Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 87.1%

Time: 19.3s

Alternatives: 16

Speedup: 3.8×

• 27.3% 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: 54.5% 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: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_2 := \frac{t}{\ell} \cdot \sin k\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot t_2\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t_2 \cdot \frac{t}{\frac{\ell}{t}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
        (t_2 (* (/ t l) (sin k))))
   (if (<= t -7.6e+25)
     (/ 2.0 (* (* (* t (/ t l)) t_2) t_1))
     (if (<= t 5.4e-73)
       (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow (/ l k) 2.0) t)))
       (/ 2.0 (* t_1 (* t_2 (/ t (/ l t)))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)));
	double t_2 = (t / l) * sin(k);
	double tmp;
	if (t <= -7.6e+25) {
		tmp = 2.0 / (((t * (t / l)) * t_2) * t_1);
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * (pow((l / k), 2.0) / t));
	} else {
		tmp = 2.0 / (t_1 * (t_2 * (t / (l / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = tan(k) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0)))
    t_2 = (t / l) * sin(k)
    if (t <= (-7.6d+25)) then
        tmp = 2.0d0 / (((t * (t / l)) * t_2) * t_1)
    else if (t <= 5.4d-73) then
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * (((l / k) ** 2.0d0) / t))
    else
        tmp = 2.0d0 / (t_1 * (t_2 * (t / (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)));
	double t_2 = (t / l) * Math.sin(k);
	double tmp;
	if (t <= -7.6e+25) {
		tmp = 2.0 / (((t * (t / l)) * t_2) * t_1);
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow((l / k), 2.0) / t));
	} else {
		tmp = 2.0 / (t_1 * (t_2 * (t / (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * (1.0 + (1.0 + math.pow((k / t), 2.0)))
	t_2 = (t / l) * math.sin(k)
	tmp = 0
	if t <= -7.6e+25:
		tmp = 2.0 / (((t * (t / l)) * t_2) * t_1)
	elif t <= 5.4e-73:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * (math.pow((l / k), 2.0) / t))
	else:
		tmp = 2.0 / (t_1 * (t_2 * (t / (l / t))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))
	t_2 = Float64(Float64(t / l) * sin(k))
	tmp = 0.0
	if (t <= -7.6e+25)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(t / l)) * t_2) * t_1));
	elseif (t <= 5.4e-73)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((Float64(l / k) ^ 2.0) / t)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t_2 * Float64(t / Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * (1.0 + (1.0 + ((k / t) ^ 2.0)));
	t_2 = (t / l) * sin(k);
	tmp = 0.0;
	if (t <= -7.6e+25)
		tmp = 2.0 / (((t * (t / l)) * t_2) * t_1);
	elseif (t <= 5.4e-73)
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * (((l / k) ^ 2.0) / t));
	else
		tmp = 2.0 / (t_1 * (t_2 * (t / (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.6e+25], N[(2.0 / N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e-73], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t$95$2 * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.6000000000000001e25

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. +-commutative 65.6%

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

   3. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. *-un-lft-identity 76.5%

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

      2. associate-/l* 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. associate-/l* 76.5%

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

      3. pow1 76.5%

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

      4. associate-*l* 79.5%

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

      5. associate-/l* 93.2%

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

      6. div-inv 93.2%

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

      7. clear-num 93.3%

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

   9. Applied egg-rr 93.3%

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

   if -7.6000000000000001e25 < t < 5.39999999999999989e-73

   1. Initial program 37.7%

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

      1. associate-/l/ 37.7%

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

      2. associate-*l/ 37.6%

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

      3. associate-*l/ 37.6%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in l around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. unpow2 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in k around inf 66.1%

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

       1. associate-/r* 65.3%

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

       2. associate-*r/ 65.3%

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

       3. unpow2 65.3%

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

       4. unpow2 65.3%

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

       5. times-frac 87.2%

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

       6. unpow2 87.2%

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

       7. associate-/r* 89.6%

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

       8. associate-*l/ 89.7%

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

       9. associate-*r/ 89.7%

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

   12. Simplified 89.7%

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

   if 5.39999999999999989e-73 < t

   1. Initial program 63.7%

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

      1. associate-*l* 63.7%

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

      2. +-commutative 63.7%

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

   3. Simplified 63.7%

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

      1. unpow3 63.7%

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

      2. times-frac 72.1%

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

   5. Applied egg-rr 72.1%

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

      1. *-un-lft-identity 72.1%

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

      2. associate-/l* 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. associate-/l* 72.1%

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

      3. pow1 72.1%

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

      4. associate-*l* 78.7%

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

      5. associate-/l* 84.0%

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

      6. div-inv 83.9%

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

      7. clear-num 83.9%

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

   9. Applied egg-rr 83.9%

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

       1. *-commutative 83.9%

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

       2. associate-/r/ 84.0%

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

   11. Applied egg-rr 84.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \frac{t}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 2: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+24} \lor \neg \left(t \leq 6.1 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.85e+24) (not (<= t 6.1e-74)))
   (/
    2.0
    (*
     (* (* t (/ t l)) (* (/ t l) (sin k)))
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))))
   (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow (/ l k) 2.0) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e+24) || !(t <= 6.1e-74)) {
		tmp = 2.0 / (((t * (t / l)) * ((t / l) * sin(k))) * (tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * (pow((l / k), 2.0) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.85d+24)) .or. (.not. (t <= 6.1d-74))) then
        tmp = 2.0d0 / (((t * (t / l)) * ((t / l) * sin(k))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0)))))
    else
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * (((l / k) ** 2.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e+24) || !(t <= 6.1e-74)) {
		tmp = 2.0 / (((t * (t / l)) * ((t / l) * Math.sin(k))) * (Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow((l / k), 2.0) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.85e+24) or not (t <= 6.1e-74):
		tmp = 2.0 / (((t * (t / l)) * ((t / l) * math.sin(k))) * (math.tan(k) * (1.0 + (1.0 + math.pow((k / t), 2.0)))))
	else:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * (math.pow((l / k), 2.0) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.85e+24) || !(t <= 6.1e-74))
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((Float64(l / k) ^ 2.0) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.85e+24) || ~((t <= 6.1e-74)))
		tmp = 2.0 / (((t * (t / l)) * ((t / l) * sin(k))) * (tan(k) * (1.0 + (1.0 + ((k / t) ^ 2.0)))));
	else
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * (((l / k) ^ 2.0) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.85e+24], N[Not[LessEqual[t, 6.1e-74]], $MachinePrecision]], N[(2.0 / N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.85e24 or 6.0999999999999998e-74 < t

