Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.2% → 87.2%

Time: 22.3s

Alternatives: 16

Speedup: 28.1×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% 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: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{t_2}}{\frac{t}{\frac{\sqrt[3]{\ell \cdot \ell}}{\sqrt[3]{\sin k}}}}\right)}^{3}}{t_1}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (/ 2.0 (tan k))))
   (if (<= t -2.1e-59)
     (/ (pow (/ (cbrt t_2) (/ t (/ (cbrt (* l l)) (cbrt (sin k))))) 3.0) t_1)
     (if (<= t 4.2e-11)
       (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (/ (pow (sin k) 2.0) (cos k))))
       (/ (/ t_2 (* (sin k) (pow (/ (pow t 1.5) l) 2.0))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 / tan(k);
	double tmp;
	if (t <= -2.1e-59) {
		tmp = pow((cbrt(t_2) / (t / (cbrt((l * l)) / cbrt(sin(k))))), 3.0) / t_1;
	} else if (t <= 4.2e-11) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = (t_2 / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (t <= -2.1e-59) {
		tmp = Math.pow((Math.cbrt(t_2) / (t / (Math.cbrt((l * l)) / Math.cbrt(Math.sin(k))))), 3.0) / t_1;
	} else if (t <= 4.2e-11) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = (t_2 / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t <= -2.1e-59)
		tmp = Float64((Float64(cbrt(t_2) / Float64(t / Float64(cbrt(Float64(l * l)) / cbrt(sin(k))))) ^ 3.0) / t_1);
	elseif (t <= 4.2e-11)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(t_2 / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1e-59], N[(N[Power[N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[(t / N[(N[Power[N[(l * l), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 4.2e-11], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.09999999999999997e-59

   1. Initial program 71.3%

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

      1. associate-/r* 71.3%

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

      2. associate-*l* 64.1%

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

      3. sqr-neg 64.1%

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

      4. associate-*l* 71.3%

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

      5. *-commutative 71.3%

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

      6. sqr-neg 71.3%

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

      7. associate-/r* 71.3%

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

   3. Simplified 71.0%

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

      1. add-cube-cbrt 70.8%

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

      2. pow3 70.8%

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

      3. cbrt-div 70.8%

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

      4. cbrt-div 70.7%

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

      5. rem-cbrt-cube 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. cbrt-div 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -2.09999999999999997e-59 < t < 4.1999999999999997e-11

   1. Initial program 40.8%

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

      1. associate-/r* 40.8%

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

      2. associate-*l* 40.8%

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

      3. sqr-neg 40.8%

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

      4. associate-*l* 40.8%

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

      5. *-commutative 40.8%

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

      6. sqr-neg 40.8%

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

      7. associate-*l/ 40.8%

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

      8. associate-*r/ 40.6%

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

      9. associate-/r/ 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in k around inf 75.2%

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

      1. associate-*r* 75.2%

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

      2. times-frac 75.7%

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

      3. unpow2 75.7%

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

      4. unpow2 75.7%

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

      5. associate-*l* 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in l around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. associate-/r* 75.8%

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

      3. associate-*l/ 75.7%

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

      4. unpow2 75.7%

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

      5. associate-*r* 80.5%

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

      6. unpow2 80.5%

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

      7. associate-/l* 80.6%

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

      8. times-frac 93.4%

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

      9. associate-/r* 95.4%

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

   9. Simplified 95.4%

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

       1. clear-num 95.5%

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

       2. frac-times 95.5%

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

       3. *-un-lft-identity 95.5%

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

   11. Applied egg-rr 95.5%

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

   if 4.1999999999999997e-11 < t

   1. Initial program 63.8%

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

      1. associate-/r* 63.8%

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

      2. associate-*l* 56.8%

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

      3. sqr-neg 56.8%

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

      4. associate-*l* 63.8%

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

      5. *-commutative 63.8%

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

      6. sqr-neg 63.8%

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

      7. associate-/r* 63.8%

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

   3. Simplified 63.6%

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

      1. associate-/r/ 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. pow2 63.8%

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

      4. sqrt-div 63.8%

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

      5. sqrt-pow1 72.1%

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

      6. metadata-eval 72.1%

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

      7. sqrt-prod 48.1%

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

      8. add-sqr-sqrt 90.7%

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

   5. Applied egg-rr 90.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\sqrt[3]{\ell \cdot \ell}}{\sqrt[3]{\sin k}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\\ \mathbf{if}\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))))
   (if (<= (/ 2.0 (* t_1 (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l)))))) 0.0)
     (/ 2.0 (* (* (tan k) (* (/ k l) (/ (pow t 3.0) l))) t_1))
     (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (/ (pow (sin k) 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (pow((k / t), 2.0) + 1.0);
	double tmp;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 0.0) {
		tmp = 2.0 / ((tan(k) * ((k / l) * (pow(t, 3.0) / l))) * t_1);
	} else {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (pow(sin(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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (((k / t) ** 2.0d0) + 1.0d0)
    if ((2.0d0 / (t_1 * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 0.0d0) then
        tmp = 2.0d0 / ((tan(k) * ((k / l) * ((t ** 3.0d0) / l))) * t_1)
    else
        tmp = 2.0d0 * (((l / k) / (t * (k / l))) / ((sin(k) ** 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (Math.pow((k / t), 2.0) + 1.0);
	double tmp;
	if ((2.0 / (t_1 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 0.0) {
		tmp = 2.0 / ((Math.tan(k) * ((k / l) * (Math.pow(t, 3.0) / l))) * t_1);
	} else {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (math.pow((k / t), 2.0) + 1.0)
	tmp = 0
	if (2.0 / (t_1 * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 0.0:
		tmp = 2.0 / ((math.tan(k) * ((k / l) * (math.pow(t, 3.0) / l))) * t_1)
	else:
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (math.pow(math.sin(k), 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0))
	tmp = 0.0
	if (Float64(2.0 / Float64(t_1 * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))) * t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (((k / t) ^ 2.0) + 1.0);
	tmp = 0.0;
	if ((2.0 / (t_1 * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 0.0)
		tmp = 2.0 / ((tan(k) * ((k / l) * ((t ^ 3.0) / l))) * t_1);
	else
		tmp = 2.0 * (((l / k) / (t * (k / l))) / ((sin(k) ^ 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < -0.0

