Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 86.1%

Time: 24.0s

Alternatives: 25

Speedup: 1.9×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 2.65 \cdot 10^{-5} \lor \neg \left(t\_m \leq 2.5 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}}{t\_2} \cdot \frac{\ell}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m)))))
   (*
    t_s
    (if (<= t_m 2.5e-178)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (or (<= t_m 2.65e-5) (not (<= t_m 2.5e+66)))
        (/
         2.0
         (pow
          (*
           (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0)))
           (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
          3.0))
        (*
         (/ (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) t_2)
         (/ l t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 2.5e-178) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if ((t_m <= 2.65e-5) || !(t_m <= 2.5e+66)) {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	} else {
		tmp = ((l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) / t_2) * (l / t_2);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 2.5e-178) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if ((t_m <= 2.65e-5) || !(t_m <= 2.5e+66)) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	} else {
		tmp = ((l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) / t_2) * (l / t_2);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	tmp = 0.0
	if (t_m <= 2.5e-178)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif ((t_m <= 2.65e-5) || !(t_m <= 2.5e+66))
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) / t_2) * Float64(l / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.5e-178], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 2.65e-5], N[Not[LessEqual[t$95$m, 2.5e+66]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.49999999999999988e-178

   1. Initial program 51.4%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-/r* 58.0%

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

      2. add-cube-cbrt 57.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. times-frac 31.6%

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

      5. pow2 31.6%

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

      6. cbrt-div 31.6%

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

      7. rem-cbrt-cube 31.6%

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

      8. cbrt-div 31.6%

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

      9. rem-cbrt-cube 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in t around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 64.3%

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

   9. Simplified 64.3%

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

   if 2.49999999999999988e-178 < t < 2.65e-5 or 2.49999999999999996e66 < t

   1. Initial program 51.9%

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

   3. Simplified 52.0%

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

      1. associate-/r* 59.4%

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

      2. add-cube-cbrt 59.2%

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

      3. add-sqr-sqrt 25.8%

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

      4. times-frac 25.8%

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

      5. pow2 25.8%

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

      6. cbrt-div 25.8%

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

      7. rem-cbrt-cube 25.8%

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

      8. cbrt-div 25.8%

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

      9. rem-cbrt-cube 38.4%

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

   6. Applied egg-rr 38.4%

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

      1. add-cube-cbrt 38.4%

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

      2. pow3 38.4%

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

   8. Applied egg-rr 90.0%

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

      1. add-cube-cbrt 89.9%

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

      2. pow3 89.9%

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

   10. Applied egg-rr 94.5%

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

   if 2.65e-5 < t < 2.49999999999999996e66

   1. Initial program 58.6%

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

   3. Simplified 58.5%

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

      1. associate-*r* 59.2%

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

      2. add-sqr-sqrt 59.0%

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

      3. times-frac 64.3%

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

   6. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-178}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-5} \lor \neg \left(t \leq 2.5 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t\_2 + 1\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{+301}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + t\_2\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;{\left(\sqrt{\frac{1}{{t\_m}^{3}}} \cdot \left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{0.5}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3
         (*
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))
          (+ 1.0 (+ t_2 1.0)))))
   (*
    t_s
    (if (<= t_3 1e+301)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) (+ 2.0 t_2))) l)))
      (if (<= t_3 INFINITY)
        (pow
         (* (sqrt (/ 1.0 (pow t_m 3.0))) (* l (/ (* (sqrt 2.0) (sqrt 0.5)) k)))
         2.0)
        (pow
         (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m)))
         2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0));
	double tmp;
	if (t_3 <= 1e+301) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow((sqrt((1.0 / pow(t_m, 3.0))) * (l * ((sqrt(2.0) * sqrt(0.5)) / k))), 2.0);
	} else {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0));
	double tmp;
	if (t_3 <= 1e+301) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * (2.0 + t_2))) / l));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt((1.0 / Math.pow(t_m, 3.0))) * (l * ((Math.sqrt(2.0) * Math.sqrt(0.5)) / k))), 2.0);
	} else {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))
	tmp = 0
	if t_3 <= 1e+301:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((math.sin(k) * (math.tan(k) * (2.0 + t_2))) / l))
	elif t_3 <= math.inf:
		tmp = math.pow((math.sqrt((1.0 / math.pow(t_m, 3.0))) * (l * ((math.sqrt(2.0) * math.sqrt(0.5)) / k))), 2.0)
	else:
		tmp = math.pow(((l * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((1.0 / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0)))
	tmp = 0.0
	if (t_3 <= 1e+301)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + t_2))) / l)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(1.0 / (t_m ^ 3.0))) * Float64(l * Float64(Float64(sqrt(2.0) * sqrt(0.5)) / k))) ^ 2.0;
	else
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0));
	tmp = 0.0;
	if (t_3 <= 1e+301)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	elseif (t_3 <= Inf)
		tmp = (sqrt((1.0 / (t_m ^ 3.0))) * (l * ((sqrt(2.0) * sqrt(0.5)) / k))) ^ 2.0;
	else
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 1e+301], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Power[N[(N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 1.00000000000000005e301

   1. Initial program 83.2%

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

   3. Simplified 83.3%

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

      1. associate-*l* 81.3%

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

      2. associate-/r* 83.8%

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

      3. associate-+r+ 83.8%

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

      4. metadata-eval 83.8%

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

      5. associate-*l* 83.8%

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

      6. associate-*l/ 86.8%

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

      7. associate-*l* 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. associate-/l* 86.9%

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

   8. Simplified 86.9%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 68.5%

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

   3. Simplified 65.8%

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

      1. add-sqr-sqrt 65.8%

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

      2. pow2 65.8%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in k around 0 69.2%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{{t}^{3}}} \cdot \frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k}\right)}}^{2} \]

      2. associate-/l* 69.3%

         \[\leadsto {\left(\sqrt{\frac{1}{{t}^{3}}} \cdot \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)}\right)}^{2} \]

      3. *-commutative 69.3%

         \[\leadsto {\left(\sqrt{\frac{1}{{t}^{3}}} \cdot \left(\ell \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{0.5}}}{k}\right)\right)}^{2} \]

   9. Simplified 69.3%

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{1}{{t}^{3}}} \cdot \left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{0.5}}{k}\right)\right)}}^{2} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   6. Applied egg-rr 13.3%

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

   7. Taylor expanded in t around 0 40.7%

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

      1. associate-/l* 40.7%

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

   9. Simplified 40.7%

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

   10. Taylor expanded in k around 0 24.8%

       \[\leadsto {\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \color{blue}{\sqrt{\frac{1}{t}}}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \leq 10^{+301}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{elif}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \leq \infty:\\ \;\;\;\;{\left(\sqrt{\frac{1}{{t}^{3}}} \cdot \left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{0.5}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot t\_2} \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t\_2\right) \cdot \left(\sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))) t_2))
         5e+288)
      (/ 2.0 (* (* (tan k) t_2) (* (sin k) (/ (/ (pow t_m 3.0) l) l))))
      (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * t_2)) <= 5e+288) {
		tmp = 2.0 / ((tan(k) * t_2) * (sin(k) * ((pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0)
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))) * t_2)) <= 5d+288) then
        tmp = 2.0d0 / ((tan(k) * t_2) * (sin(k) * (((t_m ** 3.0d0) / l) / l)))
    else
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= 5e+288) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 + (math.pow((k / t_m), 2.0) + 1.0)
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= 5e+288:
		tmp = 2.0 / ((math.tan(k) * t_2) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l)))
	else:
		tmp = math.pow(((l * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((1.0 / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * t_2)) <= 5e+288)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	else
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 + (((k / t_m) ^ 2.0) + 1.0);
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * t_2)) <= 5e+288)
		tmp = 2.0 / ((tan(k) * t_2) * (sin(k) * (((t_m ^ 3.0) / l) / l)));
	else
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 5e+288], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 5.0000000000000003e288

   1. Initial program 77.5%

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

   3. Simplified 77.6%

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

      1. associate-/r* 81.0%

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

      2. add-cube-cbrt 80.7%

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

      3. add-sqr-sqrt 40.6%

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

      4. times-frac 40.6%

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

      5. pow2 40.6%

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

      6. cbrt-div 40.6%

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

      7. rem-cbrt-cube 40.6%

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

      8. cbrt-div 40.6%

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

      9. rem-cbrt-cube 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. pow1 41.9%

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

      2. frac-times 41.2%

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

      3. unpow2 41.2%

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

      4. pow3 41.2%

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

      5. add-sqr-sqrt 81.4%

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

   8. Applied egg-rr 81.4%

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

      1. unpow1 81.4%

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

      2. *-commutative 81.4%

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

      3. cube-div 80.7%

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

      4. rem-cube-cbrt 81.0%

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

   10. Simplified 81.0%

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

   if 5.0000000000000003e288 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 20.3%