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. unpow3 64.6%

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

      2. times-frac 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. associate-/l* 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. associate-/l* 74.1%

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

      3. pow1 74.1%

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

      4. associate-*l* 79.1%

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

      5. associate-/l* 88.3%

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

      6. div-inv 88.3%

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

      7. clear-num 88.3%

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

   9. Applied egg-rr 88.3%

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

   if -1.85e24 < t < 6.0999999999999998e-74

   1. Initial program 37.7%

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

      1. associate-/l/ 37.7%

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

      2. associate-*l/ 37.6%

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

      3. associate-*l/ 37.6%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in l around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. unpow2 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in k around inf 66.1%

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

       1. associate-/r* 65.3%

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

       2. associate-*r/ 65.3%

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

       3. unpow2 65.3%

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

       4. unpow2 65.3%

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

       5. times-frac 87.2%

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

       6. unpow2 87.2%

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

       7. associate-/r* 89.6%

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

       8. associate-*l/ 89.7%

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

       9. associate-*r/ 89.7%

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

   12. Simplified 89.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+24} \lor \neg \left(t \leq 6.1 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4e+24)
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (* (sin k) (* (/ t l) (* t (/ t l))))))
   (if (<= t 5.4e-73)
     (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow (/ l k) 2.0) t)))
     (if (<= t 1.6e+37)
       (/ (/ l k) (/ (pow t 3.0) (/ l k)))
       (/
        2.0
        (*
         (* (sin k) (* (/ t l) (/ t (/ l t))))
         (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+24) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * (sin(k) * ((t / l) * (t * (t / l)))));
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * (pow((l / k), 2.0) / t));
	} else if (t <= 1.6e+37) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-4d+24)) then
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0)))) * (sin(k) * ((t / l) * (t * (t / l)))))
    else if (t <= 5.4d-73) then
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * (((l / k) ** 2.0d0) / t))
    else if (t <= 1.6d+37) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) * (k / t))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+24) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))) * (Math.sin(k) * ((t / l) * (t * (t / l)))));
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow((l / k), 2.0) / t));
	} else if (t <= 1.6e+37) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 / ((Math.sin(k) * ((t / l) * (t / (l / t)))) * (Math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -4e+24:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + math.pow((k / t), 2.0)))) * (math.sin(k) * ((t / l) * (t * (t / l)))))
	elif t <= 5.4e-73:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * (math.pow((l / k), 2.0) / t))
	elif t <= 1.6e+37:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 / ((math.sin(k) * ((t / l) * (t / (l / t)))) * (math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4e+24)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))));
	elseif (t <= 5.4e-73)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((Float64(l / k) ^ 2.0) / t)));
	elseif (t <= 1.6e+37)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t / l) * Float64(t / Float64(l / t)))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4e+24)
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t) ^ 2.0)))) * (sin(k) * ((t / l) * (t * (t / l)))));
	elseif (t <= 5.4e-73)
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * (((l / k) ^ 2.0) / t));
	elseif (t <= 1.6e+37)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4e+24], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e-73], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+37], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.9999999999999999e24

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. +-commutative 65.6%

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

   3. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 76.5%

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

   5. Applied egg-rr 76.5%

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

   6. Taylor expanded in t around 0 76.5%

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

      1. unpow2 65.0%

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

      2. associate-*l/ 74.8%

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

      3. *-commutative 74.8%

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

   8. Simplified 87.2%

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

   if -3.9999999999999999e24 < t < 5.39999999999999989e-73

   1. Initial program 37.7%

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

      1. associate-/l/ 37.7%

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

      2. associate-*l/ 37.6%

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

      3. associate-*l/ 37.6%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in l around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. unpow2 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in k around inf 66.1%

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

       1. associate-/r* 65.3%

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

       2. associate-*r/ 65.3%

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

       3. unpow2 65.3%

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

       4. unpow2 65.3%

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

       5. times-frac 87.2%

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

       6. unpow2 87.2%

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

       7. associate-/r* 89.6%

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

       8. associate-*l/ 89.7%

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

       9. associate-*r/ 89.7%

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

   12. Simplified 89.7%

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

   if 5.39999999999999989e-73 < t < 1.60000000000000007e37

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 67.9%

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

      4. associate-/r/ 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/l/ 66.0%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 92.4%

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

   9. Simplified 92.4%

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

   if 1.60000000000000007e37 < t

   1. Initial program 64.0%

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

      1. associate-*l* 64.0%

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

      2. +-commutative 64.0%

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

   3. Simplified 64.0%

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

      1. unpow3 64.0%

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

      2. times-frac 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-un-lft-identity 75.2%

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

      2. associate-/l* 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. unpow2 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]

Alternative 4: 83.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4e+24)
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (* (sin k) (* (/ t l) (* t (/ t l))))))
   (if (<= t 5.4e-73)
     (* 2.0 (/ (* (cos k) (* (/ l k) (/ l k))) (* t (pow (sin k) 2.0))))
     (if (<= t 2.8e+38)
       (/ (/ l k) (/ (pow t 3.0) (/ l k)))
       (/
        2.0
        (*
         (* (sin k) (* (/ t l) (/ t (/ l t))))
         (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+24) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * (sin(k) * ((t / l) * (t * (t / l)))));
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (t * pow(sin(k), 2.0)));
	} else if (t <= 2.8e+38) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-4d+24)) then
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0)))) * (sin(k) * ((t / l) * (t * (t / l)))))
    else if (t <= 5.4d-73) then
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / k))) / (t * (sin(k) ** 2.0d0)))
    else if (t <= 2.8d+38) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) * (k / t))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4e+24) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))) * (Math.sin(k) * ((t / l) * (t * (t / l)))));
	} else if (t <= 5.4e-73) {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / k))) / (t * Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 2.8e+38) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 / ((Math.sin(k) * ((t / l) * (t / (l / t)))) * (Math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -4e+24:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + math.pow((k / t), 2.0)))) * (math.sin(k) * ((t / l) * (t * (t / l)))))
	elif t <= 5.4e-73:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / k))) / (t * math.pow(math.sin(k), 2.0)))
	elif t <= 2.8e+38:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 / ((math.sin(k) * ((t / l) * (t / (l / t)))) * (math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4e+24)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))));
	elseif (t <= 5.4e-73)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / k))) / Float64(t * (sin(k) ^ 2.0))));
	elseif (t <= 2.8e+38)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t / l) * Float64(t / Float64(l / t)))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4e+24)
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t) ^ 2.0)))) * (sin(k) * ((t / l) * (t * (t / l)))));
	elseif (t <= 5.4e-73)
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (t * (sin(k) ^ 2.0)));
	elseif (t <= 2.8e+38)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4e+24], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e-73], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+38], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.9999999999999999e24