   1. Initial program 82.4%

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

   2. Taylor expanded in k around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. times-frac 89.1%

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

   4. Simplified 89.1%

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

   if -0.0 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 23.5%

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

      1. associate-/r* 23.5%

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

      2. associate-*l* 23.5%

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

      3. sqr-neg 23.5%

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

      4. associate-*l* 23.5%

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

      5. *-commutative 23.5%

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

      6. sqr-neg 23.5%

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

      7. associate-*l/ 23.5%

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

      8. associate-*r/ 23.5%

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

      9. associate-/r/ 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in k around inf 58.3%

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

      1. associate-*r* 58.3%

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

      2. times-frac 58.7%

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

      3. unpow2 58.7%

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

      4. unpow2 58.7%

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

      5. associate-*l* 63.4%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in l around 0 58.3%

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

      1. associate-*r* 58.3%

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

      2. associate-/r* 58.7%

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

      3. associate-*l/ 58.7%

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

      4. unpow2 58.7%

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

      5. associate-*r* 63.4%

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

      6. unpow2 63.4%

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

      7. associate-/l* 63.4%

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

      8. times-frac 77.7%

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

      9. associate-/r* 80.0%

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

   9. Simplified 80.0%

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

       1. clear-num 80.1%

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

       2. frac-times 80.1%

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

       3. *-un-lft-identity 80.1%

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

   11. Applied egg-rr 80.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 3: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{t_2}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}{t_1}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (/ 2.0 (tan k))))
   (if (<= t -2.35e-70)
     (/ (pow (/ (cbrt t_2) (/ t (cbrt (/ l (/ (sin k) l))))) 3.0) t_1)
     (if (<= t 1.15e-11)
       (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (/ (pow (sin k) 2.0) (cos k))))
       (/ (/ t_2 (* (sin k) (pow (/ (pow t 1.5) l) 2.0))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 / tan(k);
	double tmp;
	if (t <= -2.35e-70) {
		tmp = pow((cbrt(t_2) / (t / cbrt((l / (sin(k) / l))))), 3.0) / t_1;
	} else if (t <= 1.15e-11) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = (t_2 / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (t <= -2.35e-70) {
		tmp = Math.pow((Math.cbrt(t_2) / (t / Math.cbrt((l / (Math.sin(k) / l))))), 3.0) / t_1;
	} else if (t <= 1.15e-11) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = (t_2 / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t <= -2.35e-70)
		tmp = Float64((Float64(cbrt(t_2) / Float64(t / cbrt(Float64(l / Float64(sin(k) / l))))) ^ 3.0) / t_1);
	elseif (t <= 1.15e-11)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(t_2 / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.35e-70], N[(N[Power[N[(N[Power[t$95$2, 1/3], $MachinePrecision] / N[(t / N[Power[N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 1.15e-11], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3499999999999999e-70

   1. Initial program 69.9%

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

      1. associate-/r* 69.9%

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

      2. associate-*l* 63.0%

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

      3. sqr-neg 63.0%

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

      4. associate-*l* 69.9%

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

      5. *-commutative 69.9%

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

      6. sqr-neg 69.9%

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

      7. associate-/r* 69.9%

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

   3. Simplified 69.6%

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

      1. add-cube-cbrt 69.4%

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

      2. pow3 69.4%

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

      3. cbrt-div 69.4%

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

      4. cbrt-div 69.3%

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

      5. rem-cbrt-cube 72.4%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in k around inf 53.6%

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

      1. unpow1/3 72.4%

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

      2. unpow2 72.4%

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

      3. associate-/l* 75.7%

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

   8. Simplified 75.7%

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

   if -2.3499999999999999e-70 < t < 1.15000000000000007e-11

   1. Initial program 41.0%

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

      1. associate-/r* 41.0%

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

      2. associate-*l* 40.9%

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

      3. sqr-neg 40.9%

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

      4. associate-*l* 41.0%

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

      5. *-commutative 41.0%

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

      6. sqr-neg 41.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*r/ 40.8%

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

      9. associate-/r/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in k around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. times-frac 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*l* 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in l around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. associate-/r* 76.9%

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

      3. associate-*l/ 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r* 81.8%

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

      6. unpow2 81.8%

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

      7. associate-/l* 81.9%

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

      8. times-frac 95.0%

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

      9. associate-/r* 97.0%

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

   9. Simplified 97.0%

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

       1. clear-num 97.1%

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

       2. frac-times 97.2%

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

       3. *-un-lft-identity 97.2%

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

   11. Applied egg-rr 97.2%

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

   if 1.15000000000000007e-11 < t

   1. Initial program 63.8%

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

      1. associate-/r* 63.8%

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

      2. associate-*l* 56.8%

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

      3. sqr-neg 56.8%

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

      4. associate-*l* 63.8%

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

      5. *-commutative 63.8%

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

      6. sqr-neg 63.8%

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

      7. associate-/r* 63.8%

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

   3. Simplified 63.6%

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

      1. associate-/r/ 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. pow2 63.8%

