Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 86.6%

Time: 30.4s

Alternatives: 15

Speedup: 3.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{t}{\ell}\right)}^{2}\\ t_2 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ t_3 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot t_1\right)}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot t_3\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot t_3}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{\sin k \cdot \left(t \cdot t_1\right)}{\frac{\cos k}{\sin k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ t l) 2.0))
        (t_2 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
        (t_3 (/ (sin k) l)))
   (if (<= t -9.5e+103)
     (/ 2.0 (* t_2 (* (* (tan k) (* t (sin k))) t_1)))
     (if (<= t -1.2e-40)
       (/
        (* l (* (/ l (sin k)) (/ 2.0 (* (tan k) (pow t 3.0)))))
        (+ 2.0 (/ (/ k t) (/ t k))))
       (if (<= t 1.58e-58)
         (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) t_3)))
         (if (<= t 1.1e+93)
           (/
            (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) t_3))
            (+ 2.0 (/ (* k (/ k t)) t)))
           (/ 2.0 (* t_2 (/ (* (sin k) (* t t_1)) (/ (cos k) (sin k)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((t / l), 2.0);
	double t_2 = 1.0 + (1.0 + pow((k / t), 2.0));
	double t_3 = sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / (t_2 * ((tan(k) * (t * sin(k))) * t_1));
	} else if (t <= -1.2e-40) {
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 1.58e-58) {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * t_3));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * t_3)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / (t_2 * ((sin(k) * (t * t_1)) / (cos(k) / sin(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (t / l) ** 2.0d0
    t_2 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    t_3 = sin(k) / l
    if (t <= (-9.5d+103)) then
        tmp = 2.0d0 / (t_2 * ((tan(k) * (t * sin(k))) * t_1))
    else if (t <= (-1.2d-40)) then
        tmp = (l * ((l / sin(k)) * (2.0d0 / (tan(k) * (t ** 3.0d0))))) / (2.0d0 + ((k / t) / (t / k)))
    else if (t <= 1.58d-58) then
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * t_3))
    else if (t <= 1.1d+93) then
        tmp = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * t_3)) / (2.0d0 + ((k * (k / t)) / t))
    else
        tmp = 2.0d0 / (t_2 * ((sin(k) * (t * t_1)) / (cos(k) / sin(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((t / l), 2.0);
	double t_2 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double t_3 = Math.sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / (t_2 * ((Math.tan(k) * (t * Math.sin(k))) * t_1));
	} else if (t <= -1.2e-40) {
		tmp = (l * ((l / Math.sin(k)) * (2.0 / (Math.tan(k) * Math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 1.58e-58) {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * t_3));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * t_3)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / (t_2 * ((Math.sin(k) * (t * t_1)) / (Math.cos(k) / Math.sin(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((t / l), 2.0)
	t_2 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	t_3 = math.sin(k) / l
	tmp = 0
	if t <= -9.5e+103:
		tmp = 2.0 / (t_2 * ((math.tan(k) * (t * math.sin(k))) * t_1))
	elif t <= -1.2e-40:
		tmp = (l * ((l / math.sin(k)) * (2.0 / (math.tan(k) * math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)))
	elif t <= 1.58e-58:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * t_3))
	elif t <= 1.1e+93:
		tmp = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * t_3)) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = 2.0 / (t_2 * ((math.sin(k) * (t * t_1)) / (math.cos(k) / math.sin(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / l) ^ 2.0
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	t_3 = Float64(sin(k) / l)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(tan(k) * Float64(t * sin(k))) * t_1)));
	elseif (t <= -1.2e-40)
		tmp = Float64(Float64(l * Float64(Float64(l / sin(k)) * Float64(2.0 / Float64(tan(k) * (t ^ 3.0))))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	elseif (t <= 1.58e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * t_3)));
	elseif (t <= 1.1e+93)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * t_3)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(sin(k) * Float64(t * t_1)) / Float64(cos(k) / sin(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (t / l) ^ 2.0;
	t_2 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	t_3 = sin(k) / l;
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = 2.0 / (t_2 * ((tan(k) * (t * sin(k))) * t_1));
	elseif (t <= -1.2e-40)
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * (t ^ 3.0))))) / (2.0 + ((k / t) / (t / k)));
	elseif (t <= 1.58e-58)
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * t_3));
	elseif (t <= 1.1e+93)
		tmp = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * t_3)) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = 2.0 / (t_2 * ((sin(k) * (t * t_1)) / (cos(k) / sin(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, -9.5e+103], N[(2.0 / N[(t$95$2 * N[(N[(N[Tan[k], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.2e-40], N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.58e-58], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e+93], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(N[Sin[k], $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.49999999999999922e103