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. +-commutative 65.6%

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

   3. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 76.5%

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

   5. Applied egg-rr 76.5%

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

   6. Taylor expanded in t around 0 76.5%

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

      1. unpow2 65.0%

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

      2. associate-*l/ 74.8%

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

      3. *-commutative 74.8%

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

   8. Simplified 87.2%

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

   if -3.9999999999999999e24 < t < 5.39999999999999989e-73

   1. Initial program 37.7%

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

      1. associate-/l/ 37.7%

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

      2. associate-*l/ 37.6%

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

      3. associate-*l/ 37.6%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in l around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. unpow2 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 87.2%

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

   9. Simplified 87.2%

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

   if 5.39999999999999989e-73 < t < 2.8e38

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 67.9%

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

      4. associate-/r/ 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/l/ 66.0%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 92.4%

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

   9. Simplified 92.4%

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

   if 2.8e38 < t

   1. Initial program 64.0%

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

      1. associate-*l* 64.0%

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

      2. +-commutative 64.0%

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

   3. Simplified 64.0%

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

      1. unpow3 64.0%

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

      2. times-frac 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-un-lft-identity 75.2%

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

      2. associate-/l* 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. unpow2 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]

Alternative 5: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{if}\;t \leq -3750:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (sin k) (* (/ t l) (/ t (/ l t))))
           (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))))))
   (if (<= t -3750.0)
     t_1
     (if (<= t 2.1e-73)
       (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (* (/ l t) (/ l (* k k)))))
       (if (<= t 4e+37) (/ (/ l k) (/ (pow t 3.0) (/ l k))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -3750.0) {
		tmp = t_1;
	} else if (t <= 2.1e-73) {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	} else if (t <= 4e+37) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 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 = 2.0d0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) * (k / t))))))
    if (t <= (-3750.0d0)) then
        tmp = t_1
    else if (t <= 2.1d-73) then
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * ((l / t) * (l / (k * k))))
    else if (t <= 4d+37) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.sin(k) * ((t / l) * (t / (l / t)))) * (Math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -3750.0) {
		tmp = t_1;
	} else if (t <= 2.1e-73) {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	} else if (t <= 4e+37) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((math.sin(k) * ((t / l) * (t / (l / t)))) * (math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))))
	tmp = 0
	if t <= -3750.0:
		tmp = t_1
	elif t <= 2.1e-73:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * ((l / t) * (l / (k * k))))
	elif t <= 4e+37:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t / l) * Float64(t / Float64(l / t)))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))))))
	tmp = 0.0
	if (t <= -3750.0)
		tmp = t_1;
	elseif (t <= 2.1e-73)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	elseif (t <= 4e+37)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	tmp = 0.0;
	if (t <= -3750.0)
		tmp = t_1;
	elseif (t <= 2.1e-73)
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * ((l / t) * (l / (k * k))));
	elseif (t <= 4e+37)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3750.0], t$95$1, If[LessEqual[t, 2.1e-73], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+37], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3750 or 3.99999999999999982e37 < t

   1. Initial program 64.7%

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

      1. associate-*l* 64.7%

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

      2. +-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. unpow3 64.7%

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

      2. times-frac 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. associate-/l* 85.8%

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

   7. Applied egg-rr 85.8%

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

      1. unpow2 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -3750 < t < 2.0999999999999999e-73

   1. Initial program 37.4%

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

      1. associate-/l/ 37.4%

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

      2. associate-*l/ 37.4%

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

      3. associate-*l/ 37.4%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. associate-/l/ 38.2%

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

      7. associate-*r* 38.2%

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

      8. *-commutative 38.2%

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

      9. associate-*r* 38.2%

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

      10. *-commutative 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in k around inf 66.3%