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

   3. Simplified 20.3%

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

      1. add-sqr-sqrt 20.3%

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

      2. pow2 20.3%

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

   6. Applied egg-rr 33.8%

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

   7. Taylor expanded in t around 0 53.9%

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

      1. associate-/l* 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in k around 0 39.1%

       \[\leadsto {\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \color{blue}{\sqrt{\frac{1}{t}}}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)} \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-171}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.26 \cdot 10^{-5} \lor \neg \left(t\_m \leq 6.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}}{t\_2} \cdot \frac{\ell}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (hypot 1.0 (hypot 1.0 (/ k t_m)))))
   (*
    t_s
    (if (<= t_m 6e-171)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (or (<= t_m 1.26e-5) (not (<= t_m 6.2e+93)))
        (/
         2.0
         (*
          (pow (* (cbrt (sin k)) (/ (/ t_m (cbrt l)) (cbrt l))) 3.0)
          (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))))
        (*
         (/ (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) t_2)
         (/ l t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 6e-171) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if ((t_m <= 1.26e-5) || !(t_m <= 6.2e+93)) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * ((t_m / cbrt(l)) / cbrt(l))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	} else {
		tmp = ((l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) / t_2) * (l / t_2);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 6e-171) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if ((t_m <= 1.26e-5) || !(t_m <= 6.2e+93)) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * ((t_m / Math.cbrt(l)) / Math.cbrt(l))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	} else {
		tmp = ((l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) / t_2) * (l / t_2);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	tmp = 0.0
	if (t_m <= 6e-171)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif ((t_m <= 1.26e-5) || !(t_m <= 6.2e+93))
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(Float64(t_m / cbrt(l)) / cbrt(l))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	else
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) / t_2) * Float64(l / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-171], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 1.26e-5], N[Not[LessEqual[t$95$m, 6.2e+93]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.9999999999999999e-171

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

      1. associate-/r* 57.6%

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

      2. add-cube-cbrt 57.5%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. pow2 31.4%

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

      6. cbrt-div 31.4%

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

      7. rem-cbrt-cube 31.4%

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

      8. cbrt-div 31.4%

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

      9. rem-cbrt-cube 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. times-frac 64.6%

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

   9. Simplified 64.6%

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

   if 5.9999999999999999e-171 < t < 1.25999999999999996e-5 or 6.20000000000000038e93 < t

   1. Initial program 52.4%

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

   3. Simplified 52.5%

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

      1. associate-/r* 60.5%

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

      2. add-cube-cbrt 60.3%

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

      3. add-sqr-sqrt 25.2%

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

      4. times-frac 25.2%

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

      5. pow2 25.2%

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

      6. cbrt-div 25.2%

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

      7. rem-cbrt-cube 25.2%

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

      8. cbrt-div 25.2%

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

      9. rem-cbrt-cube 38.8%

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

   6. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.8%

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

      2. pow3 38.8%

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

   8. Applied egg-rr 92.4%

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

   if 1.25999999999999996e-5 < t < 6.20000000000000038e93

   1. Initial program 57.7%

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

   3. Simplified 57.4%

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

      1. associate-*r* 58.1%

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

      2. add-sqr-sqrt 57.9%

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

      3. times-frac 62.1%

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

   6. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-171}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-5} \lor \neg \left(t \leq 6.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-172}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.26 \cdot 10^{-5} \lor \neg \left(t\_m \leq 3.3 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_2 \cdot \frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 6.5e-172)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (or (<= t_m 1.26e-5) (not (<= t_m 3.3e+93)))
        (/
         2.0
         (*
          (pow (* (cbrt (sin k)) (/ (/ t_m (cbrt l)) (cbrt l))) 3.0)
          (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))))
        (* t_2 (* t_2 (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 6.5e-172) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if ((t_m <= 1.26e-5) || !(t_m <= 3.3e+93)) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * ((t_m / cbrt(l)) / cbrt(l))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	} else {
		tmp = t_2 * (t_2 * (2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 6.5e-172) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if ((t_m <= 1.26e-5) || !(t_m <= 3.3e+93)) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * ((t_m / Math.cbrt(l)) / Math.cbrt(l))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	} else {
		tmp = t_2 * (t_2 * (2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 6.5e-172)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif ((t_m <= 1.26e-5) || !(t_m <= 3.3e+93))
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(Float64(t_m / cbrt(l)) / cbrt(l))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	else
		tmp = Float64(t_2 * Float64(t_2 * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-172], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 1.26e-5], N[Not[LessEqual[t$95$m, 3.3e+93]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$2 * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.50000000000000012e-172

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

      1. associate-/r* 57.6%

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

      2. add-cube-cbrt 57.5%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. pow2 31.4%

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

      6. cbrt-div 31.4%

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

      7. rem-cbrt-cube 31.4%

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

      8. cbrt-div 31.4%

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

      9. rem-cbrt-cube 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. times-frac 64.6%

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

   9. Simplified 64.6%

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

   if 6.50000000000000012e-172 < t < 1.25999999999999996e-5 or 3.30000000000000009e93 < t

   1. Initial program 52.4%

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

   3. Simplified 52.5%

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

      1. associate-/r* 60.5%

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

      2. add-cube-cbrt 60.3%

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

      3. add-sqr-sqrt 25.2%

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

      4. times-frac 25.2%

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

      5. pow2 25.2%

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

      6. cbrt-div 25.2%

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

      7. rem-cbrt-cube 25.2%

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

      8. cbrt-div 25.2%

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

      9. rem-cbrt-cube 38.8%

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

   6. Applied egg-rr 38.8%

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

      1. add-cube-cbrt 38.8%

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

      2. pow3 38.8%

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

   8. Applied egg-rr 92.4%

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

   if 1.25999999999999996e-5 < t < 3.30000000000000009e93

   1. Initial program 57.7%

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

   3. Simplified 57.4%

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

      1. associate-*r* 58.1%

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

      2. add-sqr-sqrt 57.9%

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

      3. times-frac 62.1%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/l* 87.3%

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

      2. associate-*l* 83.4%

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

   8. Simplified 83.4%

      \[\leadsto \color{blue}{\left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right) \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-172}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-5} \lor \neg \left(t \leq 3.3 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.15 \cdot 10^{+105}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.5e-36)
    (/
     2.0
     (pow
      (* (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) (* (cbrt k) (cbrt 2.0)))
      3.0))
    (if (<= k 4.15e+105)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (/
       2.0
       (pow
        (*
         (/ (/ t_m (cbrt l)) (cbrt l))
         (cbrt (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (tan k)))))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))) * (cbrt(k) * cbrt(2.0))), 3.0);
	} else if (k <= 4.15e+105) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow((((t_m / cbrt(l)) / cbrt(l)) * cbrt(((2.0 + pow((k / t_m), 2.0)) * (sin(k) * tan(k))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))) * (Math.cbrt(k) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 4.15e+105) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow((((t_m / Math.cbrt(l)) / Math.cbrt(l)) * Math.cbrt(((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.tan(k))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.5e-36)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) * Float64(cbrt(k) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 4.15e+105)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m / cbrt(l)) / cbrt(l)) * cbrt(Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * tan(k))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.5e-36], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.15e+105], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.50000000000000002e-36

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

      1. associate-/r* 59.4%

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

      2. add-cube-cbrt 59.2%

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

      3. add-sqr-sqrt 30.4%

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

      4. times-frac 30.4%

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

      5. pow2 30.4%

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

      6. cbrt-div 30.4%

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

      7. rem-cbrt-cube 30.5%

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

      8. cbrt-div 30.4%

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

      9. rem-cbrt-cube 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. add-cube-cbrt 36.9%