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

      4. sqrt-div 63.8%

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

      5. sqrt-pow1 72.1%

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

      6. metadata-eval 72.1%

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

      7. sqrt-prod 48.1%

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

      8. add-sqr-sqrt 90.7%

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

   5. Applied egg-rr 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 4: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right) \cdot \left(1 + \left(t_1 + 1\right)\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -3.4e-68)
     (/ 2.0 (* (* (tan k) (* (/ k l) (/ (pow t 3.0) l))) (+ 1.0 (+ t_1 1.0))))
     (if (<= t 5.2e-13)
       (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (/ (pow (sin k) 2.0) (cos k))))
       (/
        (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
        (+ 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -3.4e-68) {
		tmp = 2.0 / ((tan(k) * ((k / l) * (pow(t, 3.0) / l))) * (1.0 + (t_1 + 1.0)));
	} else if (t <= 5.2e-13) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if (t <= (-3.4d-68)) then
        tmp = 2.0d0 / ((tan(k) * ((k / l) * ((t ** 3.0d0) / l))) * (1.0d0 + (t_1 + 1.0d0)))
    else if (t <= 5.2d-13) then
        tmp = 2.0d0 * (((l / k) / (t * (k / l))) / ((sin(k) ** 2.0d0) / cos(k)))
    else
        tmp = ((2.0d0 / tan(k)) / (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0))) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -3.4e-68) {
		tmp = 2.0 / ((Math.tan(k) * ((k / l) * (Math.pow(t, 3.0) / l))) * (1.0 + (t_1 + 1.0)));
	} else if (t <= 5.2e-13) {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -3.4e-68:
		tmp = 2.0 / ((math.tan(k) * ((k / l) * (math.pow(t, 3.0) / l))) * (1.0 + (t_1 + 1.0)))
	elif t <= 5.2e-13:
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (math.pow(math.sin(k), 2.0) / math.cos(k)))
	else:
		tmp = ((2.0 / math.tan(k)) / (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -3.4e-68)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(k / l) * Float64((t ^ 3.0) / l))) * Float64(1.0 + Float64(t_1 + 1.0))));
	elseif (t <= 5.2e-13)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -3.4e-68)
		tmp = 2.0 / ((tan(k) * ((k / l) * ((t ^ 3.0) / l))) * (1.0 + (t_1 + 1.0)));
	elseif (t <= 5.2e-13)
		tmp = 2.0 * (((l / k) / (t * (k / l))) / ((sin(k) ^ 2.0) / cos(k)));
	else
		tmp = ((2.0 / tan(k)) / (sin(k) * (((t ^ 1.5) / l) ^ 2.0))) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -3.4e-68], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.2e-13], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.40000000000000018e-68

   1. Initial program 69.9%

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

   2. Taylor expanded in k around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. times-frac 72.9%

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

   4. Simplified 72.9%

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

   if -3.40000000000000018e-68 < t < 5.2000000000000001e-13

   1. Initial program 41.0%

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

      1. associate-/r* 41.0%

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

      2. associate-*l* 40.9%

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

      3. sqr-neg 40.9%

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

      4. associate-*l* 41.0%

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

      5. *-commutative 41.0%

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

      6. sqr-neg 41.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*r/ 40.8%

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

      9. associate-/r/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in k around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. times-frac 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*l* 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in l around 0 76.3%

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

      1. associate-*r* 76.3%

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

      2. associate-/r* 76.9%

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

      3. associate-*l/ 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*r* 81.8%

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

      6. unpow2 81.8%

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

      7. associate-/l* 81.9%

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

      8. times-frac 95.0%

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

      9. associate-/r* 97.0%

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

   9. Simplified 97.0%

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

       1. clear-num 97.1%

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

       2. frac-times 97.2%

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

       3. *-un-lft-identity 97.2%

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

   11. Applied egg-rr 97.2%

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

   if 5.2000000000000001e-13 < t

   1. Initial program 63.8%

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

      1. associate-/r* 63.8%

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

      2. associate-*l* 56.8%

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

      3. sqr-neg 56.8%

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

      4. associate-*l* 63.8%

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

      5. *-commutative 63.8%

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

      6. sqr-neg 63.8%

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

      7. associate-/r* 63.8%

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

   3. Simplified 63.6%

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

      1. associate-/r/ 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. pow2 63.8%

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

      4. sqrt-div 63.8%

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

      5. sqrt-pow1 72.1%

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

      6. metadata-eval 72.1%

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

      7. sqrt-prod 48.1%

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

      8. add-sqr-sqrt 90.7%

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

   5. Applied egg-rr 90.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 5: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.6e-36)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (* 2.0 (* (cos k) (/ (* (/ l k) (/ (/ l k) t)) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-36) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (cos(k) * (((l / k) * ((l / 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 (k <= 5.6d-36) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = 2.0d0 * (cos(k) * (((l / k) * ((l / 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 (k <= 5.6e-36) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (Math.cos(k) * (((l / k) * ((l / k) / t)) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.6e-36:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = 2.0 * (math.cos(k) * (((l / k) * ((l / k) / t)) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.6e-36)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.6e-36)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = 2.0 * (cos(k) * (((l / k) * ((l / k) / t)) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.6e-36], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.6000000000000002e-36

   1. Initial program 62.2%

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

      1. associate-/r* 62.2%

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

      2. associate-*l* 56.3%

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

      3. sqr-neg 56.3%

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

      4. associate-*l* 62.2%

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

      5. *-commutative 62.2%

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

      6. sqr-neg 62.2%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 61.4%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. unpow2 63.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. associate-*l/ 62.1%

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

      4. unpow2 62.1%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 72.2%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if 5.6000000000000002e-36 < k

   1. Initial program 42.4%

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

      1. associate-/r* 42.4%

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

      2. associate-*l* 42.4%

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

      3. sqr-neg 42.4%

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

      4. associate-*l* 42.4%

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

      5. *-commutative 42.4%

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

      6. sqr-neg 42.4%

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

      7. associate-*l/ 42.4%

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

      8. associate-*r/ 42.4%

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

      9. associate-/r/ 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in k around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. times-frac 69.5%

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

      3. unpow2 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*l* 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in l around 0 68.9%

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

      1. associate-*r* 68.9%

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

      2. associate-/r* 69.5%

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

      3. associate-*l/ 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*r* 75.9%

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

      6. unpow2 75.9%

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

      7. associate-/l* 75.9%

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

      8. times-frac 85.7%

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

      9. associate-/r* 85.7%

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

   9. Simplified 85.7%

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

       1. clear-num 85.7%

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

       2. frac-times 85.8%

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

       3. *-un-lft-identity 85.8%

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

   11. Applied egg-rr 85.8%

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

       1. expm1-log1p-u 75.3%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)\right)} \]

       2. expm1-udef 59.7%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)} - 1\right)} \]

   13. Applied egg-rr 59.7%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 75.3%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)\right)} \]