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

      1. associate-/r* 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-/l* 65.6%

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

      4. unpow2 65.6%

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

      5. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in l around 0 66.3%

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

      1. associate-/r* 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*l/ 65.6%

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

      4. unpow2 65.6%

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

      5. unpow2 65.6%

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

      6. *-commutative 65.6%

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

      7. times-frac 67.3%

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

      8. unpow2 67.3%

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

      9. unpow2 67.3%

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

      10. associate-/l/ 66.4%

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

      11. unpow2 66.4%

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

      12. times-frac 79.6%

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

      13. unpow2 79.6%

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

   9. Simplified 79.6%

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

   if 2.0999999999999999e-73 < t < 3.99999999999999982e37

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 67.9%

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

      4. associate-/r/ 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/l/ 66.0%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3750:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 83.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (sin k) (* (/ t l) (/ t (/ l t))))
           (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))))))
   (if (<= t -4.7e+27)
     t_1
     (if (<= t 3.6e-74)
       (* 2.0 (/ (* (cos k) (* (/ l k) (/ l k))) (* t (pow (sin k) 2.0))))
       (if (<= t 2.3e+37) (/ (/ l k) (/ (pow t 3.0) (/ l k))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -4.7e+27) {
		tmp = t_1;
	} else if (t <= 3.6e-74) {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (t * pow(sin(k), 2.0)));
	} else if (t <= 2.3e+37) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 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 = 2.0d0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) * (k / t))))))
    if (t <= (-4.7d+27)) then
        tmp = t_1
    else if (t <= 3.6d-74) then
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / k))) / (t * (sin(k) ** 2.0d0)))
    else if (t <= 2.3d+37) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.sin(k) * ((t / l) * (t / (l / t)))) * (Math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -4.7e+27) {
		tmp = t_1;
	} else if (t <= 3.6e-74) {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / k))) / (t * Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 2.3e+37) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((math.sin(k) * ((t / l) * (t / (l / t)))) * (math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))))
	tmp = 0
	if t <= -4.7e+27:
		tmp = t_1
	elif t <= 3.6e-74:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / k))) / (t * math.pow(math.sin(k), 2.0)))
	elif t <= 2.3e+37:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t / l) * Float64(t / Float64(l / t)))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))))))
	tmp = 0.0
	if (t <= -4.7e+27)
		tmp = t_1;
	elseif (t <= 3.6e-74)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / k))) / Float64(t * (sin(k) ^ 2.0))));
	elseif (t <= 2.3e+37)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	tmp = 0.0;
	if (t <= -4.7e+27)
		tmp = t_1;
	elseif (t <= 3.6e-74)
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (t * (sin(k) ^ 2.0)));
	elseif (t <= 2.3e+37)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.7e+27], t$95$1, If[LessEqual[t, 3.6e-74], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e+37], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.69999999999999976e27 or 2.30000000000000002e37 < t

   1. Initial program 64.9%

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

      1. associate-*l* 64.9%

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

      2. +-commutative 64.9%

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

   3. Simplified 64.9%

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

      1. unpow3 64.9%

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

      2. times-frac 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. *-un-lft-identity 75.9%

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

      2. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. unpow2 85.6%

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

   9. Applied egg-rr 85.6%

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

   if -4.69999999999999976e27 < t < 3.6000000000000002e-74

   1. Initial program 37.7%

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

      1. associate-/l/ 37.7%

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

      2. associate-*l/ 37.6%

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

      3. associate-*l/ 37.6%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

      4. unpow2 65.3%

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

      5. unpow2 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in l around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. unpow2 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 87.2%

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

   9. Simplified 87.2%

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

   if 3.6000000000000002e-74 < t < 2.30000000000000002e37

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 67.9%

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

      4. associate-/r/ 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/l/ 66.0%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.92 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (sin k) (* (/ t l) (/ t (/ l t))))
           (* (tan k) (+ 1.0 (+ 1.0 (* (/ k t) (/ k t)))))))))
   (if (<= t -1.92e-174)
     t_1
     (if (<= t 5e-74)
       (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))
       (if (<= t 1.6e+38) (/ (/ l k) (/ (pow t 3.0) (/ l k))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -1.92e-174) {
		tmp = t_1;
	} else if (t <= 5e-74) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	} else if (t <= 1.6e+38) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 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 = 2.0d0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) * (k / t))))))
    if (t <= (-1.92d-174)) then
        tmp = t_1
    else if (t <= 5d-74) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    else if (t <= 1.6d+38) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.sin(k) * ((t / l) * (t / (l / t)))) * (Math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	double tmp;
	if (t <= -1.92e-174) {
		tmp = t_1;
	} else if (t <= 5e-74) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	} else if (t <= 1.6e+38) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((math.sin(k) * ((t / l) * (t / (l / t)))) * (math.tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))))
	tmp = 0
	if t <= -1.92e-174:
		tmp = t_1
	elif t <= 5e-74:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	elif t <= 1.6e+38:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t / l) * Float64(t / Float64(l / t)))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t)))))))
	tmp = 0.0
	if (t <= -1.92e-174)
		tmp = t_1;
	elseif (t <= 5e-74)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	elseif (t <= 1.6e+38)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / ((sin(k) * ((t / l) * (t / (l / t)))) * (tan(k) * (1.0 + (1.0 + ((k / t) * (k / t))))));
	tmp = 0.0;
	if (t <= -1.92e-174)
		tmp = t_1;
	elseif (t <= 5e-74)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	elseif (t <= 1.6e+38)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.92e-174], t$95$1, If[LessEqual[t, 5e-74], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+38], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.92e-174 or 1.59999999999999993e38 < t

   1. Initial program 61.7%

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

      1. associate-*l* 61.7%

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

      2. +-commutative 61.7%

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

   3. Simplified 61.7%

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

      1. unpow3 61.7%

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

      2. times-frac 74.4%

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

   5. Applied egg-rr 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. associate-/l* 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. unpow2 82.9%

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

   9. Applied egg-rr 82.9%

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

   if -1.92e-174 < t < 4.99999999999999998e-74

   1. Initial program 30.6%

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

      1. associate-/l/ 30.6%

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

      2. associate-*l/ 30.6%

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

      3. associate-*l/ 30.6%

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

      4. associate-/r/ 31.7%

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

      5. *-commutative 31.7%

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

      6. associate-/l/ 31.7%

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

      7. associate-*r* 31.7%

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

      8. *-commutative 31.7%

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

      9. associate-*r* 31.7%

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

      10. *-commutative 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in k around inf 68.7%

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

      1. unpow2 68.7%

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

   6. Simplified 68.7%

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

   if 4.99999999999999998e-74 < t < 1.59999999999999993e38

   1. Initial program 63.1%

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

      1. associate-/l/ 63.1%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 67.9%

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

      4. associate-/r/ 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/l/ 66.0%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. associate-*r* 66.0%