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

      2. pow3 36.9%

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

   8. Applied egg-rr 80.7%

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

      1. add-cube-cbrt 80.6%

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

      2. pow3 80.6%

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

   10. Applied egg-rr 86.6%

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

   11. Taylor expanded in k around 0 73.2%

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

   if 2.50000000000000002e-36 < k < 4.15e105

   1. Initial program 48.6%

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

   3. Simplified 48.6%

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

      1. associate-/r* 54.4%

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

      2. add-cube-cbrt 54.3%

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

      3. add-sqr-sqrt 17.1%

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

      4. times-frac 17.1%

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

      5. pow2 17.1%

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

      6. cbrt-div 17.1%

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

      7. rem-cbrt-cube 17.2%

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

      8. cbrt-div 17.2%

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

      9. rem-cbrt-cube 20.6%

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

   6. Applied egg-rr 20.6%

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

   7. Taylor expanded in t around 0 80.1%

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

      1. associate-*r* 80.2%

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

      2. times-frac 80.2%

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

   9. Simplified 80.2%

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

   if 4.15e105 < k

   1. Initial program 53.8%

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

   3. Simplified 53.7%

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

      1. associate-/r* 57.2%

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

      2. add-cube-cbrt 57.1%

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

      3. add-sqr-sqrt 33.6%

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

      4. times-frac 33.6%

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

      5. pow2 33.6%

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

      6. cbrt-div 33.5%

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

      7. rem-cbrt-cube 33.6%

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

      8. cbrt-div 33.6%

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

      9. rem-cbrt-cube 42.2%

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

   6. Applied egg-rr 42.2%

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

      1. add-cube-cbrt 42.2%

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

      2. pow3 42.3%

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

   8. Applied egg-rr 83.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.15 \cdot 10^{+105}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot t\_2\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 270000000:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot {t\_2}^{3}}}{t\_3}\\ \mathbf{elif}\;k \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}} \cdot \left({\ell}^{2} \cdot \cos k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot t\_3}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (pow (cbrt l) 2.0)))
        (t_3 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= k 3.1e-153)
      (/ 2.0 (pow (* (* (cbrt (sin k)) t_2) (* (cbrt k) (cbrt 2.0))) 3.0))
      (if (<= k 270000000.0)
        (/
         2.0
         (pow
          (*
           (/ (pow t_m 1.5) l)
           (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (sqrt (* (sin k) (tan k)))))
          2.0))
        (if (<= k 3.4e+37)
          (/ (/ 2.0 (* (sin k) (pow t_2 3.0))) t_3)
          (if (<= k 1.42e+147)
            (*
             (/ 2.0 (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))
             (* (pow l 2.0) (cos k)))
            (/
             2.0
             (pow
              (* (* t_m (pow l -0.6666666666666666)) (cbrt (* (sin k) t_3)))
              3.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / pow(cbrt(l), 2.0);
	double t_3 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (k <= 3.1e-153) {
		tmp = 2.0 / pow(((cbrt(sin(k)) * t_2) * (cbrt(k) * cbrt(2.0))), 3.0);
	} else if (k <= 270000000.0) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k / t_m))) * sqrt((sin(k) * tan(k))))), 2.0);
	} else if (k <= 3.4e+37) {
		tmp = (2.0 / (sin(k) * pow(t_2, 3.0))) / t_3;
	} else if (k <= 1.42e+147) {
		tmp = (2.0 / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))) * (pow(l, 2.0) * cos(k));
	} else {
		tmp = 2.0 / pow(((t_m * pow(l, -0.6666666666666666)) * cbrt((sin(k) * t_3))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (k <= 3.1e-153) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * t_2) * (Math.cbrt(k) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 270000000.0) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * Math.sqrt((Math.sin(k) * Math.tan(k))))), 2.0);
	} else if (k <= 3.4e+37) {
		tmp = (2.0 / (Math.sin(k) * Math.pow(t_2, 3.0))) / t_3;
	} else if (k <= 1.42e+147) {
		tmp = (2.0 / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))) * (Math.pow(l, 2.0) * Math.cos(k));
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(l, -0.6666666666666666)) * Math.cbrt((Math.sin(k) * t_3))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / (cbrt(l) ^ 2.0))
	t_3 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (k <= 3.1e-153)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * t_2) * Float64(cbrt(k) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 270000000.0)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * sqrt(Float64(sin(k) * tan(k))))) ^ 2.0));
	elseif (k <= 3.4e+37)
		tmp = Float64(Float64(2.0 / Float64(sin(k) * (t_2 ^ 3.0))) / t_3);
	elseif (k <= 1.42e+147)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))) * Float64((l ^ 2.0) * cos(k)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (l ^ -0.6666666666666666)) * cbrt(Float64(sin(k) * t_3))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.1e-153], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 270000000.0], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.4e+37], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[k, 1.42e+147], N[(N[(2.0 / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if k < 3.09999999999999995e-153

   1. Initial program 51.5%

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

   3. Simplified 51.5%

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

      1. associate-/r* 59.1%

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

      2. add-cube-cbrt 58.9%

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

      3. add-sqr-sqrt 30.4%

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

      4. times-frac 30.4%

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

      5. pow2 30.4%

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

      6. cbrt-div 30.5%

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

      7. rem-cbrt-cube 30.5%

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

      8. cbrt-div 30.4%

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

      9. rem-cbrt-cube 35.7%

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

   6. Applied egg-rr 35.7%

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

      1. add-cube-cbrt 35.7%

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

      2. pow3 35.7%

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

   8. Applied egg-rr 78.7%

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

      1. add-cube-cbrt 78.6%

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

      2. pow3 78.6%

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

   10. Applied egg-rr 85.2%

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

   11. Taylor expanded in k around 0 70.4%

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

   if 3.09999999999999995e-153 < k < 2.7e8

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

      1. associate-*l* 52.9%

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

      2. associate-/r* 56.2%

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

      3. associate-+r+ 56.2%

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

      4. metadata-eval 56.2%

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

      5. associate-*l* 56.2%

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

      6. add-sqr-sqrt 20.0%

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

      7. pow2 20.0%

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

   6. Applied egg-rr 30.7%

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

   if 2.7e8 < k < 3.40000000000000006e37

   1. Initial program 84.3%

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

   3. Simplified 84.3%

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

      1. associate-/r* 99.7%

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

      2. add-cube-cbrt 99.5%

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

      3. add-sqr-sqrt 16.1%

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

      4. times-frac 16.1%

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

      5. pow2 16.1%

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

      6. cbrt-div 16.1%

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

      7. rem-cbrt-cube 16.7%

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

      8. cbrt-div 16.7%

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

      9. rem-cbrt-cube 16.3%

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

   6. Applied egg-rr 16.3%

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

      1. add-cube-cbrt 16.7%

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

      2. pow3 16.7%

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

   8. Applied egg-rr 98.5%

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

      1. *-un-lft-identity 98.5%

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

      2. unpow-prod-down 98.5%

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

      3. unpow3 98.5%

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

      4. add-cube-cbrt 98.7%

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

      5. associate-/l/ 99.0%

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

      6. pow2 99.0%

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

      7. associate-+r+ 99.0%

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

      8. metadata-eval 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. *-lft-identity 99.0%

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

       2. associate-/r* 99.0%

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

   12. Simplified 99.0%

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

   if 3.40000000000000006e37 < k < 1.42e147

   1. Initial program 48.0%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in t around 0 77.1%

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

      1. associate-/r/ 78.6%

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

      2. associate-*r* 78.7%

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

   7. Applied egg-rr 78.7%

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

   if 1.42e147 < k

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

      1. associate-*l* 50.9%

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

      2. associate-/r* 54.9%

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

      3. associate-+r+ 54.9%

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

      4. metadata-eval 54.9%

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

      5. associate-*l* 54.9%

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

      6. add-cube-cbrt 54.9%

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

      7. pow3 54.9%

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

   6. Applied egg-rr 62.5%

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

      1. pow1/3 62.3%

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

      2. pow-pow 47.4%

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

      3. metadata-eval 47.4%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 270000000:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}} \cdot \left({\ell}^{2} \cdot \cos k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-169}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.45e-169)
    (*
     2.0
     (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ (/ t_m (cbrt l)) (cbrt l))) 3.0)
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.45e-169) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * ((t_m / cbrt(l)) / cbrt(l))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.45e-169) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * ((t_m / Math.cbrt(l)) / Math.cbrt(l))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.45e-169)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(Float64(t_m / cbrt(l)) / cbrt(l))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-169], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-169

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

      1. associate-/r* 57.6%

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

      2. add-cube-cbrt 57.5%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. pow2 31.4%

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

      6. cbrt-div 31.4%

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

      7. rem-cbrt-cube 31.4%

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

      8. cbrt-div 31.4%

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

      9. rem-cbrt-cube 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. times-frac 64.6%

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

   9. Simplified 64.6%

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

   if 1.4500000000000001e-169 < t

   1. Initial program 53.6%

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

   3. Simplified 53.7%

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

      1. associate-/r* 59.9%

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

      2. add-cube-cbrt 59.7%

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

      3. add-sqr-sqrt 26.2%

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

      4. times-frac 26.2%

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

      5. pow2 26.2%

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

      6. cbrt-div 26.2%

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

      7. rem-cbrt-cube 26.2%

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

      8. cbrt-div 26.2%

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

      9. rem-cbrt-cube 36.5%

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

   6. Applied egg-rr 36.5%

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

      1. add-cube-cbrt 36.5%

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

      2. pow3 36.5%

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

   8. Applied egg-rr 86.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-169}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.45e-36)
    (/
     2.0
     (pow
      (* (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) (* (cbrt k) (cbrt 2.0)))
      3.0))
    (if (<= k 2.75e+146)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (/
       2.0
       (pow
        (*
         (* t_m (pow l -0.6666666666666666))
         (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.45e-36) {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))) * (cbrt(k) * cbrt(2.0))), 3.0);
	} else if (k <= 2.75e+146) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m * pow(l, -0.6666666666666666)) * cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.45e-36) {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))) * (Math.cbrt(k) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 2.75e+146) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(l, -0.6666666666666666)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.45e-36)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) * Float64(cbrt(k) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 2.75e+146)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (l ^ -0.6666666666666666)) * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.45e-36], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.75e+146], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.4499999999999998e-36

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

      1. associate-/r* 59.4%

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

      2. add-cube-cbrt 59.2%

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

      3. add-sqr-sqrt 30.4%

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

      4. times-frac 30.4%

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

      5. pow2 30.4%

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

      6. cbrt-div 30.4%

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

      7. rem-cbrt-cube 30.5%

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

      8. cbrt-div 30.4%

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

      9. rem-cbrt-cube 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. add-cube-cbrt 36.9%

