-/l* 62.8%

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

   2. associate-/l* 63.2%

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

7. Simplified 63.2%

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

8. Taylor expanded in k around 0 55.5%

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

   1. associate-/l* 56.5%

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

10. Simplified 56.5%

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

    1. unpow2 56.5%

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

12. Applied egg-rr 56.5%

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

13. Final simplification 56.5%

    \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\ell \cdot \ell}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))