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

      10. *-commutative 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. unpow2 66.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.92 \cdot 10^{-174}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\frac{\ell}{t}}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := t \cdot \sin k\\ \mathbf{if}\;k \leq 10^{-138}:\\ \;\;\;\;\frac{\frac{2}{t_2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{2 \cdot k}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-44}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{elif}\;k \leq 0.026:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+97}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot t_1}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))) (t_2 (* t (sin k))))
   (if (<= k 1e-138)
     (/ (/ 2.0 (* t_2 (* (/ t l) (/ t l)))) (* 2.0 k))
     (if (<= k 7.2e-44)
       (* t_1 (/ l (pow t 3.0)))
       (if (<= k 0.026)
         (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
         (if (<= k 1.65e+97)
           (* (* l l) (/ 2.0 (* (tan k) (* (* k k) t_2))))
           (* 2.0 (/ (* l t_1) (* t (pow (sin k) 2.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double t_2 = t * sin(k);
	double tmp;
	if (k <= 1e-138) {
		tmp = (2.0 / (t_2 * ((t / l) * (t / l)))) / (2.0 * k);
	} else if (k <= 7.2e-44) {
		tmp = t_1 * (l / pow(t, 3.0));
	} else if (k <= 0.026) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else if (k <= 1.65e+97) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * t_2)));
	} else {
		tmp = 2.0 * ((l * t_1) / (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l / (k * k)
    t_2 = t * sin(k)
    if (k <= 1d-138) then
        tmp = (2.0d0 / (t_2 * ((t / l) * (t / l)))) / (2.0d0 * k)
    else if (k <= 7.2d-44) then
        tmp = t_1 * (l / (t ** 3.0d0))
    else if (k <= 0.026d0) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else if (k <= 1.65d+97) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * t_2)))
    else
        tmp = 2.0d0 * ((l * t_1) / (t * (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double t_2 = t * Math.sin(k);
	double tmp;
	if (k <= 1e-138) {
		tmp = (2.0 / (t_2 * ((t / l) * (t / l)))) / (2.0 * k);
	} else if (k <= 7.2e-44) {
		tmp = t_1 * (l / Math.pow(t, 3.0));
	} else if (k <= 0.026) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else if (k <= 1.65e+97) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * t_2)));
	} else {
		tmp = 2.0 * ((l * t_1) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (k * k)
	t_2 = t * math.sin(k)
	tmp = 0
	if k <= 1e-138:
		tmp = (2.0 / (t_2 * ((t / l) * (t / l)))) / (2.0 * k)
	elif k <= 7.2e-44:
		tmp = t_1 * (l / math.pow(t, 3.0))
	elif k <= 0.026:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	elif k <= 1.65e+97:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * t_2)))
	else:
		tmp = 2.0 * ((l * t_1) / (t * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(k * k))
	t_2 = Float64(t * sin(k))
	tmp = 0.0
	if (k <= 1e-138)
		tmp = Float64(Float64(2.0 / Float64(t_2 * Float64(Float64(t / l) * Float64(t / l)))) / Float64(2.0 * k));
	elseif (k <= 7.2e-44)
		tmp = Float64(t_1 * Float64(l / (t ^ 3.0)));
	elseif (k <= 0.026)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	elseif (k <= 1.65e+97)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * t_2))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * t_1) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k * k);
	t_2 = t * sin(k);
	tmp = 0.0;
	if (k <= 1e-138)
		tmp = (2.0 / (t_2 * ((t / l) * (t / l)))) / (2.0 * k);
	elseif (k <= 7.2e-44)
		tmp = t_1 * (l / (t ^ 3.0));
	elseif (k <= 0.026)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	elseif (k <= 1.65e+97)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * t_2)));
	else
		tmp = 2.0 * ((l * t_1) / (t * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1e-138], N[(N[(2.0 / N[(t$95$2 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.2e-44], N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.026], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.65e+97], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * t$95$1), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 1.00000000000000007e-138

   1. Initial program 54.8%

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

      1. associate-*l* 54.8%

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

      2. +-commutative 54.8%

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

   3. Simplified 54.8%

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

      1. unpow3 54.8%

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

      2. times-frac 66.0%

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

   5. Applied egg-rr 66.0%

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

   6. Taylor expanded in k around 0 62.9%

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

   7. Taylor expanded in t around 0 62.9%

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

      1. unpow2 62.9%

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

      2. associate-*l/ 67.1%

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

      3. *-commutative 67.1%

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

   9. Simplified 67.1%

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

       1. expm1-log1p-u 48.4%

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

       2. expm1-udef 50.6%

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

       3. *-commutative 50.6%

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

       4. associate-*l* 50.6%

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

       5. *-commutative 50.6%

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

   11. Applied egg-rr 50.6%

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

       1. expm1-def 48.4%

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

       2. expm1-log1p 67.1%

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

       3. associate-/r* 67.1%

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

       4. associate-*r* 72.3%

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

   13. Simplified 72.3%

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

   if 1.00000000000000007e-138 < k < 7.1999999999999998e-44

   1. Initial program 55.6%

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

      1. associate-/l/ 55.6%

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

      2. associate-*l/ 55.6%

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

      3. associate-*l/ 52.9%

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

      4. associate-/r/ 50.6%

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

      5. *-commutative 50.6%

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

      6. associate-/l/ 50.6%

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

      7. associate-*r* 50.6%

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

      8. *-commutative 50.6%

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

      9. associate-*r* 50.6%

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

      10. *-commutative 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in k around 0 66.5%

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

      1. unpow2 66.5%

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

      2. *-commutative 66.5%

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

      3. unpow2 66.5%

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

   6. Simplified 66.5%

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

      1. times-frac 86.5%

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

   8. Applied egg-rr 86.5%

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

   if 7.1999999999999998e-44 < k < 0.0259999999999999988

   1. Initial program 40.0%

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

      1. associate-/l/ 40.0%

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

      2. associate-*l/ 40.0%

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

      3. associate-*l/ 40.0%

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

      4. associate-/r/ 40.0%

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

      5. *-commutative 40.0%

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

      6. associate-/l/ 40.0%

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

      7. associate-*r* 40.0%

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

      8. *-commutative 40.0%

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

      9. associate-*r* 40.0%

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

      10. *-commutative 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in k around inf 81.0%

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

      1. associate-/r* 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-/l* 81.0%

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

      4. unpow2 81.0%

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

      5. unpow2 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in k around 0 81.0%

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

      1. unpow2 81.0%

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

      2. *-commutative 81.0%

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

      3. times-frac 100.0%

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

   9. Simplified 100.0%

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

   if 0.0259999999999999988 < k < 1.6500000000000001e97

   1. Initial program 42.7%

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

      1. associate-/l/ 42.7%

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

      2. associate-*l/ 42.7%

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

      3. associate-*l/ 42.7%

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

      4. associate-/r/ 46.6%

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

      5. *-commutative 46.6%

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

      6. associate-/l/ 46.5%

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

      7. associate-*r* 46.6%

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

      8. *-commutative 46.6%

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

      9. associate-*r* 46.7%

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

      10. *-commutative 46.7%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in k around inf 72.3%