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

      2. pow3 36.9%

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

   8. Applied egg-rr 80.7%

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

      1. add-cube-cbrt 80.6%

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

      2. pow3 80.6%

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

   10. Applied egg-rr 86.6%

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

   11. Taylor expanded in k around 0 73.2%

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

   if 2.4499999999999998e-36 < k < 2.7500000000000002e146

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

      1. associate-/r* 56.7%

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

      2. add-cube-cbrt 56.6%

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

      3. add-sqr-sqrt 19.9%

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

      4. times-frac 19.9%

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

      5. pow2 19.9%

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

      6. cbrt-div 19.9%

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

      7. rem-cbrt-cube 20.0%

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

      8. cbrt-div 20.0%

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

      9. rem-cbrt-cube 22.8%

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

   6. Applied egg-rr 22.8%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. associate-*r* 77.9%

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

      2. times-frac 77.9%

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

   9. Simplified 77.9%

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

   if 2.7500000000000002e146 < k

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

      1. associate-*l* 50.9%

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

      2. associate-/r* 54.9%

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

      3. associate-+r+ 54.9%

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

      4. metadata-eval 54.9%

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

      5. associate-*l* 54.9%

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

      6. add-cube-cbrt 54.9%

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

      7. pow3 54.9%

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

   6. Applied egg-rr 62.5%

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

      1. pow1/3 62.3%

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

      2. pow-pow 47.4%

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

      3. metadata-eval 47.4%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := 2 + t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-177}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t\_m \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+93}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot t\_3\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)) (t_3 (+ 2.0 t_2)))
   (*
    t_s
    (if (<= t_m 1e-177)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (<= t_m 2.7e-5)
        (/
         2.0
         (*
          (* (tan k) (+ 1.0 (+ t_2 1.0)))
          (pow (* (pow l -0.6666666666666666) (* t_m (cbrt k))) 3.0)))
        (if (<= t_m 6.2e+93)
          (* (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) (/ l t_3))
          (/
           2.0
           (*
            (sin k)
            (* (* (tan k) t_3) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double tmp;
	if (t_m <= 1e-177) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 2.7e-5) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * pow((pow(l, -0.6666666666666666) * (t_m * cbrt(k))), 3.0));
	} else if (t_m <= 6.2e+93) {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / t_3);
	} else {
		tmp = 2.0 / (sin(k) * ((tan(k) * t_3) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double tmp;
	if (t_m <= 1e-177) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 2.7e-5) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * Math.pow((Math.pow(l, -0.6666666666666666) * (t_m * Math.cbrt(k))), 3.0));
	} else if (t_m <= 6.2e+93) {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / t_3);
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.tan(k) * t_3) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(2.0 + t_2)
	tmp = 0.0
	if (t_m <= 1e-177)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 2.7e-5)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * (Float64((l ^ -0.6666666666666666) * Float64(t_m * cbrt(k))) ^ 3.0)));
	elseif (t_m <= 6.2e+93)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / t_3));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(tan(k) * t_3) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-177], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e-5], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[l, -0.6666666666666666], $MachinePrecision] * N[(t$95$m * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+93], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.99999999999999952e-178

   1. Initial program 51.4%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-/r* 58.0%

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

      2. add-cube-cbrt 57.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. times-frac 31.6%

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

      5. pow2 31.6%

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

      6. cbrt-div 31.6%

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

      7. rem-cbrt-cube 31.6%

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

      8. cbrt-div 31.6%

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

      9. rem-cbrt-cube 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in t around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 64.3%

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

   9. Simplified 64.3%

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

   if 9.99999999999999952e-178 < t < 2.6999999999999999e-5

   1. Initial program 52.8%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in k around 0 46.9%

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

      1. add-cube-cbrt 46.8%

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

      2. pow3 46.8%

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

      3. div-inv 46.8%

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

      4. pow-flip 46.9%

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

      5. metadata-eval 46.9%

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

      6. cbrt-prod 46.9%

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

      7. *-commutative 46.9%

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

      8. cbrt-prod 47.8%

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

      9. unpow3 47.8%

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

      10. add-cbrt-cube 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. pow1/3 63.8%

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

      2. pow-pow 28.0%

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

      3. metadata-eval 28.0%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   9. Applied egg-rr 34.7%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{k}\right) \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 2.6999999999999999e-5 < t < 6.20000000000000038e93

   1. Initial program 57.7%

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

   3. Simplified 57.4%

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

      1. associate-*r* 58.1%

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

      2. *-un-lft-identity 58.1%

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

      3. times-frac 62.2%

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

      4. associate-*r* 66.2%

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

   6. Applied egg-rr 66.2%

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

   if 6.20000000000000038e93 < t

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

      1. associate-/r* 58.1%

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

      2. add-cube-cbrt 58.1%

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

      3. add-sqr-sqrt 26.9%

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

      4. times-frac 26.9%

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

      5. pow2 26.9%

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

      6. cbrt-div 26.9%

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

      7. rem-cbrt-cube 26.9%

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

      8. cbrt-div 26.9%

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

      9. rem-cbrt-cube 45.1%

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

   6. Applied egg-rr 45.1%

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

      1. add-cube-cbrt 45.0%

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

      2. pow3 45.0%

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

   8. Applied egg-rr 94.0%

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

      1. pow1 94.0%

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

      2. unpow-prod-down 89.6%

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

      3. unpow3 89.6%

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

      4. add-cube-cbrt 89.5%

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

      5. associate-/l/ 89.6%

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

      6. pow2 89.6%

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

      7. associate-+r+ 89.6%

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

      8. metadata-eval 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. unpow1 89.6%

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

       2. associate-*l* 89.6%

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

   12. Simplified 89.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-177}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+93}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + t\_2\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= k 2.5e-36)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ t_2 1.0)))
        (pow (* (/ (/ t_m (cbrt l)) (cbrt l)) (cbrt k)) 3.0)))
      (if (<= k 3.8e+147)
        (*
         2.0
         (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
        (/
         2.0
         (pow
          (*
           (* t_m (pow l -0.6666666666666666))
           (cbrt (* (sin k) (* (tan k) (+ 2.0 t_2)))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * pow((((t_m / cbrt(l)) / cbrt(l)) * cbrt(k)), 3.0));
	} else if (k <= 3.8e+147) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m * pow(l, -0.6666666666666666)) * cbrt((sin(k) * (tan(k) * (2.0 + t_2))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * Math.pow((((t_m / Math.cbrt(l)) / Math.cbrt(l)) * Math.cbrt(k)), 3.0));
	} else if (k <= 3.8e+147) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(l, -0.6666666666666666)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + t_2))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (k <= 2.5e-36)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * (Float64(Float64(Float64(t_m / cbrt(l)) / cbrt(l)) * cbrt(k)) ^ 3.0)));
	elseif (k <= 3.8e+147)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (l ^ -0.6666666666666666)) * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + t_2))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.5e-36], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+147], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.50000000000000002e-36

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

      1. associate-/r* 59.4%

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

      2. add-cube-cbrt 59.2%

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

      3. add-sqr-sqrt 30.4%

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

      4. times-frac 30.4%

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

      5. pow2 30.4%

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

      6. cbrt-div 30.4%

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

      7. rem-cbrt-cube 30.5%

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

      8. cbrt-div 30.4%

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

      9. rem-cbrt-cube 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. add-cube-cbrt 36.9%

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

      2. pow3 36.9%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in k around 0 78.2%

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

   if 2.50000000000000002e-36 < k < 3.7999999999999997e147

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

      1. associate-/r* 56.7%

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

      2. add-cube-cbrt 56.6%

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

      3. add-sqr-sqrt 19.9%

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

      4. times-frac 19.9%

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

      5. pow2 19.9%

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

      6. cbrt-div 19.9%

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

      7. rem-cbrt-cube 20.0%

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

      8. cbrt-div 20.0%

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

      9. rem-cbrt-cube 22.8%

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

   6. Applied egg-rr 22.8%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. associate-*r* 77.9%

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

      2. times-frac 77.9%

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

   9. Simplified 77.9%

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

   if 3.7999999999999997e147 < k

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

      1. associate-*l* 50.9%

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

      2. associate-/r* 54.9%

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

      3. associate-+r+ 54.9%

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

      4. metadata-eval 54.9%

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

      5. associate-*l* 54.9%

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

      6. add-cube-cbrt 54.9%

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

      7. pow3 54.9%

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

   6. Applied egg-rr 62.5%

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

      1. pow1/3 62.3%

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

      2. pow-pow 47.4%

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

      3. metadata-eval 47.4%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t\_m \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + t\_2\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= k 2.5e-36)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ t_2 1.0)))
        (pow (* (pow l -0.6666666666666666) (* t_m (cbrt k))) 3.0)))
      (if (<= k 1.18e+147)
        (*
         2.0
         (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
        (/
         2.0
         (pow
          (*
           (* t_m (pow l -0.6666666666666666))
           (cbrt (* (sin k) (* (tan k) (+ 2.0 t_2)))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * pow((pow(l, -0.6666666666666666) * (t_m * cbrt(k))), 3.0));
	} else if (k <= 1.18e+147) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m * pow(l, -0.6666666666666666)) * cbrt((sin(k) * (tan(k) * (2.0 + t_2))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (k <= 2.5e-36) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * Math.pow((Math.pow(l, -0.6666666666666666) * (t_m * Math.cbrt(k))), 3.0));
	} else if (k <= 1.18e+147) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(l, -0.6666666666666666)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + t_2))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (k <= 2.5e-36)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * (Float64((l ^ -0.6666666666666666) * Float64(t_m * cbrt(k))) ^ 3.0)));
	elseif (k <= 1.18e+147)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (l ^ -0.6666666666666666)) * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + t_2))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.5e-36], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[l, -0.6666666666666666], $MachinePrecision] * N[(t$95$m * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.18e+147], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.50000000000000002e-36

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in k around 0 50.0%

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

      1. add-cube-cbrt 50.0%

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

      2. pow3 50.0%

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

      3. div-inv 49.8%

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

      4. pow-flip 49.9%

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

      5. metadata-eval 49.9%

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

      6. cbrt-prod 49.9%

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

      7. *-commutative 49.9%

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

      8. cbrt-prod 50.1%

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

      9. unpow3 50.1%

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

      10. add-cbrt-cube 63.3%

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

   7. Applied egg-rr 63.3%

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

      1. pow1/3 61.0%

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

      2. pow-pow 37.1%

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

      3. metadata-eval 37.1%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   9. Applied egg-rr 38.3%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{k}\right) \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 2.50000000000000002e-36 < k < 1.18000000000000006e147

