Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.8% → 82.0%

Time: 25.9s

Alternatives: 27

Speedup: 1.3×

• 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: 53.8% 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: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ t_3 := \frac{t}{\sqrt[3]{\ell}}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t_3}{\frac{\sqrt[3]{\ell}}{t_1}}\right)}^{3} \cdot \tan k\right) \cdot t_2}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot {\left(t_1 \cdot \frac{t_3}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (sin k)))
        (t_2 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
        (t_3 (/ t (cbrt l))))
   (if (<= t -4.5e-65)
     (/ 2.0 (* (* (pow (/ t_3 (/ (cbrt l) t_1)) 3.0) (tan k)) t_2))
     (if (<= t 6.4e-102)
       (*
        2.0
        (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))
       (/ 2.0 (* t_2 (* (tan k) (pow (* t_1 (/ t_3 (cbrt l))) 3.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt(sin(k));
	double t_2 = 1.0 + (1.0 + pow((k / t), 2.0));
	double t_3 = t / cbrt(l);
	double tmp;
	if (t <= -4.5e-65) {
		tmp = 2.0 / ((pow((t_3 / (cbrt(l) / t_1)), 3.0) * tan(k)) * t_2);
	} else if (t <= 6.4e-102) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * pow((t_1 * (t_3 / cbrt(l))), 3.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt(Math.sin(k));
	double t_2 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double t_3 = t / Math.cbrt(l);
	double tmp;
	if (t <= -4.5e-65) {
		tmp = 2.0 / ((Math.pow((t_3 / (Math.cbrt(l) / t_1)), 3.0) * Math.tan(k)) * t_2);
	} else if (t <= 6.4e-102) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * Math.pow((t_1 * (t_3 / Math.cbrt(l))), 3.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(sin(k))
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	t_3 = Float64(t / cbrt(l))
	tmp = 0.0
	if (t <= -4.5e-65)
		tmp = Float64(2.0 / Float64(Float64((Float64(t_3 / Float64(cbrt(l) / t_1)) ^ 3.0) * tan(k)) * t_2));
	elseif (t <= 6.4e-102)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * (Float64(t_1 * Float64(t_3 / cbrt(l))) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e-65], N[(2.0 / N[(N[(N[Power[N[(t$95$3 / N[(N[Power[l, 1/3], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e-102], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$1 * N[(t$95$3 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.4999999999999998e-65

   1. Initial program 60.8%

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

      1. associate-/r* 71.1%

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

      2. add-cube-cbrt 71.0%

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

      3. *-un-lft-identity 71.0%

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

      4. times-frac 71.0%

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

      5. pow2 71.0%

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

      6. cbrt-div 71.0%

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

      7. rem-cbrt-cube 71.0%

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

      8. cbrt-div 70.9%

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

      9. rem-cbrt-cube 83.2%

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

   3. Applied egg-rr 83.2%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.1%

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

      3. cbrt-prod 83.0%

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

      4. frac-times 79.0%

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

      5. unpow2 79.0%

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

      6. *-un-lft-identity 79.0%

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

      7. cbrt-div 80.2%

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

      8. add-cbrt-cube 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. associate-*l/ 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. associate-/l* 90.9%

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

   9. Simplified 90.9%

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

   if -4.4999999999999998e-65 < t < 6.39999999999999973e-102

   1. Initial program 38.0%

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

   3. Simplified 36.4%

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

      1. div-inv 36.4%

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

      2. pow-flip 36.4%

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

      3. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

   6. Taylor expanded in t around 0 74.2%

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

      1. times-frac 75.0%

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

      2. associate-/r* 75.0%

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

      3. associate-*r/ 77.6%

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

   8. Simplified 77.6%

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

   if 6.39999999999999973e-102 < t

   1. Initial program 64.9%

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

      1. associate-/r* 72.0%

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

      2. add-cube-cbrt 72.0%

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

      3. *-un-lft-identity 72.0%

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

      4. times-frac 72.0%

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

      5. pow2 72.0%

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

      6. cbrt-div 71.9%

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

      7. rem-cbrt-cube 71.9%

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

      8. cbrt-div 71.8%

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

      9. rem-cbrt-cube 87.0%

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

   3. Applied egg-rr 87.0%

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

      1. add-cube-cbrt 86.9%

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

      2. pow3 86.9%

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

      3. cbrt-prod 86.9%

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

      4. frac-times 80.1%

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

      5. unpow2 80.1%

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

      6. *-un-lft-identity 80.1%

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

      7. cbrt-div 81.2%

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

      8. add-cbrt-cube 89.0%

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

   5. Applied egg-rr 89.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 2: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-66} \lor \neg \left(t \leq 4.2 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -8.8e-66) (not (<= t 4.2e-93)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (pow (* (cbrt (sin k)) (/ (/ t (cbrt l)) (cbrt l))) 3.0))))
   (*
    2.0
    (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.8e-66) || !(t <= 4.2e-93)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * pow((cbrt(sin(k)) * ((t / cbrt(l)) / cbrt(l))), 3.0)));
	} else {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -8.8e-66) || !(t <= 4.2e-93)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * ((t / Math.cbrt(l)) / Math.cbrt(l))), 3.0)));
	} else {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -8.8e-66) || !(t <= 4.2e-93))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(Float64(t / cbrt(l)) / cbrt(l))) ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -8.8e-66], N[Not[LessEqual[t, 4.2e-93]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.8000000000000004e-66 or 4.2000000000000002e-93 < t

   1. Initial program 63.0%

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

      1. associate-/r* 71.6%

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

      2. add-cube-cbrt 71.5%

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

      3. *-un-lft-identity 71.5%

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

      4. times-frac 71.5%

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

      5. pow2 71.5%

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

      6. cbrt-div 71.5%

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

      7. rem-cbrt-cube 71.5%

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

      8. cbrt-div 71.4%

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

      9. rem-cbrt-cube 85.2%

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

   3. Applied egg-rr 85.2%

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

      1. add-cube-cbrt 85.1%

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

      2. pow3 85.1%

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

      3. cbrt-prod 85.1%

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

      4. frac-times 79.6%

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

      5. unpow2 79.6%

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

      6. *-un-lft-identity 79.6%

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

      7. cbrt-div 80.7%

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

      8. add-cbrt-cube 89.8%

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

   5. Applied egg-rr 89.8%

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

   if -8.8000000000000004e-66 < t < 4.2000000000000002e-93

   1. Initial program 38.0%

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

   3. Simplified 36.4%

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

      1. div-inv 36.4%

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

      2. pow-flip 36.4%

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

      3. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

   6. Taylor expanded in t around 0 74.2%

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

      1. times-frac 75.0%

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

      2. associate-/r* 75.0%

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

      3. associate-*r/ 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-66} \lor \neg \left(t \leq 4.2 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-65} \lor \neg \left(t \leq 5 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5e-65) (not (<= t 5e-106)))
   (/
    2.0
    (*
     (* (tan k) (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0)))
     (+ 1.0 (+ 1.0 (* (/ k t) (/ k t))))))
   (*
    2.0
    (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5e-65) || !(t <= 5e-106)) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5e-65) || !(t <= 5e-106)) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5e-65) || !(t <= 5e-106))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5e-65], N[Not[LessEqual[t, 5e-106]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.99999999999999983e-65 or 4.99999999999999983e-106 < t