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

      1. unpow2 72.3%

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

   6. Simplified 72.3%

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

   if 1.6500000000000001e97 < k

   1. Initial program 47.5%

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

      1. associate-/l/ 47.5%

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

      2. associate-*l/ 47.5%

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

      3. associate-*l/ 47.5%

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

      4. associate-/r/ 47.6%

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

      5. *-commutative 47.6%

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

      6. associate-/l/ 47.6%

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

      7. associate-*r* 47.6%

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

      8. *-commutative 47.6%

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

      9. associate-*r* 47.6%

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

      10. *-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in k around inf 62.5%

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

      1. associate-/r* 62.6%

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

      2. *-commutative 62.6%

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

      3. associate-/l* 62.6%

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

      4. unpow2 62.6%

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

      5. unpow2 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in k around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*r/ 67.0%

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

      3. unpow2 67.0%

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

   9. Simplified 67.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \frac{\ell}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-138}:\\ \;\;\;\;\frac{\frac{2}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}{2 \cdot k}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{elif}\;k \leq 0.026:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+97}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 9: 69.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.85e+18)
   (/ 2.0 (* (* (* t (sin k)) (* (/ t l) (/ t l))) (* 2.0 k)))
   (if (<= t 5.1e-74)
     (* 2.0 (/ (* (cos k) (* (/ l k) (/ l k))) (* k (* t k))))
     (/ (/ l k) (/ (pow t 3.0) (/ l k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.85e+18) {
		tmp = 2.0 / (((t * sin(k)) * ((t / l) * (t / l))) * (2.0 * k));
	} else if (t <= 5.1e-74) {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * k)));
	} else {
		tmp = (l / k) / (pow(t, 3.0) / (l / 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 (t <= (-3.85d+18)) then
        tmp = 2.0d0 / (((t * sin(k)) * ((t / l) * (t / l))) * (2.0d0 * k))
    else if (t <= 5.1d-74) then
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * k)))
    else
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.85e+18) {
		tmp = 2.0 / (((t * Math.sin(k)) * ((t / l) * (t / l))) * (2.0 * k));
	} else if (t <= 5.1e-74) {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / k))) / (k * (t * k)));
	} else {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.85e+18:
		tmp = 2.0 / (((t * math.sin(k)) * ((t / l) * (t / l))) * (2.0 * k))
	elif t <= 5.1e-74:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / k))) / (k * (t * k)))
	else:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.85e+18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * sin(k)) * Float64(Float64(t / l) * Float64(t / l))) * Float64(2.0 * k)));
	elseif (t <= 5.1e-74)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / k))) / Float64(k * Float64(t * k))));
	else
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.85e+18)
		tmp = 2.0 / (((t * sin(k)) * ((t / l) * (t / l))) * (2.0 * k));
	elseif (t <= 5.1e-74)
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * k)));
	else
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.85e+18], N[(2.0 / N[(N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.1e-74], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.85e18

   1. Initial program 65.2%

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

      1. associate-*l* 65.2%

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

      2. +-commutative 65.2%

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

   3. Simplified 65.2%

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

      1. unpow3 65.2%

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

      2. times-frac 77.1%

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

   5. Applied egg-rr 77.1%

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

   6. Taylor expanded in k around 0 64.6%

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

   7. Taylor expanded in t around 0 64.6%

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

      1. unpow2 64.6%

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

      2. associate-*l/ 74.1%

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

      3. *-commutative 74.1%

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

   9. Simplified 74.1%

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

       1. expm1-log1p-u 29.9%

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

       2. expm1-udef 20.2%

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

       3. *-commutative 20.2%

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

       4. associate-*l* 20.2%

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

       5. *-commutative 20.2%

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

   11. Applied egg-rr 20.2%

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

       1. expm1-def 29.9%

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

       2. expm1-log1p 74.1%

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

       3. associate-*r* 81.5%

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

   13. Simplified 81.5%

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

   if -3.85e18 < t < 5.0999999999999997e-74

   1. Initial program 37.4%

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

      1. associate-/l/ 37.4%

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

      2. associate-*l/ 37.4%

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

      3. associate-*l/ 37.4%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. associate-/l/ 38.2%

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

      7. associate-*r* 38.2%

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

      8. *-commutative 38.2%

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

      9. associate-*r* 38.2%

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

      10. *-commutative 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in k around inf 66.3%

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

      1. associate-/r* 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-/l* 65.6%

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

      4. unpow2 65.6%

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

      5. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in l around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. unpow2 65.6%

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

      3. unpow2 65.6%

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

      4. times-frac 87.0%

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

   9. Simplified 87.0%

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

       1. add-cbrt-cube 63.7%

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

       2. *-commutative 63.7%

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

       3. *-commutative 63.7%

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

       4. *-commutative 63.7%

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

   11. Applied egg-rr 63.7%

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

       1. associate-*l* 63.7%

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

       2. cube-unmult 63.7%

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

       3. *-commutative 63.7%

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

   13. Simplified 63.7%

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

   14. Taylor expanded in k around 0 68.0%

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

       1. unpow2 68.0%

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

       2. associate-*l* 68.0%

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

   16. Simplified 68.0%

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

   if 5.0999999999999997e-74 < t

   1. Initial program 63.7%

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

      1. associate-/l/ 63.7%

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

      2. associate-*l/ 66.3%

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

      3. associate-*l/ 65.4%

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

      4. associate-/r/ 64.8%

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

      5. *-commutative 64.8%

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

      6. associate-/l/ 64.8%

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

      7. associate-*r* 64.8%

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

      8. *-commutative 64.8%

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

      9. associate-*r* 64.8%

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

      10. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in k around 0 59.5%

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

      1. unpow2 59.5%

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

      2. *-commutative 59.5%

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

      3. unpow2 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in l around 0 59.5%

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

      1. associate-/r* 59.0%

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

      2. unpow2 59.0%

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

      3. unpow2 59.0%

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

      4. times-frac 70.1%

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

      5. associate-/l* 74.6%

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

   9. Simplified 74.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 10: 64.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e+31)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (* 2.0 (/ (* (cos k) (* (/ l k) (/ l k))) (* k (* t k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+31) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * 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 <= 1.3d+31) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+31) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / k) * (l / k))) / (k * (t * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.3e+31:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / k) * (l / k))) / (k * (t * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e+31)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / k) * Float64(l / k))) / Float64(k * Float64(t * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e+31)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (k * (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.3e+31], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e31