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

      1. associate-/r* 56.7%

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

      2. add-cube-cbrt 56.6%

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

      3. add-sqr-sqrt 19.9%

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

      4. times-frac 19.9%

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

      5. pow2 19.9%

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

      6. cbrt-div 19.9%

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

      7. rem-cbrt-cube 20.0%

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

      8. cbrt-div 20.0%

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

      9. rem-cbrt-cube 22.8%

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

   6. Applied egg-rr 22.8%

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

   7. Taylor expanded in t around 0 77.8%

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

      1. associate-*r* 77.9%

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

      2. times-frac 77.9%

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

   9. Simplified 77.9%

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

   if 1.18000000000000006e147 < k

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

      1. associate-*l* 50.9%

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

      2. associate-/r* 54.9%

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

      3. associate-+r+ 54.9%

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

      4. metadata-eval 54.9%

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

      5. associate-*l* 54.9%

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

      6. add-cube-cbrt 54.9%

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

      7. pow3 54.9%

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

   6. Applied egg-rr 62.5%

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

      1. pow1/3 62.3%

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

      2. pow-pow 47.4%

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

      3. metadata-eval 47.4%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\ell}^{-0.6666666666666666}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\\ t_4 := \frac{2}{t\_3 \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t\_m \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-177}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+93}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (* (tan k) (+ 1.0 (+ t_2 1.0))))
        (t_4
         (/
          2.0
          (* t_3 (pow (* (pow l -0.6666666666666666) (* t_m (cbrt k))) 3.0)))))
   (*
    t_s
    (if (<= t_m 1e-177)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (<= t_m 2.2e-5)
        t_4
        (if (<= t_m 6.2e+93)
          (*
           (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))
           (/ l (+ 2.0 t_2)))
          (if (<= t_m 1.7e+187)
            (/ 2.0 (* t_3 (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
            t_4)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = tan(k) * (1.0 + (t_2 + 1.0));
	double t_4 = 2.0 / (t_3 * pow((pow(l, -0.6666666666666666) * (t_m * cbrt(k))), 3.0));
	double tmp;
	if (t_m <= 1e-177) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 2.2e-5) {
		tmp = t_4;
	} else if (t_m <= 6.2e+93) {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else if (t_m <= 1.7e+187) {
		tmp = 2.0 / (t_3 * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = t_4;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = Math.tan(k) * (1.0 + (t_2 + 1.0));
	double t_4 = 2.0 / (t_3 * Math.pow((Math.pow(l, -0.6666666666666666) * (t_m * Math.cbrt(k))), 3.0));
	double tmp;
	if (t_m <= 1e-177) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 2.2e-5) {
		tmp = t_4;
	} else if (t_m <= 6.2e+93) {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / (2.0 + t_2));
	} else if (t_m <= 1.7e+187) {
		tmp = 2.0 / (t_3 * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = t_4;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0)))
	t_4 = Float64(2.0 / Float64(t_3 * (Float64((l ^ -0.6666666666666666) * Float64(t_m * cbrt(k))) ^ 3.0)))
	tmp = 0.0
	if (t_m <= 1e-177)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 2.2e-5)
		tmp = t_4;
	elseif (t_m <= 6.2e+93)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + t_2)));
	elseif (t_m <= 1.7e+187)
		tmp = Float64(2.0 / Float64(t_3 * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = t_4;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 / N[(t$95$3 * N[Power[N[(N[Power[l, -0.6666666666666666], $MachinePrecision] * N[(t$95$m * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-177], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-5], t$95$4, If[LessEqual[t$95$m, 6.2e+93], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+187], N[(2.0 / N[(t$95$3 * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 9.99999999999999952e-178

   1. Initial program 51.4%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-/r* 58.0%

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

      2. add-cube-cbrt 57.9%

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

      3. add-sqr-sqrt 31.6%

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

      4. times-frac 31.6%

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

      5. pow2 31.6%

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

      6. cbrt-div 31.6%

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

      7. rem-cbrt-cube 31.6%

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

      8. cbrt-div 31.6%

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

      9. rem-cbrt-cube 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in t around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 64.3%

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

   9. Simplified 64.3%

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

   if 9.99999999999999952e-178 < t < 2.1999999999999999e-5 or 1.7e187 < t

   1. Initial program 53.6%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in k around 0 49.9%

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

      1. add-cube-cbrt 49.9%

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

      2. pow3 49.9%

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

      3. div-inv 49.9%

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

      4. pow-flip 50.0%

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

      5. metadata-eval 50.0%

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

      6. cbrt-prod 50.0%

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

      7. *-commutative 50.0%

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

      8. cbrt-prod 50.5%

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

      9. unpow3 50.5%

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

      10. add-cbrt-cube 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. pow1/3 62.2%

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

      2. pow-pow 29.3%

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

      3. metadata-eval 29.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}} \]

   9. Applied egg-rr 33.5%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \sqrt[3]{k}\right) \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 2.1999999999999999e-5 < t < 6.20000000000000038e93

   1. Initial program 57.7%

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

   3. Simplified 57.4%

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

      1. associate-*r* 58.1%

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

      2. *-un-lft-identity 58.1%

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

      3. times-frac 62.2%

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

      4. associate-*r* 66.2%

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

   6. Applied egg-rr 66.2%

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

   if 6.20000000000000038e93 < t < 1.7e187

   1. Initial program 46.1%

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

   3. Simplified 46.1%

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

      1. associate-/r* 51.8%

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

      2. add-sqr-sqrt 51.8%

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

      3. pow2 51.8%

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

      4. associate-/r* 46.1%

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

      5. sqrt-div 46.1%

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

      6. sqrt-pow1 66.4%

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

      7. metadata-eval 66.4%

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

      8. sqrt-prod 60.0%

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

      9. add-sqr-sqrt 90.3%

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

   6. Applied egg-rr 90.3%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-177}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+93}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left({\ell}^{-0.6666666666666666} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-169}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-55} \lor \neg \left(t\_m \leq 5.2 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + t\_2\right)\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.95e-169)
      (*
       2.0
       (* (/ (pow l 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
      (if (or (<= t_m 1.2e-55) (not (<= t_m 5.2e+92)))
        (/
         2.0
         (*
          (* (tan k) (+ 1.0 (+ t_2 1.0)))
          (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
        (/
         2.0
         (*
          (/ (pow t_m 3.0) l)
          (/ (* (sin k) (* (tan k) (+ 2.0 t_2))) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.95e-169) {
		tmp = 2.0 * ((pow(l, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if ((t_m <= 1.2e-55) || !(t_m <= 5.2e+92)) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 1.95d-169) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (t_m * (k ** 2.0d0))) * (cos(k) / (sin(k) ** 2.0d0)))
    else if ((t_m <= 1.2d-55) .or. (.not. (t_m <= 5.2d+92))) then
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (t_2 + 1.0d0))) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((sin(k) * (tan(k) * (2.0d0 + t_2))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.95e-169) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if ((t_m <= 1.2e-55) || !(t_m <= 5.2e+92)) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * (2.0 + t_2))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 1.95e-169:
		tmp = 2.0 * ((math.pow(l, 2.0) / (t_m * math.pow(k, 2.0))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	elif (t_m <= 1.2e-55) or not (t_m <= 5.2e+92):
		tmp = 2.0 / ((math.tan(k) * (1.0 + (t_2 + 1.0))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)))
	else:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((math.sin(k) * (math.tan(k) * (2.0 + t_2))) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.95e-169)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif ((t_m <= 1.2e-55) || !(t_m <= 5.2e+92))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + t_2))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.95e-169)
		tmp = 2.0 * (((l ^ 2.0) / (t_m * (k ^ 2.0))) * (cos(k) / (sin(k) ^ 2.0)));
	elseif ((t_m <= 1.2e-55) || ~((t_m <= 5.2e+92)))
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-169], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$m, 1.2e-55], N[Not[LessEqual[t$95$m, 5.2e+92]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.94999999999999988e-169

   1. Initial program 51.0%

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

   3. Simplified 51.0%

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

      1. associate-/r* 57.6%

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

      2. add-cube-cbrt 57.5%

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

      3. add-sqr-sqrt 31.4%

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

      4. times-frac 31.4%

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

      5. pow2 31.4%

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

      6. cbrt-div 31.4%

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

      7. rem-cbrt-cube 31.4%

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

      8. cbrt-div 31.4%

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

      9. rem-cbrt-cube 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in t around 0 62.7%

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

      1. associate-*r* 62.7%

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

      2. times-frac 64.6%

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

   9. Simplified 64.6%

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

   if 1.94999999999999988e-169 < t < 1.19999999999999996e-55 or 5.1999999999999998e92 < t

   1. Initial program 50.4%

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

   3. Simplified 50.5%

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

      1. associate-/r* 60.2%

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

      2. add-sqr-sqrt 60.2%

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

      3. pow2 60.2%

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

      4. associate-/r* 50.5%

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

      5. sqrt-div 50.5%

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

      6. sqrt-pow1 63.2%

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

      7. metadata-eval 63.2%

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

      8. sqrt-prod 40.7%

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

      9. add-sqr-sqrt 81.3%

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

   6. Applied egg-rr 81.3%

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

   if 1.19999999999999996e-55 < t < 5.1999999999999998e92

   1. Initial program 59.3%

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

   3. Simplified 59.2%

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

      1. associate-*l* 56.1%

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

      2. associate-/r* 56.3%

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

      3. associate-+r+ 56.3%

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

      4. metadata-eval 56.3%

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

      5. associate-*l* 56.4%

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

      6. associate-*l/ 59.4%

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

      7. associate-*l* 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. associate-/l* 62.4%