   1. Initial program 63.0%

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

      1. unpow3 63.0%

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

      2. *-un-lft-identity 63.0%

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

      3. times-frac 67.5%

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

      4. pow2 67.5%

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

      5. pow2 67.5%

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

   3. Applied egg-rr 67.5%

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

      1. unpow2 56.1%

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

   5. Applied egg-rr 67.5%

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

      1. add-cube-cbrt 67.3%

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

      2. pow3 67.4%

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

      3. pow2 67.4%

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

      4. frac-times 62.9%

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

      5. unpow2 62.9%

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

      6. unpow3 63.0%

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

      7. *-un-lft-identity 63.0%

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

      8. cbrt-div 62.9%

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

      9. unpow3 62.9%

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

      10. add-cbrt-cube 69.2%

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

      11. cbrt-prod 86.1%

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

      12. pow2 86.1%

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

   7. Applied egg-rr 86.1%

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

   if -4.99999999999999983e-65 < t < 4.99999999999999983e-106

   1. Initial program 38.0%

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

   3. Simplified 36.4%

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

      1. div-inv 36.4%

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

      2. pow-flip 36.4%

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

      3. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

   6. Taylor expanded in t around 0 74.2%

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

      1. times-frac 75.0%

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

      2. associate-/r* 75.0%

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

      3. associate-*r/ 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-65} \lor \neg \left(t \leq 5 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 4: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + t_1}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t_1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -7e-66)
     (/
      (/ 2.0 (* (tan k) (* (sin k) (/ (pow (/ t (cbrt l)) 3.0) l))))
      (+ 2.0 t_1))
     (if (<= t 2.8e-119)
       (*
        2.0
        (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 t_1))
         (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -7e-66) {
		tmp = (2.0 / (tan(k) * (sin(k) * (pow((t / cbrt(l)), 3.0) / l)))) / (2.0 + t_1);
	} else if (t <= 2.8e-119) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -7e-66) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)))) / (2.0 + t_1);
	} else if (t <= 2.8e-119) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -7e-66)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)))) / Float64(2.0 + t_1));
	elseif (t <= 2.8e-119)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -7e-66], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e-119], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.0000000000000001e-66

   1. Initial program 60.8%

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

   3. Simplified 71.1%

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

      1. add-cube-cbrt 71.0%

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

      2. pow3 71.0%

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

      3. cbrt-div 70.9%

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

      4. rem-cbrt-cube 79.0%

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

   5. Applied egg-rr 79.0%

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

   if -7.0000000000000001e-66 < t < 2.8e-119

   1. Initial program 35.7%

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

   3. Simplified 34.6%

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

      1. div-inv 34.6%

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

      2. pow-flip 34.6%

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

      3. metadata-eval 34.6%

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

   5. Applied egg-rr 34.6%

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

   6. Taylor expanded in t around 0 72.6%

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

      1. times-frac 73.5%

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

      2. associate-/r* 73.5%

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

      3. associate-*r/ 76.3%

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

   8. Simplified 76.3%

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

   if 2.8e-119 < t

   1. Initial program 65.7%

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

      1. add-sqr-sqrt 65.7%

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

      2. pow2 65.7%

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

      3. sqrt-div 65.7%

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

      4. sqrt-pow1 74.4%

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

      5. metadata-eval 74.4%

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

      6. sqrt-prod 43.6%

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

      7. add-sqr-sqrt 87.1%

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

   3. Applied egg-rr 87.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 5: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-65} \lor \neg \left(t \leq 8.8 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.3e-65) (not (<= t 8.8e-119)))
   (/
    (/ 2.0 (* (tan k) (* (sin k) (/ (pow (/ t (cbrt l)) 3.0) l))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (*
    2.0
    (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-65) || !(t <= 8.8e-119)) {
		tmp = (2.0 / (tan(k) * (sin(k) * (pow((t / cbrt(l)), 3.0) / l)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-65) || !(t <= 8.8e-119)) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.3e-65) || !(t <= 8.8e-119))
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.3e-65], N[Not[LessEqual[t, 8.8e-119]], $MachinePrecision]], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.3000000000000001e-65 or 8.8000000000000002e-119 < t

   1. Initial program 63.5%

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

   3. Simplified 71.8%

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

      1. add-cube-cbrt 71.7%

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

      2. pow3 71.7%

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

      3. cbrt-div 71.6%

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

      4. rem-cbrt-cube 80.4%

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

   5. Applied egg-rr 80.4%

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

   if -3.3000000000000001e-65 < t < 8.8000000000000002e-119

   1. Initial program 35.7%

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

   3. Simplified 34.6%

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

      1. div-inv 34.6%

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

      2. pow-flip 34.6%

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

      3. metadata-eval 34.6%

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

   5. Applied egg-rr 34.6%

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

   6. Taylor expanded in t around 0 72.6%

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

      1. times-frac 73.5%

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

      2. associate-/r* 73.5%

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

      3. associate-*r/ 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-65} \lor \neg \left(t \leq 8.8 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 6: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-81} \lor \neg \left(t \leq 1.45 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{t} \cdot \left({\ell}^{2} \cdot {k}^{-2}\right)\right) \cdot {\sin k}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.5e-81) (not (<= t 1.45e-118)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
   (*
    2.0
    (* (* (/ (cos k) t) (* (pow l 2.0) (pow k -2.0))) (pow (sin k) -2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.5e-81) || !(t <= 1.45e-118)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * (((cos(k) / t) * (pow(l, 2.0) * pow(k, -2.0))) * pow(sin(k), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-6.5d-81)) .or. (.not. (t <= 1.45d-118))) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else
        tmp = 2.0d0 * (((cos(k) / t) * ((l ** 2.0d0) * (k ** (-2.0d0)))) * (sin(k) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.5e-81) || !(t <= 1.45e-118)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * (((Math.cos(k) / t) * (Math.pow(l, 2.0) * Math.pow(k, -2.0))) * Math.pow(Math.sin(k), -2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.5e-81) or not (t <= 1.45e-118):
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	else:
		tmp = 2.0 * (((math.cos(k) / t) * (math.pow(l, 2.0) * math.pow(k, -2.0))) * math.pow(math.sin(k), -2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.5e-81) || !(t <= 1.45e-118))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / t) * Float64((l ^ 2.0) * (k ^ -2.0))) * (sin(k) ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.5e-81) || ~((t <= 1.45e-118)))
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	else
		tmp = 2.0 * (((cos(k) / t) * ((l ^ 2.0) * (k ^ -2.0))) * (sin(k) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.5e-81], N[Not[LessEqual[t, 1.45e-118]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000002e-81 or 1.4499999999999999e-118 < t