   1. Initial program 53.8%

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

      1. associate-/l/ 53.8%

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

      2. associate-*l/ 54.8%

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

      3. associate-*l/ 54.4%

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

      4. associate-/r/ 54.7%

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

      5. *-commutative 54.7%

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

      6. associate-/l/ 54.7%

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

      7. associate-*r* 54.7%

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

      8. *-commutative 54.7%

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

      9. associate-*r* 54.7%

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

      10. *-commutative 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in k around 0 50.7%

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

      1. unpow2 50.7%

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

      2. *-commutative 50.7%

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

      3. unpow2 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in l around 0 50.7%

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

      1. associate-/r* 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. times-frac 62.9%

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

      5. associate-/l* 67.4%

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

   9. Simplified 67.4%

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

   if 1.3e31 < k

   1. Initial program 46.8%

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

      1. associate-/l/ 46.8%

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

      2. associate-*l/ 46.8%

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

      3. associate-*l/ 46.8%

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

      4. associate-/r/ 46.8%

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

      5. *-commutative 46.8%

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

      6. associate-/l/ 46.8%

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

      7. associate-*r* 46.8%

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

      8. *-commutative 46.8%

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

      9. associate-*r* 46.9%

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

      10. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in k around inf 65.6%

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

      1. associate-/r* 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-/l* 64.3%

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

      4. unpow2 64.3%

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

      5. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in l around 0 64.3%

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

      1. associate-*r/ 64.3%

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

      2. unpow2 64.3%

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

      3. unpow2 64.3%

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

      4. times-frac 88.6%

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

   9. Simplified 88.6%

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

       1. add-cbrt-cube 64.0%

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

       2. *-commutative 64.0%

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

       3. *-commutative 64.0%

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

       4. *-commutative 64.0%

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

   11. Applied egg-rr 64.0%

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

       1. associate-*l* 64.0%

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

       2. cube-unmult 64.0%

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

       3. *-commutative 64.0%

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

   13. Simplified 64.0%

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

   14. Taylor expanded in k around 0 61.0%

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

       1. unpow2 61.0%

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

       2. associate-*l* 61.0%

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

   16. Simplified 61.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 11: 61.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-68} \lor \neg \left(t \leq 6.6 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.6e-68) (not (<= t 6.6e-74)))
   (* (/ l (* k k)) (/ l (pow t 3.0)))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.6e-68) || !(t <= 6.6e-74)) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.6d-68)) .or. (.not. (t <= 6.6d-74))) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.6e-68) || !(t <= 6.6e-74)) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.6e-68) or not (t <= 6.6e-74):
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.6e-68) || !(t <= 6.6e-74))
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.6e-68) || ~((t <= 6.6e-74)))
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.6e-68], N[Not[LessEqual[t, 6.6e-74]], $MachinePrecision]], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.60000000000000007e-68 or 6.59999999999999992e-74 < t

   1. Initial program 63.6%

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

      1. associate-/l/ 63.6%

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

      2. associate-*l/ 64.8%

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

      3. associate-*l/ 64.4%

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

      4. associate-/r/ 64.1%

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

      5. *-commutative 64.1%

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

      6. associate-/l/ 64.1%

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

      7. associate-*r* 64.1%

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

      8. *-commutative 64.1%

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

      9. associate-*r* 64.1%

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

      10. *-commutative 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in k around 0 53.2%

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

      1. unpow2 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

   6. Simplified 53.2%

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

      1. times-frac 60.8%

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

   8. Applied egg-rr 60.8%

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

   if -3.60000000000000007e-68 < t < 6.59999999999999992e-74

   1. Initial program 34.7%

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

      1. associate-/l/ 34.7%

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

      2. associate-*l/ 34.7%

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

      3. associate-*l/ 34.7%

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

      4. associate-/r/ 35.6%

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

      5. *-commutative 35.6%

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

      6. associate-/l/ 35.6%

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

      7. associate-*r* 35.6%

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

      8. *-commutative 35.6%

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

      9. associate-*r* 35.6%

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

      10. *-commutative 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in k around inf 67.8%

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

      1. associate-/r* 66.9%

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

      2. *-commutative 66.9%

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

      3. associate-/l* 66.9%

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

      4. unpow2 66.9%

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

      5. unpow2 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in k around 0 54.8%

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

      1. unpow2 54.8%

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

      2. *-commutative 54.8%

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

      3. times-frac 66.2%

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

   9. Simplified 66.2%

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

       1. associate-*l/ 66.4%

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

   11. Applied egg-rr 66.4%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-68} \lor \neg \left(t \leq 6.6 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 12: 67.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-78} \lor \neg \left(t \leq 1.65 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2e-78) (not (<= t 1.65e-74)))
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2e-78) || !(t <= 1.65e-74)) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-2d-78)) .or. (.not. (t <= 1.65d-74))) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2e-78) || !(t <= 1.65e-74)) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -2e-78) or not (t <= 1.65e-74):
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2e-78) || !(t <= 1.65e-74))
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -2e-78) || ~((t <= 1.65e-74)))
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2e-78], N[Not[LessEqual[t, 1.65e-74]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2e-78 or 1.64999999999999998e-74 < t

   1. Initial program 63.9%

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

      1. associate-/l/ 63.9%

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

      2. associate-*l/ 65.1%

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

      3. associate-*l/ 64.6%

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

      4. associate-/r/ 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-/l/ 64.3%

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

      7. associate-*r* 64.3%

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

      8. *-commutative 64.3%

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

      9. associate-*r* 64.4%

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

      10. *-commutative 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in k around 0 53.1%

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

      1. unpow2 53.1%

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

      2. *-commutative 53.1%

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

      3. unpow2 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in l around 0 53.1%

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

      1. associate-/r* 52.3%

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

      2. unpow2 52.3%

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

      3. unpow2 52.3%

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

      4. times-frac 63.7%

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

      5. associate-/l* 68.5%

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

   9. Simplified 68.5%

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

   if -2e-78 < t < 1.64999999999999998e-74

   1. Initial program 33.1%

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

      1. associate-/l/ 33.1%

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

      2. associate-*l/ 33.1%

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

      3. associate-*l/ 33.1%

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

      4. associate-/r/ 34.0%

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

      5. *-commutative 34.0%

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

      6. associate-/l/ 34.0%

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

      7. associate-*r* 34.0%

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

      8. *-commutative 34.0%

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

      9. associate-*r* 34.0%

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

      10. *-commutative 34.0%

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

   3. Simplified 34.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 67.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 66.5%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 66.5%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      3. associate-/l* 66.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 66.5%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