   1. Initial program 53.7%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. unpow2 71.8%

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

      2. add-sqr-sqrt 52.4%

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

      3. times-frac 67.5%

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

   5. Applied egg-rr 67.5%

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

   7. Applied egg-rr 94.1%

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

      1. distribute-rgt1-in 94.1%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -9.49999999999999922e103 < t < -1.19999999999999996e-40

   1. Initial program 75.3%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

   3. Simplified 91.8%

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

      1. unpow2 91.8%

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

      2. clear-num 91.8%

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

      3. un-div-inv 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.1%

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

      3. associate-/r* 96.3%

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

      4. *-commutative 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. associate-/r/ 96.2%

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

      2. associate-/l* 85.7%

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

      3. associate-/r/ 96.2%

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

      4. associate-/l/ 96.4%

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

   9. Simplified 96.4%

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

   if -1.19999999999999996e-40 < t < 1.57999999999999997e-58

   1. Initial program 38.0%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.1%

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

   4. Simplified 72.1%

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

      1. associate-*l/ 75.5%

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

      2. unpow2 75.5%

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

      3. clear-num 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 75.5%

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

      7. clear-num 75.5%

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

      8. unpow2 75.5%

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

      9. associate-*r/ 75.5%

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

      10. tan-quot 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-un-lft-identity 75.4%

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

      4. associate-/r/ 75.4%

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

      5. /-rgt-identity 75.4%

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

      6. times-frac 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 86.8%

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

      3. associate-/l* 86.8%

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

      4. associate-/r/ 86.7%

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

   10. Simplified 86.7%

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

   if 1.57999999999999997e-58 < t < 1.10000000000000011e93

   1. Initial program 78.0%

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

      1. associate-/r* 78.6%

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

      2. sqr-neg 78.6%

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

   3. Simplified 87.8%

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

      1. frac-times 90.2%

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

      2. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. unpow2 90.2%

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

      2. associate-*r/ 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 1.10000000000000011e93 < t

   1. Initial program 61.1%

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

      1. unpow3 61.1%

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

      2. times-frac 80.6%

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

      3. pow1 80.6%

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

      4. pow1 80.6%

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

      5. pow-sqr 80.6%

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

      6. metadata-eval 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. unpow2 80.6%

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

      2. add-sqr-sqrt 48.1%

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

      3. times-frac 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. frac-times 48.1%

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

      2. unpow2 48.1%

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

      3. add-sqr-sqrt 80.6%

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

      4. tan-quot 80.6%

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

      5. associate-*r/ 80.6%

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

   7. Applied egg-rr 95.1%

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

      1. associate-/l* 95.0%

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

      2. associate-*l* 95.0%

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

   9. Simplified 95.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot \left(t \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}{\frac{\cos k}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}{\frac{\cos k}{\sin k}}}\\ \end{array} \]