   1. Initial program 64.8%

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

      1. unpow3 64.9%

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

      2. times-frac 78.4%

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

      3. pow2 78.4%

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

   3. Applied egg-rr 78.4%

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

   if -6.5000000000000002e-81 < t < 1.4499999999999999e-118

   1. Initial program 31.5%

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

   3. Simplified 30.4%

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

      1. div-inv 30.4%

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

      2. pow-flip 30.4%

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

      3. metadata-eval 30.4%

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

   5. Applied egg-rr 30.4%

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

   6. Taylor expanded in t around 0 70.9%

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

      1. times-frac 71.8%

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

      2. associate-/r* 71.8%

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

      3. associate-*r/ 74.8%

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

   8. Simplified 74.8%

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

      1. div-inv 74.8%

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

      2. div-inv 74.8%

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

      3. pow-flip 74.8%

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

      4. metadata-eval 74.8%

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

      5. pow-flip 74.8%

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

      6. metadata-eval 74.8%

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

   10. Applied egg-rr 74.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-81} \lor \neg \left(t \leq 1.45 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{t} \cdot \left({\ell}^{2} \cdot {k}^{-2}\right)\right) \cdot {\sin k}^{-2}\right)\\ \end{array} \]

Alternative 7: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-65} \lor \neg \left(t \leq 1.6 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.2e-65) (not (<= t 1.6e-119)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
   (*
    2.0
    (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.2e-65) || !(t <= 1.6e-119)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.2d-65)) .or. (.not. (t <= 1.6d-119))) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else
        tmp = 2.0d0 * ((((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / t)) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.2e-65) || !(t <= 1.6e-119)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.2e-65) or not (t <= 1.6e-119):
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	else:
		tmp = 2.0 * (((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / t)) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.2e-65) || !(t <= 1.6e-119))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.2e-65) || ~((t <= 1.6e-119)))
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	else
		tmp = 2.0 * ((((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / t)) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.2e-65], N[Not[LessEqual[t, 1.6e-119]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999999e-65 or 1.59999999999999997e-119 < t

   1. Initial program 63.5%

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

      1. unpow3 63.5%

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

      2. times-frac 77.6%

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

      3. pow2 77.6%

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

   3. Applied egg-rr 77.6%

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

   if -3.1999999999999999e-65 < t < 1.59999999999999997e-119

   1. Initial program 35.7%

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

   3. Simplified 34.6%

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

      1. div-inv 34.6%

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

      2. pow-flip 34.6%

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

      3. metadata-eval 34.6%

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

   5. Applied egg-rr 34.6%

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

   6. Taylor expanded in t around 0 72.6%

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

      1. times-frac 73.5%

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

      2. associate-/r* 73.5%

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

      3. associate-*r/ 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-65} \lor \neg \left(t \leq 1.6 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 8: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-126} \lor \neg \left(t \leq 1.6 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.8e-126) (not (<= t 1.6e-119)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k))
     (* (pow k 2.0) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.8e-126) || !(t <= 1.6e-119)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.8d-126)) .or. (.not. (t <= 1.6d-119))) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l)))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.8e-126) || !(t <= 1.6e-119)) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.8e-126) or not (t <= 1.6e-119):
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l)))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.8e-126) || !(t <= 1.6e-119))
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.8e-126) || ~((t <= 1.6e-119)))
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (sin(k) * (((t ^ 2.0) / l) * (t / l)))));
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.8e-126], N[Not[LessEqual[t, 1.6e-119]], $MachinePrecision]], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7999999999999999e-126 or 1.59999999999999997e-119 < t

   1. Initial program 63.5%

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

      1. unpow3 63.6%

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

      2. times-frac 77.5%

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

      3. pow2 77.5%

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

   3. Applied egg-rr 77.5%

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

   if -3.7999999999999999e-126 < t < 1.59999999999999997e-119

   1. Initial program 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in t around 0 72.1%

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

      1. unpow2 75.3%

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

      2. sin-mult 69.7%

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

   6. Applied egg-rr 71.0%

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

      1. div-sub 69.7%

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

      2. +-inverses 69.7%

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

      3. cos-0 69.7%

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

      4. metadata-eval 69.7%

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

      5. count-2 69.7%

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

      6. *-commutative 69.7%

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

   8. Simplified 71.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-126} \lor \neg \left(t \leq 1.6 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \end{array} \]

Alternative 9: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (*
          (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (tan k)))
          (/ l (+ 2.0 (pow (/ k t) 2.0))))))
   (if (<= t -1.32e-65)
     t_1
     (if (<= t 4e-59)
       (*
        2.0
        (/
         (* (pow l 2.0) (cos k))
         (* (pow k 2.0) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
       (if (<= t 4.3e+95)
         t_1
         (/
          2.0
          (pow
           (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t))))
           2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * tan(k))) * (l / (2.0 + pow((k / t), 2.0)));
	double tmp;
	if (t <= -1.32e-65) {
		tmp = t_1;
	} else if (t <= 4e-59) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t <= 4.3e+95) {
		tmp = t_1;
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * Math.tan(k))) * (l / (2.0 + Math.pow((k / t), 2.0)));
	double tmp;
	if (t <= -1.32e-65) {
		tmp = t_1;
	} else if (t <= 4e-59) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t <= 4.3e+95) {
		tmp = t_1;
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * math.tan(k))) * (l / (2.0 + math.pow((k / t), 2.0)))
	tmp = 0
	if t <= -1.32e-65:
		tmp = t_1
	elif t <= 4e-59:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t <= 4.3e+95:
		tmp = t_1
	else:
		tmp = 2.0 / math.pow(((k * (math.pow(t, 1.5) / l)) * math.hypot(1.0, math.hypot(1.0, (k / t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * tan(k))) * Float64(l / Float64(2.0 + (Float64(k / t) ^ 2.0))))
	tmp = 0.0
	if (t <= -1.32e-65)
		tmp = t_1;
	elseif (t <= 4e-59)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t <= 4.3e+95)
		tmp = t_1;
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * tan(k))) * (l / (2.0 + ((k / t) ^ 2.0)));
	tmp = 0.0;
	if (t <= -1.32e-65)
		tmp = t_1;
	elseif (t <= 4e-59)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t <= 4.3e+95)
		tmp = t_1;
	else
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.32e-65], t$95$1, If[LessEqual[t, 4e-59], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+95], t$95$1, N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.32e-65 or 4.0000000000000001e-59 < t < 4.3e95