      5. unpow2 66.5%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

   6. Simplified 66.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{{\sin k}^{2} \cdot t}} \]

   7. Taylor expanded in k around 0 54.9%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 54.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 54.9%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 66.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 66.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 67.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 67.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-78} \lor \neg \left(t \leq 1.65 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 13: 62.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.32e-45)
   (/ (* l l) (* k (* k (pow t 3.0))))
   (if (<= t 1.75e-74)
     (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
     (* (/ l (* k k)) (/ l (pow t 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.32e-45) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else if (t <= 1.75e-74) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = (l / (k * k)) * (l / 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 <= (-1.32d-45)) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else if (t <= 1.75d-74) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.32e-45) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else if (t <= 1.75e-74) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.32e-45:
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	elif t <= 1.75e-74:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.32e-45)
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	elseif (t <= 1.75e-74)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.32e-45)
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	elseif (t <= 1.75e-74)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.32e-45], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-74], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.32000000000000005e-45

   1. Initial program 62.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 62.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 62.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 62.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 62.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 62.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 62.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 62.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 62.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 62.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 62.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 62.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 47.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 47.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 47.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 47.4%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in t around 0 47.4%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. unpow2 47.4%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* 56.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot k\right) \cdot k}} \]

      4. *-commutative 56.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot k} \]

      5. *-commutative 56.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   9. Simplified 56.8%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   if -1.32000000000000005e-45 < t < 1.75000000000000007e-74

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 36.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 36.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 36.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 37.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 37.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 37.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 37.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 37.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 37.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 37.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 37.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 67.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 67.2%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      3. associate-/l* 67.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 67.2%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

      5. unpow2 67.2%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

   6. Simplified 67.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{{\sin k}^{2} \cdot t}} \]

   7. Taylor expanded in k around 0 54.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 54.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 54.6%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 65.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 65.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 65.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   11. Applied egg-rr 65.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

   if 1.75000000000000007e-74 < t

   1. Initial program 63.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 63.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 66.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 65.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 64.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 64.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 64.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 64.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 64.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 59.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 59.5%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 59.5%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 59.5%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. times-frac 68.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   8. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 14: 61.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2e-162)
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (if (<= k 8.2e+116)
     (* (/ l (* k k)) (/ l (pow t 3.0)))
     (* 2.0 (* l (/ l (* t (pow k 4.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-162) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else if (k <= 8.2e+116) {
		tmp = (l / (k * k)) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * (l * (l / (t * pow(k, 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d-162) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else if (k <= 8.2d+116) then
        tmp = (l / (k * k)) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (l * (l / (t * (k ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-162) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else if (k <= 8.2e+116) {
		tmp = (l / (k * k)) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (l * (l / (t * Math.pow(k, 4.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2e-162:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	elif k <= 8.2e+116:
		tmp = (l / (k * k)) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (l * (l / (t * math.pow(k, 4.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2e-162)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	elseif (k <= 8.2e+116)
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2e-162)
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	elseif (k <= 8.2e+116)
		tmp = (l / (k * k)) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * (l * (l / (t * (k ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2e-162], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.2e+116], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999991e-162

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.9%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 54.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 54.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 54.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 54.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 54.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 54.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 54.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 54.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 47.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 47.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 47.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 47.8%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in l around 0 47.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 47.2%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 47.2%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 47.2%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 62.4%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   9. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]

   if 1.99999999999999991e-162 < k < 8.1999999999999996e116

   1. Initial program 50.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.9%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.9%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.9%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 52.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 52.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. unpow2 52.9%

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{t}^{3} \cdot \left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. times-frac 60.2%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   8. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]

   if 8.1999999999999996e116 < k

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 61.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 61.9%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 61.9%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

      3. associate-/l* 61.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2} \cdot t} \]

      4. unpow2 61.9%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

      5. unpow2 61.9%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

   6. Simplified 61.9%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{{\sin k}^{2} \cdot t}} \]

   7. Taylor expanded in k around 0 61.9%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 61.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 61.9%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   9. Simplified 61.9%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

   10. Taylor expanded in l around 0 61.9%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 61.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. *-commutative 61.9%

          \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

       3. associate-*l/ 66.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t \cdot {k}^{4}} \cdot \ell\right)} \]

       4. *-commutative 66.7%

          \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

   12. Simplified 66.7%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \]

Alternative 15: 54.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* l (/ l (* t (pow k 4.0))))))

C

double code(double t, double l, double k) {
	return 2.0 * (l * (l / (t * pow(k, 4.0))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l * (l / (t * (k ** 4.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l * (l / (t * Math.pow(k, 4.0))));
}

Python

def code(t, l, k):
	return 2.0 * (l * (l / (t * math.pow(k, 4.0))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l * Float64(l / Float64(t * (k ^ 4.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l * (l / (t * (k ^ 4.0))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 51.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 52.6%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 52.3%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 52.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 52.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 52.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 52.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 57.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. associate-/r* 56.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

   2. *-commutative 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

   3. associate-/l* 56.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2} \cdot t} \]

   4. unpow2 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

   5. unpow2 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

6. Simplified 56.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{{\sin k}^{2} \cdot t}} \]

7. Taylor expanded in k around 0 50.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 50.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 50.0%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 50.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Taylor expanded in l around 0 50.0%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 50.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 50.0%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. associate-*l/ 54.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t \cdot {k}^{4}} \cdot \ell\right)} \]

    4. *-commutative 54.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

12. Simplified 54.8%

    \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

13. Final simplification 54.8%

    \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \]

Alternative 16: 55.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 51.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 52.6%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 52.3%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 52.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 52.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 52.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 52.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 52.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 57.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. associate-/r* 56.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

   2. *-commutative 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]

   3. associate-/l* 56.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2} \cdot t} \]

   4. unpow2 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

   5. unpow2 56.0%

      \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2} \cdot t} \]

6. Simplified 56.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{{\sin k}^{2} \cdot t}} \]

7. Taylor expanded in k around 0 50.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 50.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 50.0%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 56.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 56.0%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 56.0%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Reproduce

herbie shell --seed 2023224 
(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))))