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

   8. Simplified 62.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-169}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-55} \lor \neg \left(t \leq 5.2 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{-212}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t\_m \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot t\_2\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.14 \cdot 10^{+123}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t\_m}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 8.2e-212)
      (*
       (* (pow l 2.0) (cos k))
       (/ 2.0 (* (* t_m (pow k 2.0)) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))
      (if (<= t_m 3.6e-90)
        (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
        (if (<= t_m 5.8e+102)
          (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) t_2)) l)))
          (if (<= t_m 1.14e+123)
            (pow (/ (cbrt (* (pow l 2.0) (pow k -2.0))) t_m) 3.0)
            (*
             (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))
             (/ l t_2)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 8.2e-212) {
		tmp = (pow(l, 2.0) * cos(k)) * (2.0 / ((t_m * pow(k, 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.6e-90) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 5.8e+102) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * t_2)) / l));
	} else if (t_m <= 1.14e+123) {
		tmp = pow((cbrt((pow(l, 2.0) * pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / t_2);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 8.2e-212) {
		tmp = (Math.pow(l, 2.0) * Math.cos(k)) * (2.0 / ((t_m * Math.pow(k, 2.0)) * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.6e-90) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 5.8e+102) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * t_2)) / l));
	} else if (t_m <= 1.14e+123) {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) * Math.pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / t_2);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 8.2e-212)
		tmp = Float64(Float64((l ^ 2.0) * cos(k)) * Float64(2.0 / Float64(Float64(t_m * (k ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	elseif (t_m <= 3.6e-90)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (t_m <= 5.8e+102)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * t_2)) / l)));
	elseif (t_m <= 1.14e+123)
		tmp = Float64(cbrt(Float64((l ^ 2.0) * (k ^ -2.0))) / t_m) ^ 3.0;
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.2e-212], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e-90], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 5.8e+102], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.14e+123], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 8.20000000000000028e-212

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in t around 0 61.1%

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

      1. associate-/r/ 60.7%

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

      2. associate-*r* 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. sin-mult 56.6%

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

   9. Applied egg-rr 56.6%

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

       1. div-sub 56.6%

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

       2. +-inverses 56.6%

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

       3. cos-0 56.6%

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

       4. metadata-eval 56.6%

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

       5. count-2 56.6%

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

   11. Simplified 56.6%

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

   if 8.20000000000000028e-212 < t < 3.59999999999999981e-90

   1. Initial program 45.9%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

   6. Applied egg-rr 38.5%

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

   7. Taylor expanded in t around 0 54.6%

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

      1. associate-/l* 54.7%

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

   9. Simplified 54.7%

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

   10. Taylor expanded in k around 0 80.7%

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

   if 3.59999999999999981e-90 < t < 5.8000000000000005e102

   1. Initial program 62.4%

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

   3. Simplified 62.6%

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

      1. associate-*l* 60.0%

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

      2. associate-/r* 62.4%

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

      3. associate-+r+ 62.4%

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

      4. metadata-eval 62.4%

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

      5. associate-*l* 62.5%

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

      6. associate-*l/ 65.1%

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

      7. associate-*l* 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. associate-/l* 67.7%

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

   8. Simplified 67.7%

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

   if 5.8000000000000005e102 < t < 1.14000000000000001e123

   1. Initial program 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in k around 0 26.4%

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

      1. associate-/r* 26.4%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

   7. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 26.4%

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

      2. pow3 26.4%

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

      3. cbrt-div 26.4%

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

      4. div-inv 26.4%

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

      5. pow-flip 26.4%

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

      6. metadata-eval 26.4%

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

      7. unpow3 26.4%

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

      8. add-cbrt-cube 75.9%

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

   9. Applied egg-rr 75.9%

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

   if 1.14000000000000001e123 < t

   1. Initial program 53.3%

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

   3. Simplified 50.2%

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

      1. associate-*r* 58.1%

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

      2. *-un-lft-identity 58.1%

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

      3. times-frac 58.1%

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

      4. associate-*r* 61.3%

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

   6. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-212}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.14 \cdot 10^{+123}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-213}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t\_m \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot t\_2\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t\_m}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 4.8e-213)
      (*
       (* (pow l 2.0) (cos k))
       (/ 2.0 (* (* t_m (pow k 2.0)) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))
      (if (<= t_m 3.6e-90)
        (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
        (if (<= t_m 5.6e+102)
          (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) t_2)) l)))
          (if (<= t_m 4.5e+153)
            (pow (/ (cbrt (* (pow l 2.0) (pow k -2.0))) t_m) 3.0)
            (*
             (/ l t_2)
             (* l (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.8e-213) {
		tmp = (pow(l, 2.0) * cos(k)) * (2.0 / ((t_m * pow(k, 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.6e-90) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * t_2)) / l));
	} else if (t_m <= 4.5e+153) {
		tmp = pow((cbrt((pow(l, 2.0) * pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = (l / t_2) * (l * (2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.8e-213) {
		tmp = (Math.pow(l, 2.0) * Math.cos(k)) * (2.0 / ((t_m * Math.pow(k, 2.0)) * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.6e-90) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * t_2)) / l));
	} else if (t_m <= 4.5e+153) {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) * Math.pow(k, -2.0))) / t_m), 3.0);
	} else {
		tmp = (l / t_2) * (l * (2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.8e-213)
		tmp = Float64(Float64((l ^ 2.0) * cos(k)) * Float64(2.0 / Float64(Float64(t_m * (k ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	elseif (t_m <= 3.6e-90)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * t_2)) / l)));
	elseif (t_m <= 4.5e+153)
		tmp = Float64(cbrt(Float64((l ^ 2.0) * (k ^ -2.0))) / t_m) ^ 3.0;
	else
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.8e-213], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e-90], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+153], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 4.79999999999999991e-213

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in t around 0 61.1%

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

      1. associate-/r/ 60.7%

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

      2. associate-*r* 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. sin-mult 56.6%

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

   9. Applied egg-rr 56.6%

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

       1. div-sub 56.6%

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

       2. +-inverses 56.6%

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

       3. cos-0 56.6%

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

       4. metadata-eval 56.6%

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

       5. count-2 56.6%

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

   11. Simplified 56.6%

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

   if 4.79999999999999991e-213 < t < 3.59999999999999981e-90

   1. Initial program 45.9%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

   6. Applied egg-rr 38.5%

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

   7. Taylor expanded in t around 0 54.6%

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

      1. associate-/l* 54.7%

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

   9. Simplified 54.7%

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

   10. Taylor expanded in k around 0 80.7%

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

   if 3.59999999999999981e-90 < t < 5.60000000000000037e102

   1. Initial program 62.4%

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

   3. Simplified 62.6%

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

      1. associate-*l* 60.0%

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

      2. associate-/r* 62.4%

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

      3. associate-+r+ 62.4%

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

      4. metadata-eval 62.4%

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

      5. associate-*l* 62.5%

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

      6. associate-*l/ 65.1%

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

      7. associate-*l* 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. associate-/l* 67.7%

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

   8. Simplified 67.7%

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

   if 5.60000000000000037e102 < t < 4.5000000000000001e153

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in k around 0 53.7%

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

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

   7. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 53.7%

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

      2. pow3 53.7%

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

      3. cbrt-div 53.7%

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

      4. div-inv 53.7%

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

      5. pow-flip 53.7%

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

      6. metadata-eval 53.7%

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

      7. unpow3 53.7%

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

      8. add-cbrt-cube 73.8%

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

   9. Applied egg-rr 73.8%

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

   if 4.5000000000000001e153 < t

   1. Initial program 48.9%

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

   3. Simplified 44.8%

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

      1. associate-*r* 55.7%

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

      2. *-un-lft-identity 55.7%

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

      3. times-frac 55.7%

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

      4. associate-*r* 60.0%

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

   6. Applied egg-rr 60.0%

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

      1. /-rgt-identity 60.0%

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

      2. *-commutative 60.0%

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

      3. associate-*l* 55.7%

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

   8. Simplified 55.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-213}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-210}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t\_m \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t\_m \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + t\_2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.15e-210)
      (*
       (* (pow l 2.0) (cos k))
       (/ 2.0 (* (* t_m (pow k 2.0)) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))
      (if (<= t_m 3.7e-90)
        (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
        (if (<= t_m 4.5e+77)
          (/
           2.0
           (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) (+ 2.0 t_2))) l)))
          (/
           2.0
           (*
            (* (tan k) (+ 1.0 (+ t_2 1.0)))
            (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.15e-210) {
		tmp = (pow(l, 2.0) * cos(k)) * (2.0 / ((t_m * pow(k, 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.7e-90) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 4.5e+77) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 1.15d-210) then
        tmp = ((l ** 2.0d0) * cos(k)) * (2.0d0 / ((t_m * (k ** 2.0d0)) * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    else if (t_m <= 3.7d-90) then
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if (t_m <= 4.5d+77) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((sin(k) * (tan(k) * (2.0d0 + t_2))) / l))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (t_2 + 1.0d0))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.15e-210) {
		tmp = (Math.pow(l, 2.0) * Math.cos(k)) * (2.0 / ((t_m * Math.pow(k, 2.0)) * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 3.7e-90) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 4.5e+77) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * (2.0 + t_2))) / l));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 1.15e-210:
		tmp = (math.pow(l, 2.0) * math.cos(k)) * (2.0 / ((t_m * math.pow(k, 2.0)) * (0.5 - (math.cos((2.0 * k)) / 2.0))))
	elif t_m <= 3.7e-90:
		tmp = math.pow(((l * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((1.0 / t_m))), 2.0)
	elif t_m <= 4.5e+77:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((math.sin(k) * (math.tan(k) * (2.0 + t_2))) / l))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (t_2 + 1.0))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.15e-210)
		tmp = Float64(Float64((l ^ 2.0) * cos(k)) * Float64(2.0 / Float64(Float64(t_m * (k ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	elseif (t_m <= 3.7e-90)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (t_m <= 4.5e+77)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + t_2))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.15e-210)
		tmp = ((l ^ 2.0) * cos(k)) * (2.0 / ((t_m * (k ^ 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	elseif (t_m <= 3.7e-90)
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif (t_m <= 4.5e+77)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((sin(k) * (tan(k) * (2.0 + t_2))) / l));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.15e-210], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.7e-90], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.5e+77], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.15e-210