   1. Initial program 64.9%

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

   3. Simplified 58.5%

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

      1. associate-*r* 67.5%

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

      2. *-commutative 67.5%

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

      3. times-frac 71.5%

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

   5. Applied egg-rr 71.5%

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

   if -1.32e-65 < t < 4.0000000000000001e-59

   1. Initial program 39.7%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in t around 0 73.3%

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

      1. unpow2 76.6%

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

      2. sin-mult 71.5%

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

   6. Applied egg-rr 72.5%

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

      1. div-sub 71.5%

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

      2. +-inverses 71.5%

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

      3. cos-0 71.5%

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

      4. metadata-eval 71.5%

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

      5. count-2 71.5%

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

      6. *-commutative 71.5%

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

   8. Simplified 72.5%

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

   if 4.3e95 < t

   1. Initial program 58.8%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

   3. Applied egg-rr 51.5%

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

   4. Taylor expanded in k around 0 74.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-65}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 10: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-103}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (*
          (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
          (/ l (tan k)))))
   (if (<= t -7.2e-66)
     t_1
     (if (<= t 4.6e-103)
       (*
        2.0
        (/
         (* (pow l 2.0) (cos k))
         (* (pow k 2.0) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
       (if (<= t 4.3e+95)
         t_1
         (/
          2.0
          (pow
           (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t))))
           2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * (2.0 + pow((k / t), 2.0)))) * (l / tan(k));
	double tmp;
	if (t <= -7.2e-66) {
		tmp = t_1;
	} else if (t <= 4.6e-103) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else if (t <= 4.3e+95) {
		tmp = t_1;
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))) * (l / Math.tan(k));
	double tmp;
	if (t <= -7.2e-66) {
		tmp = t_1;
	} else if (t <= 4.6e-103) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else if (t <= 4.3e+95) {
		tmp = t_1;
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * (2.0 + math.pow((k / t), 2.0)))) * (l / math.tan(k))
	tmp = 0
	if t <= -7.2e-66:
		tmp = t_1
	elif t <= 4.6e-103:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	elif t <= 4.3e+95:
		tmp = t_1
	else:
		tmp = 2.0 / math.pow(((k * (math.pow(t, 1.5) / l)) * math.hypot(1.0, math.hypot(1.0, (k / t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))) * Float64(l / tan(k)))
	tmp = 0.0
	if (t <= -7.2e-66)
		tmp = t_1;
	elseif (t <= 4.6e-103)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	elseif (t <= 4.3e+95)
		tmp = t_1;
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * (2.0 + ((k / t) ^ 2.0)))) * (l / tan(k));
	tmp = 0.0;
	if (t <= -7.2e-66)
		tmp = t_1;
	elseif (t <= 4.6e-103)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	elseif (t <= 4.3e+95)
		tmp = t_1;
	else
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.2e-66], t$95$1, If[LessEqual[t, 4.6e-103], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+95], t$95$1, N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.20000000000000025e-66 or 4.6000000000000001e-103 < t < 4.3e95

   1. Initial program 64.7%

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

   3. Simplified 58.8%

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

      1. associate-*r* 67.2%

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

      2. associate-*r* 67.2%

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

      3. times-frac 79.0%

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

   5. Applied egg-rr 79.0%

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

   if -7.20000000000000025e-66 < t < 4.6000000000000001e-103

   1. Initial program 38.0%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around 0 74.2%

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

      1. unpow2 77.6%

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

      2. sin-mult 72.2%

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

   6. Applied egg-rr 73.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

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

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   8. Simplified 73.2%

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

   if 4.3e95 < t

   1. Initial program 58.8%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

   3. Applied egg-rr 51.5%

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

   4. Taylor expanded in k around 0 74.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-103}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 11: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4.2e-32)
   (/ (pow (/ (cbrt (* 2.0 (pow l 2.0))) t) 3.0) (* 2.0 (pow k 2.0)))
   (if (<= t 4e-59)
     (*
      2.0
      (/
       (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t))
       (- 0.5 (/ (cos (* 2.0 k)) 2.0))))
     (/
      2.0
      (pow (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t)))) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.2e-32) {
		tmp = pow((cbrt((2.0 * pow(l, 2.0))) / t), 3.0) / (2.0 * pow(k, 2.0));
	} else if (t <= 4e-59) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / (0.5 - (cos((2.0 * k)) / 2.0)));
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.2e-32) {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow(l, 2.0))) / t), 3.0) / (2.0 * Math.pow(k, 2.0));
	} else if (t <= 4e-59) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / (0.5 - (Math.cos((2.0 * k)) / 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4.2e-32)
		tmp = Float64((Float64(cbrt(Float64(2.0 * (l ^ 2.0))) / t) ^ 3.0) / Float64(2.0 * (k ^ 2.0)));
	elseif (t <= 4e-59)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4.2e-32], N[(N[Power[N[(N[Power[N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-59], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.1999999999999998e-32

   1. Initial program 56.8%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in k around 0 45.3%

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

      1. add-cube-cbrt 45.2%

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

      2. pow2 45.2%

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

      3. cbrt-div 45.2%

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

      4. associate-*l/ 45.2%

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

      5. cbrt-div 45.1%

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

      6. pow2 45.1%

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

      7. unpow3 45.1%

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

      8. add-cbrt-cube 45.1%

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

      9. cbrt-div 45.1%

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

   6. Applied egg-rr 49.2%

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

      1. pow-plus 49.2%

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

      2. metadata-eval 49.2%

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

      3. cube-div 49.0%

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

      4. rem-cube-cbrt 49.1%

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

   8. Simplified 49.1%

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

   if -4.1999999999999998e-32 < t < 4.0000000000000001e-59

   1. Initial program 43.4%

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

   3. Simplified 41.4%

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

      1. div-inv 41.4%

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

      2. pow-flip 41.4%

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

      3. metadata-eval 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in t around 0 73.3%

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

      1. times-frac 74.0%

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

      2. associate-/r* 74.0%

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

      3. associate-*r/ 76.2%

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

   8. Simplified 76.2%

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

      1. unpow2 76.2%

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

      2. sin-mult 71.7%

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

   10. Applied egg-rr 71.7%

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

       1. div-sub 71.7%

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

       2. +-inverses 71.7%

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

       3. cos-0 71.7%

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

       4. metadata-eval 71.7%

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

       5. count-2 71.7%

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

       6. *-commutative 71.7%

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

   12. Simplified 71.7%

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

   if 4.0000000000000001e-59 < t

   1. Initial program 65.2%

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

      1. add-sqr-sqrt 51.3%

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

      2. pow2 51.3%

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

   3. Applied egg-rr 64.5%

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

   4. Taylor expanded in k around 0 73.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 12: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -9.2e-45)
   (/ (pow (/ (cbrt (* 2.0 (pow l 2.0))) t) 3.0) (* 2.0 (pow k 2.0)))
   (if (<= t 4.5e-59)
     (*
      2.0
      (/
       (* (pow l 2.0) (cos k))
       (* (pow k 2.0) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))
     (/
      2.0
      (pow (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t)))) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.2e-45) {
		tmp = pow((cbrt((2.0 * pow(l, 2.0))) / t), 3.0) / (2.0 * pow(k, 2.0));
	} else if (t <= 4.5e-59) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.2e-45) {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow(l, 2.0))) / t), 3.0) / (2.0 * Math.pow(k, 2.0));
	} else if (t <= 4.5e-59) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -9.2e-45)
		tmp = Float64((Float64(cbrt(Float64(2.0 * (l ^ 2.0))) / t) ^ 3.0) / Float64(2.0 * (k ^ 2.0)));
	elseif (t <= 4.5e-59)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -9.2e-45], N[(N[Power[N[(N[Power[N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-59], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.19999999999999967e-45