       2. expm1-log1p 85.8%

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

       3. associate-/r/ 85.7%

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

       4. *-commutative 85.7%

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

       5. associate-/r* 83.5%

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

       6. *-commutative 83.5%

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

       7. associate-*l* 84.7%

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

       8. *-commutative 84.7%

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

       9. associate-/r/ 84.7%

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

       10. associate-/r/ 85.8%

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

       11. associate-/r* 85.6%

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

   15. Simplified 85.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k}}{{\sin k}^{2} \cdot \left(t \cdot \frac{k}{\ell}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-17)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (* 2.0 (* (cos k) (/ (/ l k) (* (pow (sin k) 2.0) (* t (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-17) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (cos(k) * ((l / k) / (pow(sin(k), 2.0) * (t * (k / l)))));
	}
	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 <= 1.2d-17) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = 2.0d0 * (cos(k) * ((l / k) / ((sin(k) ** 2.0d0) * (t * (k / l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-17) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (Math.cos(k) * ((l / k) / (Math.pow(Math.sin(k), 2.0) * (t * (k / l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.2e-17:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = 2.0 * (math.cos(k) * ((l / k) / (math.pow(math.sin(k), 2.0) * (t * (k / l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-17)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(l / k) / Float64((sin(k) ^ 2.0) * Float64(t * Float64(k / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e-17)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = 2.0 * (cos(k) * ((l / k) / ((sin(k) ^ 2.0) * (t * (k / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2e-17], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999993e-17

   1. Initial program 62.3%

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

      1. associate-/r* 62.3%

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

      2. associate-*l* 56.5%

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

      3. sqr-neg 56.5%

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

      4. associate-*l* 62.3%

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

      5. *-commutative 62.3%

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

      6. sqr-neg 62.3%

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

      7. associate-*l/ 62.3%

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

      8. associate-*r/ 61.5%

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

      9. associate-/r/ 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in k around 0 57.6%

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

      1. unpow2 57.6%

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

      2. times-frac 63.1%

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

      3. unpow2 63.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.6%

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

      1. unpow2 57.6%

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

      2. times-frac 63.1%

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

      3. associate-*l/ 62.2%

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

      4. unpow2 62.2%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 72.1%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if 1.19999999999999993e-17 < k

   1. Initial program 41.4%

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

      1. associate-/r* 41.4%

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

      2. associate-*l* 41.4%

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

      3. sqr-neg 41.4%

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

      4. associate-*l* 41.4%

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

      5. *-commutative 41.4%

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

      6. sqr-neg 41.4%

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

      7. associate-*l/ 41.5%

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

      8. associate-*r/ 41.4%

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

      9. associate-/r/ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in k around inf 67.7%

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

      1. associate-*r* 67.7%

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

      2. times-frac 68.3%

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

      3. unpow2 68.3%

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

      4. unpow2 68.3%

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

      5. associate-*l* 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in l around 0 67.7%

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

      1. associate-*r* 67.7%

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

      2. associate-/r* 68.3%

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

      3. associate-*l/ 68.3%

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

      4. unpow2 68.3%

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

      5. associate-*r* 75.0%

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

      6. unpow2 75.0%

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

      7. associate-/l* 75.0%

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

      8. times-frac 85.2%

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

      9. associate-/r* 85.1%

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

   9. Simplified 85.1%

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

       1. clear-num 85.2%

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

       2. frac-times 85.2%

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

       3. *-un-lft-identity 85.2%

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

   11. Applied egg-rr 85.2%

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

       1. expm1-log1p-u 75.7%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)\right)} \]

       2. expm1-udef 59.5%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)} - 1\right)} \]

   13. Applied egg-rr 59.5%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 75.7%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\right)\right)} \]

       2. expm1-log1p 85.2%

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

       3. associate-/r/ 85.2%

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

       4. *-commutative 85.2%

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

       5. associate-/l/ 85.2%

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

       6. *-commutative 85.2%

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

   15. Simplified 85.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k}}{{\sin k}^{2} \cdot \left(t \cdot \frac{k}{\ell}\right)}\right)\\ \end{array} \]

Alternative 7: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.6e-35)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (* 2.0 (* (cos k) (/ (/ (/ l k) (* t (/ k l))) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.6e-35) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (cos(k) * (((l / k) / (t * (k / l))) / 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 (k <= 8.6d-35) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = 2.0d0 * (cos(k) * (((l / k) / (t * (k / l))) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.6e-35) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (Math.cos(k) * (((l / k) / (t * (k / l))) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.6e-35:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = 2.0 * (math.cos(k) * (((l / k) / (t * (k / l))) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.6e-35)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.6e-35)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = 2.0 * (cos(k) * (((l / k) / (t * (k / l))) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.6e-35], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.6000000000000004e-35

   1. Initial program 62.2%

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

      1. associate-/r* 62.2%

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

      2. associate-*l* 56.3%

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

      3. sqr-neg 56.3%

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

      4. associate-*l* 62.2%

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

      5. *-commutative 62.2%

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

      6. sqr-neg 62.2%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 61.4%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. unpow2 63.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. associate-*l/ 62.1%

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

      4. unpow2 62.1%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 72.2%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if 8.6000000000000004e-35 < k

   1. Initial program 42.4%

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

      1. associate-/r* 42.4%

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

      2. associate-*l* 42.4%

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

      3. sqr-neg 42.4%

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

      4. associate-*l* 42.4%

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

      5. *-commutative 42.4%

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

      6. sqr-neg 42.4%

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

      7. associate-*l/ 42.4%

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

      8. associate-*r/ 42.4%

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

      9. associate-/r/ 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in k around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. times-frac 69.5%

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

      3. unpow2 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*l* 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in l around 0 68.9%

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

      1. associate-*r* 68.9%

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

      2. associate-/r* 69.5%

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

      3. associate-*l/ 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*r* 75.9%

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

      6. unpow2 75.9%

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

      7. associate-/l* 75.9%

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

      8. times-frac 85.7%

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

      9. associate-/r* 85.7%

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

   9. Simplified 85.7%

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

       1. clear-num 85.7%

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

       2. frac-times 85.8%

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

       3. *-un-lft-identity 85.8%

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

   11. Applied egg-rr 85.8%

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

       1. associate-/r/ 85.7%

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

   13. Applied egg-rr 85.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.02e-35)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (/ (pow (sin k) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.02e-35) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (pow(sin(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 (k <= 1.02d-35) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = 2.0d0 * (((l / k) / (t * (k / l))) / ((sin(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 (k <= 1.02e-35) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.02e-35:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = 2.0 * (((l / k) / (t * (k / l))) / (math.pow(math.sin(k), 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.02e-35)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64((sin(k) ^ 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.02e-35)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = 2.0 * (((l / k) / (t * (k / l))) / ((sin(k) ^ 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.02e-35], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999995e-35