   1. Initial program 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in t around 0 61.1%

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

      1. associate-/r/ 60.7%

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

      2. associate-*r* 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. unpow2 60.7%

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

      2. sin-mult 56.6%

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

   9. Applied egg-rr 56.6%

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

       1. div-sub 56.6%

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

       2. +-inverses 56.6%

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

       3. cos-0 56.6%

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

       4. metadata-eval 56.6%

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

       5. count-2 56.6%

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

   11. Simplified 56.6%

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

   if 1.15e-210 < t < 3.70000000000000018e-90

   1. Initial program 45.9%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

   6. Applied egg-rr 38.5%

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

   7. Taylor expanded in t around 0 54.6%

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

      1. associate-/l* 54.7%

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

   9. Simplified 54.7%

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

   10. Taylor expanded in k around 0 80.7%

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

   if 3.70000000000000018e-90 < t < 4.50000000000000024e77

   1. Initial program 62.4%

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

   3. Simplified 62.6%

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

      1. associate-*l* 60.0%

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

      2. associate-/r* 62.4%

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

      3. associate-+r+ 62.4%

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

      4. metadata-eval 62.4%

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

      5. associate-*l* 62.5%

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

      6. associate-*l/ 65.1%

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

      7. associate-*l* 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. associate-/l* 67.7%

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

   8. Simplified 67.7%

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

   if 4.50000000000000024e77 < t

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

      1. unpow3 50.7%

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

      2. times-frac 71.8%

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

      3. pow2 71.8%

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

   6. Applied egg-rr 71.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-210}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.06 \cdot 10^{-52}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102} \lor \neg \left(t\_m \leq 5.4 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t\_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.06e-52)
    (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (or (<= t_m 5.6e+102) (not (<= t_m 5.4e+154)))
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* l (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k))))))
      (pow (/ (cbrt (* (pow l 2.0) (pow k -2.0))) t_m) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.06e-52) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((t_m <= 5.6e+102) || !(t_m <= 5.4e+154)) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))));
	} else {
		tmp = pow((cbrt((pow(l, 2.0) * pow(k, -2.0))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.06e-52) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((t_m <= 5.6e+102) || !(t_m <= 5.4e+154)) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))));
	} else {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) * Math.pow(k, -2.0))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.06e-52)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif ((t_m <= 5.6e+102) || !(t_m <= 5.4e+154))
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k))))));
	else
		tmp = Float64(cbrt(Float64((l ^ 2.0) * (k ^ -2.0))) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.06e-52], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[Or[LessEqual[t$95$m, 5.6e+102], N[Not[LessEqual[t$95$m, 5.4e+154]], $MachinePrecision]], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.06e-52

   1. Initial program 50.9%

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

   3. Simplified 49.2%

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

      1. add-sqr-sqrt 34.7%

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

      2. pow2 34.7%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in t around 0 34.7%

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

      1. associate-/l* 34.7%

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

   9. Simplified 34.7%

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

   10. Taylor expanded in k around 0 20.3%

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

   if 1.06e-52 < t < 5.60000000000000037e102 or 5.40000000000000011e154 < t

   1. Initial program 55.0%

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

   3. Simplified 50.1%

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

      1. associate-*r* 55.3%

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

      2. *-un-lft-identity 55.3%

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

      3. times-frac 57.0%

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

      4. associate-*r* 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. /-rgt-identity 60.4%

         \[\leadsto \color{blue}{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-commutative 60.4%

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l* 57.0%

         \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   if 5.60000000000000037e102 < t < 5.40000000000000011e154

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

   7. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 53.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}} \]

      2. pow3 53.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}}\right)}^{3}} \]

      3. cbrt-div 53.7%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      4. div-inv 53.7%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{2}}}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      5. pow-flip 53.7%

         \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot \color{blue}{{k}^{\left(-2\right)}}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      6. metadata-eval 53.7%

         \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{\color{blue}{-2}}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      7. unpow3 53.7%

         \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \]

      8. add-cbrt-cube 73.8%

         \[\leadsto {\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{\color{blue}{t}}\right)}^{3} \]

   9. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.06 \cdot 10^{-52}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102} \lor \neg \left(t \leq 5.4 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2} \cdot {k}^{-2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-211}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t\_m \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t\_m \leq 6.3 \cdot 10^{-53}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-211)
    (*
     (* (pow l 2.0) (cos k))
     (/ 2.0 (* (* t_m (pow k 2.0)) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))
    (if (<= t_m 6.3e-53)
      (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* l (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-211) {
		tmp = (pow(l, 2.0) * cos(k)) * (2.0 / ((t_m * pow(k, 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 6.3e-53) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.8d-211) then
        tmp = ((l ** 2.0d0) * cos(k)) * (2.0d0 / ((t_m * (k ** 2.0d0)) * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    else if (t_m <= 6.3d-53) then
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l / (2.0d0 + ((k / t_m) ** 2.0d0))) * (l * (2.0d0 / ((t_m ** 3.0d0) * (sin(k) * tan(k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-211) {
		tmp = (Math.pow(l, 2.0) * Math.cos(k)) * (2.0 / ((t_m * Math.pow(k, 2.0)) * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	} else if (t_m <= 6.3e-53) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.8e-211:
		tmp = (math.pow(l, 2.0) * math.cos(k)) * (2.0 / ((t_m * math.pow(k, 2.0)) * (0.5 - (math.cos((2.0 * k)) / 2.0))))
	elif t_m <= 6.3e-53:
		tmp = math.pow(((l * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l / (2.0 + math.pow((k / t_m), 2.0))) * (l * (2.0 / (math.pow(t_m, 3.0) * (math.sin(k) * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.8e-211)
		tmp = Float64(Float64((l ^ 2.0) * cos(k)) * Float64(2.0 / Float64(Float64(t_m * (k ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	elseif (t_m <= 6.3e-53)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.8e-211)
		tmp = ((l ^ 2.0) * cos(k)) * (2.0 / ((t_m * (k ^ 2.0)) * (0.5 - (cos((2.0 * k)) / 2.0))));
	elseif (t_m <= 6.3e-53)
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l / (2.0 + ((k / t_m) ^ 2.0))) * (l * (2.0 / ((t_m ^ 3.0) * (sin(k) * tan(k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-211], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.3e-53], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.8000000000000004e-211

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 61.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 60.7%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left({\ell}^{2} \cdot \cos k\right)} \]

      2. associate-*r* 60.7%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

   7. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\frac{2}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}} \cdot \left({\ell}^{2} \cdot \cos k\right)} \]
   8. Step-by-step derivation

      1. unpow2 60.7%

         \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

      2. sin-mult 56.6%

         \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

   9. Applied egg-rr 56.6%

      \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]
   10. Step-by-step derivation

       1. div-sub 56.6%

          \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

       2. +-inverses 56.6%

          \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

       3. cos-0 56.6%

          \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

       4. metadata-eval 56.6%

          \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

       5. count-2 56.6%

          \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

   11. Simplified 56.6%

       \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}} \cdot \left({\ell}^{2} \cdot \cos k\right) \]

   if 4.8000000000000004e-211 < t < 6.29999999999999979e-53

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

      2. pow2 48.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

   6. Applied egg-rr 49.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 61.8%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 61.9%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 61.9%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 82.0%

       \[\leadsto {\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \color{blue}{\sqrt{\frac{1}{t}}}\right)}^{2} \]

   if 6.29999999999999979e-53 < t

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 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* 54.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 54.4%

         \[\leadsto \frac{\left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 55.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-*r* 58.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 58.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. /-rgt-identity 58.5%

         \[\leadsto \color{blue}{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-commutative 58.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-*l* 55.8%

         \[\leadsto \left(\ell \cdot \frac{2}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Simplified 55.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-211}:\\ \;\;\;\;\left({\ell}^{2} \cdot \cos k\right) \cdot \frac{2}{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-53}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t\_m}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-45)
    (pow (* (* l (/ (sqrt 2.0) (* k (sin k)))) (sqrt (/ 1.0 t_m))) 2.0)
    (/ 2.0 (* (* k (pow t_m 3.0)) (/ (* 2.0 k) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-45) {
		tmp = pow(((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((k * pow(t_m, 3.0)) * ((2.0 * k) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.8d-45) then
        tmp = ((l * (sqrt(2.0d0) / (k * sin(k)))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k * (t_m ** 3.0d0)) * ((2.0d0 * k) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-45) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / (k * Math.sin(k)))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((k * Math.pow(t_m, 3.0)) * ((2.0 * k) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.8e-45:
		tmp = math.pow(((l * (math.sqrt(2.0) / (k * math.sin(k)))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 / ((k * math.pow(t_m, 3.0)) * ((2.0 * k) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.8e-45)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * (t_m ^ 3.0)) * Float64(Float64(2.0 * k) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.8e-45)
		tmp = ((l * (sqrt(2.0) / (k * sin(k)))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((k * (t_m ^ 3.0)) * ((2.0 * k) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-45], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999997e-45