   1. Initial program 60.8%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in k around 0 48.9%

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

      1. add-cube-cbrt 48.8%

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

      2. pow2 48.8%

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

      3. cbrt-div 48.8%

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

      4. associate-*l/ 48.8%

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

      5. cbrt-div 48.7%

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

      6. pow2 48.7%

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

      7. unpow3 48.7%

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

      8. add-cbrt-cube 48.8%

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

      9. cbrt-div 48.8%

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

   6. Applied egg-rr 52.4%

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

      1. pow-plus 52.4%

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

      2. metadata-eval 52.4%

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

      3. cube-div 52.2%

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

      4. rem-cube-cbrt 52.4%

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

   8. Simplified 52.4%

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

   if -9.19999999999999967e-45 < t < 4.50000000000000012e-59

   1. Initial program 40.6%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in t around 0 72.8%

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

      1. unpow2 75.8%

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

      2. sin-mult 71.0%

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

   6. Applied egg-rr 71.9%

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

      1. div-sub 71.0%

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

      2. +-inverses 71.0%

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

      3. cos-0 71.0%

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

      4. metadata-eval 71.0%

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

      5. count-2 71.0%

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

      6. *-commutative 71.0%

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

   8. Simplified 71.9%

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

   if 4.50000000000000012e-59 < t

   1. Initial program 65.2%

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

      1. add-sqr-sqrt 51.3%

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

      2. pow2 51.3%

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

   3. Applied egg-rr 64.5%

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

   4. Taylor expanded in k around 0 73.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{t}\right)}^{3}}{2 \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 13: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-81)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t 1.45e-263)
     (* 2.0 (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow k 2.0)))
     (if (<= t 5.5e-31)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/
        2.0
        (pow
         (* (* k (/ (pow t 1.5) l)) (hypot 1.0 (hypot 1.0 (/ k t))))
         2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-81) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 1.45e-263) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(k, 2.0));
	} else if (t <= 5.5e-31) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow(((k * (pow(t, 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-81) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 1.45e-263) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(k, 2.0));
	} else if (t <= 5.5e-31) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.pow(t, 1.5) / l)) * Math.hypot(1.0, Math.hypot(1.0, (k / t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.5e-81:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= 1.45e-263:
		tmp = 2.0 * (((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / t)) / math.pow(k, 2.0))
	elif t <= 5.5e-31:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow(((k * (math.pow(t, 1.5) / l)) * math.hypot(1.0, math.hypot(1.0, (k / t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-81)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= 1.45e-263)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (k ^ 2.0)));
	elseif (t <= 5.5e-31)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, Float64(k / t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.5e-81)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= 1.45e-263)
		tmp = 2.0 * ((((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / t)) / (k ^ 2.0));
	elseif (t <= 5.5e-31)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) * hypot(1.0, hypot(1.0, (k / t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.5e-81], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e-263], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-31], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.49999999999999986e-81

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -3.49999999999999986e-81 < t < 1.45000000000000002e-263

   1. Initial program 27.5%

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

   3. Simplified 25.4%

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

      1. div-inv 25.4%

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

      2. pow-flip 25.4%

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

      3. metadata-eval 25.4%

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

   5. Applied egg-rr 25.4%

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

   6. Taylor expanded in t around 0 66.6%

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

      1. times-frac 68.2%

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

      2. associate-/r* 68.2%

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

      3. associate-*r/ 73.5%

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

   8. Simplified 73.5%

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

   9. Taylor expanded in k around 0 64.7%

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

   if 1.45000000000000002e-263 < t < 5.49999999999999958e-31

   1. Initial program 46.9%

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

      1. add-sqr-sqrt 45.9%

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

      2. pow2 45.9%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in k around inf 70.5%

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

   if 5.49999999999999958e-31 < t

   1. Initial program 65.1%

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

      1. add-sqr-sqrt 49.9%

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

      2. pow2 49.9%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in k around 0 75.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}\\ \end{array} \]

Alternative 14: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-81)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t -3e-309)
     (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
     (if (<= t 5.4e-30)
       (/ 2.0 (pow (* (sqrt (/ t (cos k))) (/ k (/ l (sin k)))) 2.0))
       (/ 2.0 (pow (* (/ (* k (sqrt 2.0)) l) (sqrt (pow t 3.0))) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-81) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= -3e-309) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else if (t <= 5.4e-30) {
		tmp = 2.0 / pow((sqrt((t / cos(k))) * (k / (l / sin(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) / l) * sqrt(pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-3.5d-81)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else if (t <= (-3d-309)) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else if (t <= 5.4d-30) then
        tmp = 2.0d0 / ((sqrt((t / cos(k))) * (k / (l / sin(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) / l) * sqrt((t ** 3.0d0))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-81) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= -3e-309) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else if (t <= 5.4e-30) {
		tmp = 2.0 / Math.pow((Math.sqrt((t / Math.cos(k))) * (k / (l / Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.5e-81:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= -3e-309:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	elif t <= 5.4e-30:
		tmp = 2.0 / math.pow((math.sqrt((t / math.cos(k))) * (k / (l / math.sin(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t, 3.0))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-81)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= -3e-309)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	elseif (t <= 5.4e-30)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t / cos(k))) * Float64(k / Float64(l / sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.5e-81)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= -3e-309)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	elseif (t <= 5.4e-30)
		tmp = 2.0 / ((sqrt((t / cos(k))) * (k / (l / sin(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.5e-81], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3e-309], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e-30], N[(2.0 / N[Power[N[(N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.49999999999999986e-81

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -3.49999999999999986e-81 < t < -3.000000000000001e-309

   1. Initial program 30.4%

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

   3. Simplified 28.1%

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

      1. div-inv 28.1%

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

      2. pow-flip 28.1%

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

      3. metadata-eval 28.1%

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

   5. Applied egg-rr 28.1%

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

   6. Taylor expanded in t around 0 63.1%

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

      1. times-frac 66.8%

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

      2. associate-/r* 66.8%

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

      3. associate-*r/ 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in k around 0 50.6%

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

       1. associate-/r* 59.9%

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

   11. Simplified 59.9%

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

   if -3.000000000000001e-309 < t < 5.39999999999999975e-30

   1. Initial program 43.3%

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

      1. add-sqr-sqrt 42.4%

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

      2. pow2 42.4%

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

   3. Applied egg-rr 63.7%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. associate-/l* 71.1%

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

   6. Simplified 71.1%

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

   if 5.39999999999999975e-30 < t

   1. Initial program 65.1%

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

      1. add-sqr-sqrt 49.9%

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

      2. pow2 49.9%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in k around 0 67.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]