   1. Initial program 62.2%

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

      1. associate-/r* 62.2%

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

      2. associate-*l* 56.3%

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

      3. sqr-neg 56.3%

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

      4. associate-*l* 62.2%

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

      5. *-commutative 62.2%

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

      6. sqr-neg 62.2%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 61.4%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. unpow2 63.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. associate-*l/ 62.1%

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

      4. unpow2 62.1%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 72.2%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if 1.01999999999999995e-35 < k

   1. Initial program 42.4%

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

      1. associate-/r* 42.4%

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

      2. associate-*l* 42.4%

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

      3. sqr-neg 42.4%

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

      4. associate-*l* 42.4%

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

      5. *-commutative 42.4%

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

      6. sqr-neg 42.4%

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

      7. associate-*l/ 42.4%

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

      8. associate-*r/ 42.4%

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

      9. associate-/r/ 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in k around inf 68.9%

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

      1. associate-*r* 68.9%

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

      2. times-frac 69.5%

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

      3. unpow2 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*l* 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in l around 0 68.9%

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

      1. associate-*r* 68.9%

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

      2. associate-/r* 69.5%

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

      3. associate-*l/ 69.5%

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

      4. unpow2 69.5%

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

      5. associate-*r* 75.9%

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

      6. unpow2 75.9%

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

      7. associate-/l* 75.9%

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

      8. times-frac 85.7%

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

      9. associate-/r* 85.7%

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

   9. Simplified 85.7%

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

       1. clear-num 85.7%

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

       2. frac-times 85.8%

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

       3. *-un-lft-identity 85.8%

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

   11. Applied egg-rr 85.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{\frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 9: 69.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;t \leq -3.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t_1 \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (/ l k))))
   (if (<= t -3.15e-68)
     (* (/ l k) (/ (/ l (pow t 3.0)) k))
     (if (<= t 3.4e-55)
       (/ 2.0 (* t_1 (* t_1 (/ t (cos k)))))
       (* l (/ (/ l k) (* k (pow t 3.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = k / (l / k);
	double tmp;
	if (t <= -3.15e-68) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (t <= 3.4e-55) {
		tmp = 2.0 / (t_1 * (t_1 * (t / cos(k))));
	} else {
		tmp = l * ((l / k) / (k * 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) :: t_1
    real(8) :: tmp
    t_1 = k / (l / k)
    if (t <= (-3.15d-68)) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (t <= 3.4d-55) then
        tmp = 2.0d0 / (t_1 * (t_1 * (t / cos(k))))
    else
        tmp = l * ((l / k) / (k * (t ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k / (l / k);
	double tmp;
	if (t <= -3.15e-68) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (t <= 3.4e-55) {
		tmp = 2.0 / (t_1 * (t_1 * (t / Math.cos(k))));
	} else {
		tmp = l * ((l / k) / (k * Math.pow(t, 3.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k / (l / k)
	tmp = 0
	if t <= -3.15e-68:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif t <= 3.4e-55:
		tmp = 2.0 / (t_1 * (t_1 * (t / math.cos(k))))
	else:
		tmp = l * ((l / k) / (k * math.pow(t, 3.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / Float64(l / k))
	tmp = 0.0
	if (t <= -3.15e-68)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (t <= 3.4e-55)
		tmp = Float64(2.0 / Float64(t_1 * Float64(t_1 * Float64(t / cos(k)))));
	else
		tmp = Float64(l * Float64(Float64(l / k) / Float64(k * (t ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k / (l / k);
	tmp = 0.0;
	if (t <= -3.15e-68)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (t <= 3.4e-55)
		tmp = 2.0 / (t_1 * (t_1 * (t / cos(k))));
	else
		tmp = l * ((l / k) / (k * (t ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.15e-68], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-55], N[(2.0 / N[(t$95$1 * N[(t$95$1 * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / k), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1499999999999999e-68

   1. Initial program 69.9%

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

      1. associate-/r* 69.9%

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

      2. associate-*l* 63.0%

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

      3. sqr-neg 63.0%

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

      4. associate-*l* 69.9%

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

      5. *-commutative 69.9%

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

      6. sqr-neg 69.9%

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

      7. associate-*l/ 69.9%

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

      8. associate-*r/ 68.2%

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

      9. associate-/r/ 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in k around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. unpow2 61.0%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. associate-*l/ 59.8%

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

      4. unpow2 59.8%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 71.7%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if -3.1499999999999999e-68 < t < 3.39999999999999973e-55

   1. Initial program 38.8%

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

   2. Taylor expanded in t around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      3. unpow2 76.6%

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

      4. associate-*r* 76.6%

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

      5. times-frac 85.2%

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

      6. unpow2 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in k around 0 72.4%

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

      1. unpow2 72.4%

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

   7. Simplified 72.4%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}} \]

      2. expm1-udef 33.6%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{k \cdot k}{\ell}\right)} - 1}} \]

      3. *-commutative 33.6%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{k \cdot k}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k \cdot \ell}}\right)} - 1} \]

      4. associate-/l* 33.6%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\cos k \cdot \ell}\right)} - 1} \]

      5. times-frac 33.6%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\frac{k}{\frac{\ell}{k}} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot \frac{k \cdot k}{\ell}\right)}\right)} - 1} \]

      6. associate-/l* 33.7%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{t}{\cos k} \cdot \color{blue}{\frac{k}{\frac{\ell}{k}}}\right)\right)} - 1} \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{t}{\cos k} \cdot \frac{k}{\frac{\ell}{k}}\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 59.4%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{t}{\cos k} \cdot \frac{k}{\frac{\ell}{k}}\right)\right)\right)}} \]

       2. expm1-log1p 75.2%

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

       3. *-commutative 75.2%

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

   11. Simplified 75.2%

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

   if 3.39999999999999973e-55 < t

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

      2. associate-*l* 58.1%

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

      3. sqr-neg 58.1%

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

      4. associate-*l* 64.5%

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

      5. *-commutative 64.5%

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

      6. sqr-neg 64.5%

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

      7. associate-*l/ 64.5%

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

      8. associate-*r/ 64.3%

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

      9. associate-/r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 54.4%

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

      1. unpow2 54.4%

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

      2. times-frac 61.1%

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

      3. unpow2 61.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   8. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   9. Taylor expanded in l around 0 54.4%