   1. Initial program 51.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

      2. pow2 34.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

   6. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 34.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 34.4%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 34.4%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 20.1%

       \[\leadsto {\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \color{blue}{\sqrt{\frac{1}{t}}}\right)}^{2} \]

   if 3.79999999999999997e-45 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 48.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{{\ell}^{2}}}} \]

      2. associate-+r+ 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{{\ell}^{2}}} \]

      3. metadata-eval 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}} \]

   7. Applied egg-rr 48.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 51.5%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   10. Taylor expanded in k around 0 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\left(2 \cdot \frac{k}{{\ell}^{2}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 51.5%

          \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]

   12. Simplified 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-46}:\\ \;\;\;\;{\left(\frac{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{t\_m}}\right)}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t\_m}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-46)
    (pow (/ (* l (* (sqrt 2.0) (sqrt (/ 1.0 t_m)))) (pow k 2.0)) 2.0)
    (/ 2.0 (* (* k (pow t_m 3.0)) (/ (* 2.0 k) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-46) {
		tmp = pow(((l * (sqrt(2.0) * sqrt((1.0 / t_m)))) / pow(k, 2.0)), 2.0);
	} else {
		tmp = 2.0 / ((k * pow(t_m, 3.0)) * ((2.0 * k) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.2d-46) then
        tmp = ((l * (sqrt(2.0d0) * sqrt((1.0d0 / t_m)))) / (k ** 2.0d0)) ** 2.0d0
    else
        tmp = 2.0d0 / ((k * (t_m ** 3.0d0)) * ((2.0d0 * k) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-46) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) * Math.sqrt((1.0 / t_m)))) / Math.pow(k, 2.0)), 2.0);
	} else {
		tmp = 2.0 / ((k * Math.pow(t_m, 3.0)) * ((2.0 * k) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.2e-46:
		tmp = math.pow(((l * (math.sqrt(2.0) * math.sqrt((1.0 / t_m)))) / math.pow(k, 2.0)), 2.0)
	else:
		tmp = 2.0 / ((k * math.pow(t_m, 3.0)) * ((2.0 * k) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.2e-46)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / t_m)))) / (k ^ 2.0)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * (t_m ^ 3.0)) * Float64(Float64(2.0 * k) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.2e-46)
		tmp = ((l * (sqrt(2.0) * sqrt((1.0 / t_m)))) / (k ^ 2.0)) ^ 2.0;
	else
		tmp = 2.0 / ((k * (t_m ^ 3.0)) * ((2.0 * k) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-46], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.20000000000000007e-46

   1. Initial program 51.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

      2. pow2 34.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

   6. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 34.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 34.4%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 34.4%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 19.7%

       \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 19.7%

          \[\leadsto {\color{blue}{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]

       2. associate-*l* 19.7%

          \[\leadsto {\left(\frac{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{t}}\right)}}{{k}^{2}}\right)}^{2} \]

   12. Simplified 19.7%

       \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{t}}\right)}{{k}^{2}}\right)}}^{2} \]

   if 1.20000000000000007e-46 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 48.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{{\ell}^{2}}}} \]

      2. associate-+r+ 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{{\ell}^{2}}} \]

      3. metadata-eval 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}} \]

   7. Applied egg-rr 48.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 51.5%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   10. Taylor expanded in k around 0 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\left(2 \cdot \frac{k}{{\ell}^{2}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 51.5%

          \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]

   12. Simplified 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-45}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t\_m}} \cdot \frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t\_m}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.5e-45)
    (pow (* (sqrt (/ 1.0 t_m)) (/ (* l (sqrt 2.0)) (pow k 2.0))) 2.0)
    (/ 2.0 (* (* k (pow t_m 3.0)) (/ (* 2.0 k) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-45) {
		tmp = pow((sqrt((1.0 / t_m)) * ((l * sqrt(2.0)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = 2.0 / ((k * pow(t_m, 3.0)) * ((2.0 * k) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.5d-45) then
        tmp = (sqrt((1.0d0 / t_m)) * ((l * sqrt(2.0d0)) / (k ** 2.0d0))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k * (t_m ** 3.0d0)) * ((2.0d0 * k) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-45) {
		tmp = Math.pow((Math.sqrt((1.0 / t_m)) * ((l * Math.sqrt(2.0)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = 2.0 / ((k * Math.pow(t_m, 3.0)) * ((2.0 * k) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.5e-45:
		tmp = math.pow((math.sqrt((1.0 / t_m)) * ((l * math.sqrt(2.0)) / math.pow(k, 2.0))), 2.0)
	else:
		tmp = 2.0 / ((k * math.pow(t_m, 3.0)) * ((2.0 * k) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.5e-45)
		tmp = Float64(sqrt(Float64(1.0 / t_m)) * Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * (t_m ^ 3.0)) * Float64(Float64(2.0 * k) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.5e-45)
		tmp = (sqrt((1.0 / t_m)) * ((l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = 2.0 / ((k * (t_m ^ 3.0)) * ((2.0 * k) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-45], N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.4999999999999995e-45

   1. Initial program 51.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 34.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

      2. pow2 34.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

   6. Applied egg-rr 36.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 34.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 34.4%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 34.4%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 19.7%

       \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 6.4999999999999995e-45 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 48.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{{\ell}^{2}}}} \]

      2. associate-+r+ 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{{\ell}^{2}}} \]

      3. metadata-eval 48.9%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}} \]

   7. Applied egg-rr 48.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 51.5%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   9. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   10. Taylor expanded in k around 0 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\left(2 \cdot \frac{k}{{\ell}^{2}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 51.5%

          \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]

   12. Simplified 51.5%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-45}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t\_m}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-90)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (/ 2.0 (* (* k (pow t_m 3.0)) (/ (* 2.0 k) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.9e-90) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((k * pow(t_m, 3.0)) * ((2.0 * k) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.9d-90) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / ((k * (t_m ** 3.0d0)) * ((2.0d0 * k) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.9e-90) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((k * Math.pow(t_m, 3.0)) * ((2.0 * k) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.9e-90:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / ((k * math.pow(t_m, 3.0)) * ((2.0 * k) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.9e-90)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * (t_m ^ 3.0)) * Float64(Float64(2.0 * k) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.9e-90)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	else
		tmp = 2.0 / ((k * (t_m ^ 3.0)) * ((2.0 * k) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-90], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.90000000000000005e-90

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 62.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   6. Taylor expanded in k around 0 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/l* 55.2%

         \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   8. Simplified 55.2%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 3.90000000000000005e-90 < t

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 52.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 49.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{{\ell}^{2}}}} \]

      2. associate-+r+ 49.6%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{{\ell}^{2}}} \]

      3. metadata-eval 49.6%

         \[\leadsto \frac{2}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}} \]

   7. Applied egg-rr 49.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot {t}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 52.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   9. Simplified 52.0%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{{\ell}^{2}}}} \]

   10. Taylor expanded in k around 0 52.0%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\left(2 \cdot \frac{k}{{\ell}^{2}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 52.0%

          \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]

   12. Simplified 52.0%

       \[\leadsto \frac{2}{\left(k \cdot {t}^{3}\right) \cdot \color{blue}{\frac{2 \cdot k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{k}^{4} \cdot \left(t\_m \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow k 4.0) (* t_m (pow l -2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow(k, 4.0) * (t_m * pow(l, -2.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((k ** 4.0d0) * (t_m * (l ** (-2.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow(k, 4.0) * (t_m * Math.pow(l, -2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow(k, 4.0) * (t_m * math.pow(l, -2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m * (l ^ -2.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((k ^ 4.0) * (t_m * (l ^ -2.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 52.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 58.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

6. Taylor expanded in k around 0 51.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
7. Step-by-step derivation

   1. associate-/l* 52.5%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

8. Simplified 52.5%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
9. Step-by-step derivation

   1. pow1 52.5%

      \[\leadsto \frac{2}{\color{blue}{{\left({k}^{4} \cdot \frac{t}{{\ell}^{2}}\right)}^{1}}} \]

   2. div-inv 52.5%

      \[\leadsto \frac{2}{{\left({k}^{4} \cdot \color{blue}{\left(t \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

   3. pow-flip 52.6%

      \[\leadsto \frac{2}{{\left({k}^{4} \cdot \left(t \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

   4. metadata-eval 52.6%

      \[\leadsto \frac{2}{{\left({k}^{4} \cdot \left(t \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

10. Applied egg-rr 52.6%

    \[\leadsto \frac{2}{\color{blue}{{\left({k}^{4} \cdot \left(t \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
11. Step-by-step derivation

    1. unpow1 52.6%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t \cdot {\ell}^{-2}\right)}} \]

12. Simplified 52.6%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t \cdot {\ell}^{-2}\right)}} \]
13. Add Preprocessing

Alternative 25: 52.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) * Math.pow(k, -4.0)) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 52.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 58.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

6. Taylor expanded in k around 0 51.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
7. Step-by-step derivation

   1. associate-/l* 52.5%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

8. Simplified 52.5%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

9. Taylor expanded in k around 0 51.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
10. Step-by-step derivation

    1. associate-/r* 51.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

11. Simplified 51.9%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
12. Step-by-step derivation

    1. div-inv 51.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}}{t} \]

    2. pow-flip 51.9%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}}{t} \]

    3. metadata-eval 51.9%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{\color{blue}{-4}}}{t} \]

13. Applied egg-rr 51.9%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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))))