Alternative 15: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2e-82)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t -1.05e-307)
     (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
     (if (<= t 5.5e-31)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (* (/ (* k (sqrt 2.0)) l) (sqrt (pow t 3.0))) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-82) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= -1.05e-307) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else if (t <= 5.5e-31) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) / l) * sqrt(pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-2d-82)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else if (t <= (-1.05d-307)) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else if (t <= 5.5d-31) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) / l) * sqrt((t ** 3.0d0))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-82) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= -1.05e-307) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else if (t <= 5.5e-31) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2e-82:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= -1.05e-307:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	elif t <= 5.5e-31:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t, 3.0))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2e-82)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= -1.05e-307)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	elseif (t <= 5.5e-31)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2e-82)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= -1.05e-307)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	elseif (t <= 5.5e-31)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2e-82], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.05e-307], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-31], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2e-82

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -2e-82 < t < -1.0500000000000001e-307

   1. Initial program 30.4%

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

   3. Simplified 28.1%

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

      1. div-inv 28.1%

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

      2. pow-flip 28.1%

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

      3. metadata-eval 28.1%

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

   5. Applied egg-rr 28.1%

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

   6. Taylor expanded in t around 0 63.1%

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

      1. times-frac 66.8%

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

      2. associate-/r* 66.8%

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

      3. associate-*r/ 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in k around 0 50.6%

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

       1. associate-/r* 59.9%

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

   11. Simplified 59.9%

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

   if -1.0500000000000001e-307 < t < 5.49999999999999958e-31

   1. Initial program 43.3%

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

      1. add-sqr-sqrt 42.4%

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

      2. pow2 42.4%

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

   3. Applied egg-rr 63.7%

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

   4. Taylor expanded in k around inf 71.2%

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

   if 5.49999999999999958e-31 < t

   1. Initial program 65.1%

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

      1. add-sqr-sqrt 49.9%

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

      2. pow2 49.9%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in k around 0 67.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]

Alternative 16: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -7e-82)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t 5.5e-263)
     (* 2.0 (/ (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) t)) (pow k 2.0)))
     (if (<= t 3.3e-26)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (* (/ (* k (sqrt 2.0)) l) (sqrt (pow t 3.0))) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -7e-82) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 5.5e-263) {
		tmp = 2.0 * (((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / t)) / pow(k, 2.0));
	} else if (t <= 3.3e-26) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) / l) * sqrt(pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-7d-82)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else if (t <= 5.5d-263) then
        tmp = 2.0d0 * ((((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / t)) / (k ** 2.0d0))
    else if (t <= 3.3d-26) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) / l) * sqrt((t ** 3.0d0))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -7e-82) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 5.5e-263) {
		tmp = 2.0 * (((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / t)) / Math.pow(k, 2.0));
	} else if (t <= 3.3e-26) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -7e-82:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= 5.5e-263:
		tmp = 2.0 * (((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / t)) / math.pow(k, 2.0))
	elif t <= 3.3e-26:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t, 3.0))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -7e-82)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= 5.5e-263)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / t)) / (k ^ 2.0)));
	elseif (t <= 3.3e-26)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -7e-82)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= 5.5e-263)
		tmp = 2.0 * ((((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / t)) / (k ^ 2.0));
	elseif (t <= 3.3e-26)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -7e-82], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-263], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-26], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.9999999999999997e-82

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -6.9999999999999997e-82 < t < 5.49999999999999971e-263

   1. Initial program 27.5%

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

   3. Simplified 25.4%

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

      1. div-inv 25.4%

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

      2. pow-flip 25.4%

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

      3. metadata-eval 25.4%

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

   5. Applied egg-rr 25.4%

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

   6. Taylor expanded in t around 0 66.6%

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

      1. times-frac 68.2%

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

      2. associate-/r* 68.2%

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

      3. associate-*r/ 73.5%

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

   8. Simplified 73.5%

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

   9. Taylor expanded in k around 0 64.7%

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

   if 5.49999999999999971e-263 < t < 3.2999999999999998e-26

   1. Initial program 46.9%

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

      1. add-sqr-sqrt 45.9%

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

      2. pow2 45.9%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in k around inf 70.5%

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

   if 3.2999999999999998e-26 < t

   1. Initial program 65.1%

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

      1. add-sqr-sqrt 49.9%

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

      2. pow2 49.9%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in k around 0 67.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-263}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]

Alternative 17: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.8e-82)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t 3.4e-54)
     (* 2.0 (/ (/ (pow l 2.0) (* t (pow k 2.0))) (pow (sin k) 2.0)))
     (/ 2.0 (pow (* (/ (* k (sqrt 2.0)) l) (sqrt (pow t 3.0))) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.8e-82) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 3.4e-54) {
		tmp = 2.0 * ((pow(l, 2.0) / (t * pow(k, 2.0))) / pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) / l) * sqrt(pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-3.8d-82)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else if (t <= 3.4d-54) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (t * (k ** 2.0d0))) / (sin(k) ** 2.0d0))
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) / l) * sqrt((t ** 3.0d0))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.8e-82) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 3.4e-54) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t * Math.pow(k, 2.0))) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.8e-82:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= 3.4e-54:
		tmp = 2.0 * ((math.pow(l, 2.0) / (t * math.pow(k, 2.0))) / math.pow(math.sin(k), 2.0))
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t, 3.0))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.8e-82)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= 3.4e-54)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t * (k ^ 2.0))) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.8e-82)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= 3.4e-54)
		tmp = 2.0 * (((l ^ 2.0) / (t * (k ^ 2.0))) / (sin(k) ^ 2.0));
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.8e-82], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-54], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8000000000000002e-82

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -3.8000000000000002e-82 < t < 3.39999999999999987e-54

   1. Initial program 37.5%

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

   3. Simplified 36.0%

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

      1. div-inv 36.0%

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

      2. pow-flip 36.0%

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

      3. metadata-eval 36.0%

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

   5. Applied egg-rr 36.0%

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

   6. Taylor expanded in t around 0 72.4%

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

      1. times-frac 73.1%

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

      2. associate-/r* 73.1%

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

      3. associate-*r/ 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in k around 0 57.2%

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

   if 3.39999999999999987e-54 < t

   1. Initial program 64.3%

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

      1. add-sqr-sqrt 50.0%

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

      2. pow2 50.0%

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

   3. Applied egg-rr 63.5%

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

   4. Taylor expanded in k around 0 67.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]

Alternative 18: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -8e-79)
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (if (<= t 3.4e-54)
     (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
     (/ 2.0 (pow (* (/ (* k (sqrt 2.0)) l) (sqrt (pow t 3.0))) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e-79) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 3.4e-54) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) / l) * sqrt(pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-8d-79)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else if (t <= 3.4d-54) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) / l) * sqrt((t ** 3.0d0))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e-79) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else if (t <= 3.4e-54) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t, 3.0))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -8e-79:
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	elif t <= 3.4e-54:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t, 3.0))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -8e-79)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	elseif (t <= 3.4e-54)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8e-79)
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	elseif (t <= 3.4e-54)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) / l) * sqrt((t ^ 3.0))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -8e-79], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-54], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8e-79