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

       1. associate-/r* 54.3%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 54.3%

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

       3. associate-*l/ 59.7%

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

       4. unpow2 59.7%

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}} \]

       5. associate-*l/ 61.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k}}{{t}^{3}} \cdot \ell} \]

       6. *-commutative 61.1%

          \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}} \]

       7. associate-/r* 64.8%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       8. associate-/l/ 70.1%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

   11. Simplified 70.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \frac{t}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 10: 69.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\frac{\ell}{k}}{t}}}{\ell} \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.1e-73)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= t 3.4e-55)
     (/ 2.0 (* (/ (/ k (/ (/ l k) t)) l) (/ (* k k) (cos k))))
     (* l (/ (/ l k) (* k (pow t 3.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.1e-73) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (t <= 3.4e-55) {
		tmp = 2.0 / (((k / ((l / k) / t)) / l) * ((k * k) / cos(k)));
	} else {
		tmp = l * ((l / k) / (k * 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 <= (-3.1d-73)) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (t <= 3.4d-55) then
        tmp = 2.0d0 / (((k / ((l / k) / t)) / l) * ((k * k) / cos(k)))
    else
        tmp = l * ((l / k) / (k * (t ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.1e-73) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (t <= 3.4e-55) {
		tmp = 2.0 / (((k / ((l / k) / t)) / l) * ((k * k) / Math.cos(k)));
	} else {
		tmp = l * ((l / k) / (k * Math.pow(t, 3.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.1e-73:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif t <= 3.4e-55:
		tmp = 2.0 / (((k / ((l / k) / t)) / l) * ((k * k) / math.cos(k)))
	else:
		tmp = l * ((l / k) / (k * math.pow(t, 3.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.1e-73)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (t <= 3.4e-55)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / Float64(Float64(l / k) / t)) / l) * Float64(Float64(k * k) / cos(k))));
	else
		tmp = Float64(l * Float64(Float64(l / k) / Float64(k * (t ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.1e-73)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (t <= 3.4e-55)
		tmp = 2.0 / (((k / ((l / k) / t)) / l) * ((k * k) / cos(k)));
	else
		tmp = l * ((l / k) / (k * (t ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.1e-73], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-55], N[(2.0 / N[(N[(N[(k / N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / k), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.09999999999999969e-73

   1. Initial program 69.9%

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

      1. associate-/r* 69.9%

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

      2. associate-*l* 63.0%

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

      3. sqr-neg 63.0%

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

      4. associate-*l* 69.9%

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

      5. *-commutative 69.9%

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

      6. sqr-neg 69.9%

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

      7. associate-*l/ 69.9%

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

      8. associate-*r/ 68.2%

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

      9. associate-/r/ 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in k around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. unpow2 61.0%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. associate-*l/ 59.8%

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

      4. unpow2 59.8%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 71.7%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if -3.09999999999999969e-73 < t < 3.39999999999999973e-55

   1. Initial program 38.8%

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

   2. Taylor expanded in t around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      3. unpow2 76.6%

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

      4. associate-*r* 76.6%

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

      5. times-frac 85.2%

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

      6. unpow2 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in t around 0 76.6%

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

      1. unpow2 76.6%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

      4. *-commutative 76.6%

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

      5. times-frac 85.2%

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

      6. unpow2 85.2%

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

      7. associate-/l* 86.4%

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

      8. associate-/l* 89.1%

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

      9. associate-*l/ 91.9%

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

      10. *-commutative 91.9%

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

      11. associate-*l/ 94.3%

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

      12. associate-/l* 92.9%

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

      13. *-commutative 92.9%

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

      14. associate-*r* 86.3%

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

      15. unpow2 86.3%

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

      16. associate-/l* 86.3%

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

      17. associate-/r/ 86.3%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in k around 0 76.0%

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

      1. unpow2 72.4%

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

   10. Simplified 76.0%

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

   if 3.39999999999999973e-55 < t

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

      2. associate-*l* 58.1%

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

      3. sqr-neg 58.1%

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

      4. associate-*l* 64.5%

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

      5. *-commutative 64.5%

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

      6. sqr-neg 64.5%

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

      7. associate-*l/ 64.5%

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

      8. associate-*r/ 64.3%

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

      9. associate-/r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 54.4%

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

      1. unpow2 54.4%

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

      2. times-frac 61.1%

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

      3. unpow2 61.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   8. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   9. Taylor expanded in l around 0 54.4%

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

       1. associate-/r* 54.3%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 54.3%

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

       3. associate-*l/ 59.7%

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

       4. unpow2 59.7%

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}} \]

       5. associate-*l/ 61.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k}}{{t}^{3}} \cdot \ell} \]

       6. *-commutative 61.1%

          \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}} \]

       7. associate-/r* 64.8%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       8. associate-/l/ 70.1%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

   11. Simplified 70.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\frac{\ell}{k}}{t}}}{\ell} \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 11: 63.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \cos k} \cdot \frac{k \cdot k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.4e-35)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (/ 2.0 (* (/ (* t (* k k)) (* l (cos k))) (/ (* k k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.4e-35) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 / (((t * (k * k)) / (l * cos(k))) * ((k * k) / l));
	}
	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 <= 1.4d-35) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = 2.0d0 / (((t * (k * k)) / (l * cos(k))) * ((k * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.4e-35) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = 2.0 / (((t * (k * k)) / (l * Math.cos(k))) * ((k * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.4e-35:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = 2.0 / (((t * (k * k)) / (l * math.cos(k))) * ((k * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.4e-35)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k * k)) / Float64(l * cos(k))) * Float64(Float64(k * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.4e-35)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = 2.0 / (((t * (k * k)) / (l * cos(k))) * ((k * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.4e-35], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-35

   1. Initial program 62.2%

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

      1. associate-/r* 62.2%

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

      2. associate-*l* 56.3%

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

      3. sqr-neg 56.3%

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

      4. associate-*l* 62.2%

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

      5. *-commutative 62.2%

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

      6. sqr-neg 62.2%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 61.4%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. unpow2 63.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.5%