   1. Initial program 63.8%

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

   3. Simplified 54.4%

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

      1. div-inv 54.4%

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

      2. pow-flip 54.5%

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

      3. metadata-eval 54.5%

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

   5. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

   if -8e-79 < t < 3.39999999999999987e-54

   1. Initial program 37.5%

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

   3. Simplified 36.0%

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

      1. div-inv 36.0%

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

      2. pow-flip 36.0%

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

      3. metadata-eval 36.0%

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

   5. Applied egg-rr 36.0%

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

   6. Taylor expanded in t around 0 72.4%

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

      1. times-frac 73.1%

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

      2. associate-/r* 73.1%

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

      3. associate-*r/ 75.7%

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

   8. Simplified 75.7%

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

   9. Taylor expanded in k around 0 52.9%

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

       1. associate-/r* 56.9%

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

   11. Simplified 56.9%

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

   if 3.39999999999999987e-54 < t

   1. Initial program 64.3%

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

      1. add-sqr-sqrt 50.0%

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

      2. pow2 50.0%

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

   3. Applied egg-rr 63.5%

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

   4. Taylor expanded in k around 0 67.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]

Alternative 19: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-80} \lor \neg \left(t \leq 1.65 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7.2e-80) (not (<= t 1.65e-99)))
   (/
    (* (* 2.0 (pow t -3.0)) (* l l))
    (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t) (/ k t)))))
   (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.2e-80) || !(t <= 1.65e-99)) {
		tmp = ((2.0 * pow(t, -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-7.2d-80)) .or. (.not. (t <= 1.65d-99))) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) * (l * l)) / ((sin(k) * tan(k)) * (2.0d0 + ((k / t) * (k / t))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.2e-80) || !(t <= 1.65e-99)) {
		tmp = ((2.0 * Math.pow(t, -3.0)) * (l * l)) / ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t) * (k / t))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -7.2e-80) or not (t <= 1.65e-99):
		tmp = ((2.0 * math.pow(t, -3.0)) * (l * l)) / ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t) * (k / t))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7.2e-80) || !(t <= 1.65e-99))
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) * Float64(l * l)) / Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t)))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7.2e-80) || ~((t <= 1.65e-99)))
		tmp = ((2.0 * (t ^ -3.0)) * (l * l)) / ((sin(k) * tan(k)) * (2.0 + ((k / t) * (k / t))));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7.2e-80], N[Not[LessEqual[t, 1.65e-99]], $MachinePrecision]], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2e-80 or 1.64999999999999993e-99 < t

   1. Initial program 64.4%

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

   3. Simplified 57.7%

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

      1. div-inv 57.7%

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

      2. pow-flip 57.8%

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

      3. metadata-eval 57.8%

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

   5. Applied egg-rr 57.8%

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

      1. unpow2 57.8%

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

   7. Applied egg-rr 57.8%

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

   if -7.2e-80 < t < 1.64999999999999993e-99

   1. Initial program 34.2%

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

   3. Simplified 32.6%

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

      1. div-inv 32.6%

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

      2. pow-flip 32.6%

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

      3. metadata-eval 32.6%

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

   5. Applied egg-rr 32.6%

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

   6. Taylor expanded in t around 0 72.6%

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

      1. times-frac 73.5%

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

      2. associate-/r* 73.5%

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

      3. associate-*r/ 76.3%

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

   8. Simplified 76.3%

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

   9. Taylor expanded in k around 0 54.0%

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

       1. associate-/r* 58.4%

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

   11. Simplified 58.4%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-80} \lor \neg \left(t \leq 1.65 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\left(2 \cdot {t}^{-3}\right) \cdot \left(\ell \cdot \ell\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 20: 53.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\frac{2 \cdot {k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.2e-41)
   (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
   (/ (* (/ 2.0 (pow t 3.0)) (* l l)) (/ (* 2.0 (pow k 2.0)) (cos k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.2e-41) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else {
		tmp = ((2.0 / pow(t, 3.0)) * (l * l)) / ((2.0 * pow(k, 2.0)) / cos(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.2d-41) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else
        tmp = ((2.0d0 / (t ** 3.0d0)) * (l * l)) / ((2.0d0 * (k ** 2.0d0)) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.2e-41) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else {
		tmp = ((2.0 / Math.pow(t, 3.0)) * (l * l)) / ((2.0 * Math.pow(k, 2.0)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.2e-41:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	else:
		tmp = ((2.0 / math.pow(t, 3.0)) * (l * l)) / ((2.0 * math.pow(k, 2.0)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.2e-41)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	else
		tmp = Float64(Float64(Float64(2.0 / (t ^ 3.0)) * Float64(l * l)) / Float64(Float64(2.0 * (k ^ 2.0)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.2e-41)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	else
		tmp = ((2.0 / (t ^ 3.0)) * (l * l)) / ((2.0 * (k ^ 2.0)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.2e-41], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2e-41

   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 51.3%

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

      1. div-inv 51.3%

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

      2. pow-flip 51.3%

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

      3. metadata-eval 51.3%

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

   5. Applied egg-rr 51.3%

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

   6. Taylor expanded in t around 0 64.2%

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

      1. times-frac 63.8%

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

      2. associate-/r* 63.8%

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

      3. associate-*r/ 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 54.9%

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

       1. associate-/r* 56.8%

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

   11. Simplified 56.8%

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

   if 2.2e-41 < l

   1. Initial program 39.1%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around inf 42.6%

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

      1. associate-*r/ 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in k around 0 42.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\frac{2 \cdot {k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 21: 54.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t}\\ \mathbf{elif}\;k \leq 14500:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e-162)
   (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t))
   (if (<= k 14500.0)
     (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))
     (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-162) {
		tmp = 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t);
	} else if (k <= 14500.0) {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t * pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d-162) then
        tmp = 2.0d0 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t)
    else if (k <= 14500.0d0) then
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-162) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.pow(k, -4.0)) / t);
	} else if (k <= 14500.0) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e-162:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t)
	elif k <= 14500.0:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e-162)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t));
	elseif (k <= 14500.0)
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e-162)
		tmp = 2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t);
	elseif (k <= 14500.0)
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e-162], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 14500.0], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4500000000000001e-162

   1. Initial program 58.1%

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

   3. Simplified 50.7%

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

      1. div-inv 50.7%

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

      2. pow-flip 50.7%

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

      3. metadata-eval 50.7%

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

   5. Applied egg-rr 50.7%

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

   6. Taylor expanded in t around 0 56.7%

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

      1. times-frac 58.0%

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

      2. associate-/r* 57.9%

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

      3. associate-*r/ 59.7%

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

   8. Simplified 59.7%

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

   9. Taylor expanded in k around 0 48.7%

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

       1. associate-/r* 51.5%

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

   11. Simplified 51.5%

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

       1. expm1-log1p-u 51.4%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

       2. expm1-udef 51.3%

          \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

       3. div-inv 50.9%

          \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

       4. pow-flip 50.9%

          \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

       5. metadata-eval 50.9%

          \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   13. Applied egg-rr 50.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   14. Step-by-step derivation

       1. expm1-def 51.0%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 51.0%

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

   15. Simplified 51.0%

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

   if 1.4500000000000001e-162 < k < 14500

   1. Initial program 63.0%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in k around 0 79.9%

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

   if 14500 < k

   1. Initial program 35.8%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in t around 0 64.0%

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

   5. Taylor expanded in k around 0 47.4%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t}\\ \mathbf{elif}\;k \leq 14500:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 22: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7e-54)
   (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
   (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-54) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 7d-54) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-54) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 7e-54:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	else:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7e-54)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	else
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7e-54)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	else
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7e-54], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.99999999999999964e-54

   1. Initial program 48.3%

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

   3. Simplified 43.6%

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

      1. div-inv 43.6%

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

      2. pow-flip 43.6%

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

      3. metadata-eval 43.6%

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

   5. Applied egg-rr 43.6%

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

   6. Taylor expanded in t around 0 63.1%

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

      1. times-frac 63.7%

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

      2. associate-/r* 63.7%

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

      3. associate-*r/ 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 49.5%

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

       1. associate-/r* 52.0%

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

   11. Simplified 52.0%

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

   if 6.99999999999999964e-54 < t

   1. Initial program 64.3%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in k around 0 57.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-54}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]

Alternative 23: 53.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 \cdot {k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.8e-52)
   (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t))
   (/ (* (/ 2.0 (pow t 3.0)) (* l l)) (* 2.0 (pow k 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.8e-52) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
	} else {
		tmp = ((2.0 / pow(t, 3.0)) * (l * l)) / (2.0 * pow(k, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.8d-52) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
    else
        tmp = ((2.0d0 / (t ** 3.0d0)) * (l * l)) / (2.0d0 * (k ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.8e-52) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t);
	} else {
		tmp = ((2.0 / Math.pow(t, 3.0)) * (l * l)) / (2.0 * Math.pow(k, 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.8e-52:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)
	else:
		tmp = ((2.0 / math.pow(t, 3.0)) * (l * l)) / (2.0 * math.pow(k, 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.8e-52)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t));
	else
		tmp = Float64(Float64(Float64(2.0 / (t ^ 3.0)) * Float64(l * l)) / Float64(2.0 * (k ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.8e-52)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
	else
		tmp = ((2.0 / (t ^ 3.0)) * (l * l)) / (2.0 * (k ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.8e-52], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.8000000000000003e-52

   1. Initial program 48.3%

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

   3. Simplified 43.6%

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

      1. div-inv 43.6%

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

      2. pow-flip 43.6%

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

      3. metadata-eval 43.6%

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

   5. Applied egg-rr 43.6%

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

   6. Taylor expanded in t around 0 63.1%

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

      1. times-frac 63.7%

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

      2. associate-/r* 63.7%

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

      3. associate-*r/ 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 49.5%

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

       1. associate-/r* 52.0%

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

   11. Simplified 52.0%

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

   if 3.8000000000000003e-52 < t

   1. Initial program 64.3%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in k around 0 55.7%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 \cdot {k}^{2}}\\ \end{array} \]

Alternative 24: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({k}^{-4} \cdot \frac{{\ell}^{2}}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (pow k -4.0) (/ (pow l 2.0) t))))

C

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

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 * ((k ** (-4.0d0)) * ((l ** 2.0d0) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[Power[k, -4.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 48.0%

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

   1. div-inv 48.0%

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

   2. pow-flip 48.0%

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

   3. metadata-eval 48.0%

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

5. Applied egg-rr 48.0%

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

6. Taylor expanded in t around 0 59.8%

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

   1. times-frac 59.5%

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

   2. associate-/r* 59.5%

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

   3. associate-*r/ 60.6%

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

8. Simplified 60.6%

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

9. Taylor expanded in k around 0 49.2%

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

    1. associate-/r* 50.5%

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

11. Simplified 50.5%

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

    1. div-inv 50.5%

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

    2. div-inv 50.3%

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

    3. pow-flip 50.3%

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

    4. metadata-eval 50.3%

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

13. Applied egg-rr 50.3%

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

    1. associate-*r/ 50.3%

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

    2. *-rgt-identity 50.3%

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

    3. associate-/l* 49.2%

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

    4. associate-/r/ 50.2%

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

15. Simplified 50.2%

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

16. Final simplification 50.2%

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

Alternative 25: 51.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t)))

C

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

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 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 48.0%

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

   1. div-inv 48.0%

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

   2. pow-flip 48.0%

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

   3. metadata-eval 48.0%

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

5. Applied egg-rr 48.0%

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

6. Taylor expanded in t around 0 59.8%

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

   1. times-frac 59.5%

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

   2. associate-/r* 59.5%

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

   3. associate-*r/ 60.6%

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

8. Simplified 60.6%

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

9. Taylor expanded in k around 0 49.2%

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

    1. associate-/r* 50.5%

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

11. Simplified 50.5%

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

    1. expm1-log1p-u 50.5%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

    2. expm1-udef 50.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

    3. div-inv 49.8%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

    4. pow-flip 49.8%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

    5. metadata-eval 49.8%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

13. Applied egg-rr 49.8%

    \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
14. Step-by-step derivation

    1. expm1-def 50.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 50.3%

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

15. Simplified 50.3%

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

16. Final simplification 50.3%

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

Alternative 26: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (/ (pow l 2.0) t) (pow k 4.0))))

C

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

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 * (((l ** 2.0d0) / t) / (k ** 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 48.0%

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

4. Taylor expanded in t around 0 59.8%

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

5. Taylor expanded in k around 0 49.2%

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

   1. add-sqr-sqrt 39.3%

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

   2. pow2 39.3%

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

   3. *-commutative 39.3%

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

7. Applied egg-rr 39.3%

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

   1. unpow2 39.3%

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

   2. add-sqr-sqrt 49.2%

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

   3. associate-/r* 50.4%

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

9. Applied egg-rr 50.4%

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

10. Final simplification 50.4%

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

Alternative 27: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t)))

C

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

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 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Simplified 48.0%

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

   1. div-inv 48.0%

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

   2. pow-flip 48.0%

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

   3. metadata-eval 48.0%

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

5. Applied egg-rr 48.0%

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

6. Taylor expanded in t around 0 59.8%

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

   1. times-frac 59.5%

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

   2. associate-/r* 59.5%

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

   3. associate-*r/ 60.6%

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

8. Simplified 60.6%

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

9. Taylor expanded in k around 0 49.2%

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

    1. associate-/r* 50.5%

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

11. Simplified 50.5%

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

12. Final simplification 50.5%

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

Reproduce

herbie shell --seed 2023321 
(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))))