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

      1. unpow2 57.5%

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

      2. times-frac 63.1%

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

      3. associate-*l/ 62.1%

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

      4. unpow2 62.1%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 72.2%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if 1.4e-35 < k

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. *-commutative 68.8%

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

      3. unpow2 68.8%

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

      4. associate-*r* 68.9%

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

      5. times-frac 72.3%

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

      6. unpow2 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in k around 0 57.4%

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

      1. unpow2 57.4%

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

   7. Simplified 57.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \cos k} \cdot \frac{k \cdot k}{\ell}}\\ \end{array} \]

Alternative 12: 68.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-62} \lor \neg \left(t \leq 7.2 \cdot 10^{-55}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\frac{\ell}{k}}{t}} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.55e-62) (not (<= t 7.2e-55)))
   (* l (/ (/ l k) (* k (pow t 3.0))))
   (/ 2.0 (* (/ k (/ (/ l k) t)) (* k (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.55e-62) || !(t <= 7.2e-55)) {
		tmp = l * ((l / k) / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	}
	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 <= (-1.55d-62)) .or. (.not. (t <= 7.2d-55))) then
        tmp = l * ((l / k) / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 / ((k / ((l / k) / t)) * (k * (k / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.55e-62) || !(t <= 7.2e-55)) {
		tmp = l * ((l / k) / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.55e-62) or not (t <= 7.2e-55):
		tmp = l * ((l / k) / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.55e-62) || !(t <= 7.2e-55))
		tmp = Float64(l * Float64(Float64(l / k) / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(Float64(l / k) / t)) * Float64(k * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.55e-62) || ~((t <= 7.2e-55)))
		tmp = l * ((l / k) / (k * (t ^ 3.0)));
	else
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.55e-62], N[Not[LessEqual[t, 7.2e-55]], $MachinePrecision]], N[(l * N[(N[(l / k), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55e-62 or 7.2000000000000001e-55 < t

   1. Initial program 67.5%

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

      1. associate-/r* 67.5%

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

      2. associate-*l* 60.8%

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

      3. sqr-neg 60.8%

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

      4. associate-*l* 67.5%

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

      5. *-commutative 67.5%

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

      6. sqr-neg 67.5%

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

      7. associate-*l/ 67.6%

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

      8. associate-*r/ 66.6%

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

      9. associate-/r/ 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in k around 0 56.4%

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

      1. unpow2 56.4%

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

      2. times-frac 61.4%

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

      3. unpow2 61.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   8. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   9. Taylor expanded in l around 0 56.4%

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

       1. associate-/r* 57.1%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 57.1%

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

       3. associate-*l/ 60.1%

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

       4. unpow2 60.1%

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}} \]

       5. associate-*l/ 61.4%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k}}{{t}^{3}} \cdot \ell} \]

       6. *-commutative 61.4%

          \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}} \]

       7. associate-/r* 65.9%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       8. associate-/l/ 70.7%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

   11. Simplified 70.7%

       \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

   if -1.55e-62 < t < 7.2000000000000001e-55

   1. Initial program 38.5%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. unpow2 75.9%

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

      4. associate-*r* 75.9%

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

      5. times-frac 84.5%

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

      6. unpow2 84.5%

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

   4. Simplified 84.5%

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

   5. Taylor expanded in k around 0 70.4%

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

      1. unpow2 70.4%

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

      2. associate-*r* 70.4%

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

      3. associate-/l* 70.7%

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

      4. associate-/r* 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in k around 0 73.1%

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

      1. unpow2 73.1%

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

      2. associate-*l/ 73.1%

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

      3. *-commutative 73.1%

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

   10. Simplified 73.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-62} \lor \neg \left(t \leq 7.2 \cdot 10^{-55}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\frac{\ell}{k}}{t}} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 13: 68.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\frac{\ell}{k}}{t}} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.6e-75)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= t 8e-55)
     (/ 2.0 (* (/ k (/ (/ l k) t)) (* k (/ k l))))
     (* l (/ (/ l k) (* k (pow t 3.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.6e-75) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (t <= 8e-55) {
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	} else {
		tmp = l * ((l / k) / (k * 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 <= (-5.6d-75)) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (t <= 8d-55) then
        tmp = 2.0d0 / ((k / ((l / k) / t)) * (k * (k / l)))
    else
        tmp = l * ((l / k) / (k * (t ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.6e-75) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (t <= 8e-55) {
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	} else {
		tmp = l * ((l / k) / (k * Math.pow(t, 3.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -5.6e-75:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif t <= 8e-55:
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)))
	else:
		tmp = l * ((l / k) / (k * math.pow(t, 3.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.6e-75)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (t <= 8e-55)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(Float64(l / k) / t)) * Float64(k * Float64(k / l))));
	else
		tmp = Float64(l * Float64(Float64(l / k) / Float64(k * (t ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.6e-75)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (t <= 8e-55)
		tmp = 2.0 / ((k / ((l / k) / t)) * (k * (k / l)));
	else
		tmp = l * ((l / k) / (k * (t ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.6e-75], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-55], N[(2.0 / N[(N[(k / N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / k), $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.59999999999999996e-75

   1. Initial program 69.9%

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

      1. associate-/r* 69.9%

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

      2. associate-*l* 63.0%

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

      3. sqr-neg 63.0%

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

      4. associate-*l* 69.9%

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

      5. *-commutative 69.9%

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

      6. sqr-neg 69.9%

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

      7. associate-*l/ 69.9%

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

      8. associate-*r/ 68.2%

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

      9. associate-/r/ 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in k around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. unpow2 61.0%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 61.0%

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

      3. associate-*l/ 59.8%

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

      4. unpow2 59.8%

         \[\leadsto \frac{\ell \cdot \frac{\ell}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

      5. times-frac 71.7%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   9. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}} \]

   if -5.59999999999999996e-75 < t < 7.99999999999999996e-55

   1. Initial program 38.8%

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

   2. Taylor expanded in t around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. *-commutative 76.6%

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

      3. unpow2 76.6%

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

      4. associate-*r* 76.6%

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

      5. times-frac 85.2%

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

      6. unpow2 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in k around 0 71.0%

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

      1. unpow2 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-/l* 71.3%

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

      4. associate-/r* 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in k around 0 73.7%

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

      1. unpow2 73.7%

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

      2. associate-*l/ 73.7%

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

      3. *-commutative 73.7%

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

   10. Simplified 73.7%

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

   if 7.99999999999999996e-55 < t

   1. Initial program 64.5%

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

      1. associate-/r* 64.5%

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

      2. associate-*l* 58.1%

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

      3. sqr-neg 58.1%

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

      4. associate-*l* 64.5%

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

      5. *-commutative 64.5%

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

      6. sqr-neg 64.5%

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

      7. associate-*l/ 64.5%

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

      8. associate-*r/ 64.3%

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

      9. associate-/r/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 54.4%

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

      1. unpow2 54.4%

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

      2. times-frac 61.1%

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

      3. unpow2 61.1%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   8. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k} \cdot \ell}{{t}^{3}}} \]

   9. Taylor expanded in l around 0 54.4%

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

       1. associate-/r* 54.3%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

       2. unpow2 54.3%

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

       3. associate-*l/ 59.7%

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

       4. unpow2 59.7%

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \ell}{{t}^{3}} \]

       5. associate-*l/ 61.1%

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot k}}{{t}^{3}} \cdot \ell} \]

       6. *-commutative 61.1%

          \[\leadsto \color{blue}{\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}} \]

       7. associate-/r* 64.8%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       8. associate-/l/ 70.1%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

   11. Simplified 70.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\frac{\ell}{k}}{t}} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}}\\ \end{array} \]

Alternative 14: 58.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (/ (/ l k) (* t (/ k l))) (* k k))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / k) / (t * (k / l))) / (k * k));
}

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 / k) / (t * (k / l))) / (k * k))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) / (t * (k / l))) / (k * k));
}

Python

def code(t, l, k):
	return 2.0 * (((l / k) / (t * (k / l))) / (k * k))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t * Float64(k / l))) / Float64(k * k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) / (t * (k / l))) / (k * k));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

   1. associate-/r* 56.0%

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

   2. associate-*l* 51.9%

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

   3. sqr-neg 51.9%

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

   4. associate-*l* 56.0%

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

   5. *-commutative 56.0%

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

   6. sqr-neg 56.0%

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

   7. associate-*l/ 55.9%

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

   8. associate-*r/ 55.4%

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

   9. associate-/r/ 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in k around inf 61.9%

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

   1. associate-*r* 61.9%

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

   2. times-frac 62.2%

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

   3. unpow2 62.2%

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

   4. unpow2 62.2%

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

   5. associate-*l* 64.3%

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

6. Simplified 64.3%

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

7. Taylor expanded in l around 0 61.9%

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

   1. associate-*r* 61.9%

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

   2. associate-/r* 62.6%

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

   3. associate-*l/ 62.6%

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

   4. unpow2 62.6%

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

   5. associate-*r* 64.6%

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

   6. unpow2 64.6%

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

   7. associate-/l* 64.7%

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

   8. times-frac 72.0%

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

   9. associate-/r* 73.0%

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

9. Simplified 73.0%

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

    1. clear-num 73.1%

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

    2. frac-times 73.1%

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

    3. *-un-lft-identity 73.1%

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

11. Applied egg-rr 73.1%

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

12. Taylor expanded in k around 0 60.0%

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

    1. unpow2 60.0%

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

14. Simplified 60.0%

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

15. Final simplification 60.0%

    \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k} \]

Alternative 15: 58.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (/ (* k k) l) (/ (* k k) (/ l t)))))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) / (l / 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 * k) / l) * ((k * k) / (l / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
}

Python

def code(t, l, k):
	return 2.0 / (((k * k) / l) * ((k * k) / (l / t)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) / Float64(l / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in t around 0 61.9%

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

   1. *-commutative 61.9%

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

   2. *-commutative 61.9%

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

   3. unpow2 61.9%

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

   4. associate-*r* 61.9%

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

   5. times-frac 65.7%

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

   6. unpow2 65.7%

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

4. Simplified 65.7%

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

5. Taylor expanded in k around 0 58.6%

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

   1. associate-/l* 61.1%

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

   2. unpow2 61.1%

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

7. Simplified 61.1%

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

8. Final simplification 61.1%

   \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \]

Alternative 16: 60.7% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k}{\frac{\frac{\frac{\ell}{t}}{k}}{\frac{k}{\frac{\ell}{k}}}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ k (/ (/ (/ l t) k) (/ k (/ l k))))))

C

double code(double t, double l, double k) {
	return 2.0 / (k / (((l / t) / k) / (k / (l / k))));
}

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 / (((l / t) / k) / (k / (l / k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (k / (((l / t) / k) / (k / (l / k))));
}

Python

def code(t, l, k):
	return 2.0 / (k / (((l / t) / k) / (k / (l / k))))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(k / Float64(Float64(Float64(l / t) / k) / Float64(k / Float64(l / k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (k / (((l / t) / k) / (k / (l / k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(k / N[(N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision] / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in t around 0 61.9%

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

   1. *-commutative 61.9%

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

   2. *-commutative 61.9%

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

   3. unpow2 61.9%

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

   4. associate-*r* 61.9%

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

   5. times-frac 65.7%

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

   6. unpow2 65.7%

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

4. Simplified 65.7%

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

5. Taylor expanded in k around 0 58.6%

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

   1. unpow2 58.6%

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

   2. associate-*r* 58.6%

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

   3. associate-/l* 59.1%

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

   4. associate-/r* 59.7%

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

7. Simplified 59.7%

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

   1. associate-*l/ 59.7%

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

   2. associate-/l* 59.7%

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

   3. associate-/l/ 59.1%

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

9. Applied egg-rr 59.1%

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

    1. associate-/l* 59.5%

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

    2. associate-/r* 63.4%

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

11. Simplified 63.4%

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

12. Final simplification 63.4%

    \[\leadsto \frac{2}{\frac{k}{\frac{\frac{\frac{\ell}{t}}{k}}{\frac{k}{\frac{\ell}{k}}}}} \]

Reproduce

herbie shell --seed 2023297 
(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))))