Alternative 2: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{k}{t}}{\frac{t}{k}}\\ t_2 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + t_1}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot t_2}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot {\left(\frac{t}{\sqrt{\ell}}\right)}^{2}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ k t) (/ t k))) (t_2 (/ (sin k) l)))
   (if (<= t -9.5e+103)
     (/
      2.0
      (*
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
       (* (* (tan k) (* t (sin k))) (pow (/ t l) 2.0))))
     (if (<= t -1.7e-41)
       (/ (* l (* (/ l (sin k)) (/ 2.0 (* (tan k) (pow t 3.0))))) (+ 2.0 t_1))
       (if (<= t 7.8e-59)
         (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) t_2)))
         (if (<= t 4.4e+91)
           (/
            (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) t_2))
            (+ 2.0 (/ (* k (/ k t)) t)))
           (/
            2.0
            (*
             (* (tan k) (* (sin k) (* (/ t l) (pow (/ t (sqrt l)) 2.0))))
             (+ 1.0 (+ 1.0 t_1))))))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / t) / (t / k);
	double t_2 = sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * ((tan(k) * (t * sin(k))) * pow((t / l), 2.0)));
	} else if (t <= -1.7e-41) {
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * pow(t, 3.0))))) / (2.0 + t_1);
	} else if (t <= 7.8e-59) {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * t_2));
	} else if (t <= 4.4e+91) {
		tmp = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * pow((t / sqrt(l)), 2.0)))) * (1.0 + (1.0 + t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (k / t) / (t / k)
    t_2 = sin(k) / l
    if (t <= (-9.5d+103)) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * ((tan(k) * (t * sin(k))) * ((t / l) ** 2.0d0)))
    else if (t <= (-1.7d-41)) then
        tmp = (l * ((l / sin(k)) * (2.0d0 / (tan(k) * (t ** 3.0d0))))) / (2.0d0 + t_1)
    else if (t <= 7.8d-59) then
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * t_2))
    else if (t <= 4.4d+91) then
        tmp = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * t_2)) / (2.0d0 + ((k * (k / t)) / t))
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * ((t / sqrt(l)) ** 2.0d0)))) * (1.0d0 + (1.0d0 + t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / t) / (t / k);
	double t_2 = Math.sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * ((Math.tan(k) * (t * Math.sin(k))) * Math.pow((t / l), 2.0)));
	} else if (t <= -1.7e-41) {
		tmp = (l * ((l / Math.sin(k)) * (2.0 / (Math.tan(k) * Math.pow(t, 3.0))))) / (2.0 + t_1);
	} else if (t <= 7.8e-59) {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * t_2));
	} else if (t <= 4.4e+91) {
		tmp = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((t / l) * Math.pow((t / Math.sqrt(l)), 2.0)))) * (1.0 + (1.0 + t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / t) / (t / k)
	t_2 = math.sin(k) / l
	tmp = 0
	if t <= -9.5e+103:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * ((math.tan(k) * (t * math.sin(k))) * math.pow((t / l), 2.0)))
	elif t <= -1.7e-41:
		tmp = (l * ((l / math.sin(k)) * (2.0 / (math.tan(k) * math.pow(t, 3.0))))) / (2.0 + t_1)
	elif t <= 7.8e-59:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * t_2))
	elif t <= 4.4e+91:
		tmp = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * math.pow((t / math.sqrt(l)), 2.0)))) * (1.0 + (1.0 + t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / t) / Float64(t / k))
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(Float64(tan(k) * Float64(t * sin(k))) * (Float64(t / l) ^ 2.0))));
	elseif (t <= -1.7e-41)
		tmp = Float64(Float64(l * Float64(Float64(l / sin(k)) * Float64(2.0 / Float64(tan(k) * (t ^ 3.0))))) / Float64(2.0 + t_1));
	elseif (t <= 7.8e-59)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * t_2)));
	elseif (t <= 4.4e+91)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * t_2)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * (Float64(t / sqrt(l)) ^ 2.0)))) * Float64(1.0 + Float64(1.0 + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) / (t / k);
	t_2 = sin(k) / l;
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * ((tan(k) * (t * sin(k))) * ((t / l) ^ 2.0)));
	elseif (t <= -1.7e-41)
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * (t ^ 3.0))))) / (2.0 + t_1);
	elseif (t <= 7.8e-59)
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * t_2));
	elseif (t <= 4.4e+91)
		tmp = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * ((t / sqrt(l)) ^ 2.0)))) * (1.0 + (1.0 + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, -9.5e+103], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.7e-41], N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-59], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.4e+91], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Power[N[(t / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.49999999999999922e103

   1. Initial program 53.7%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. unpow2 71.8%

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

      2. add-sqr-sqrt 52.4%

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

      3. times-frac 67.5%

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

   5. Applied egg-rr 67.5%

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

   7. Applied egg-rr 94.1%

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

      1. distribute-rgt1-in 94.1%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -9.49999999999999922e103 < t < -1.6999999999999999e-41

   1. Initial program 75.3%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

   3. Simplified 91.8%

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

      1. unpow2 91.8%

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

      2. clear-num 91.8%

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

      3. un-div-inv 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.1%

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

      3. associate-/r* 96.3%

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

      4. *-commutative 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. associate-/r/ 96.2%

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

      2. associate-/l* 85.7%

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

      3. associate-/r/ 96.2%

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

      4. associate-/l/ 96.4%

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

   9. Simplified 96.4%

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

   if -1.6999999999999999e-41 < t < 7.80000000000000038e-59

   1. Initial program 38.0%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.1%

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

   4. Simplified 72.1%

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

      1. associate-*l/ 75.5%

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

      2. unpow2 75.5%

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

      3. clear-num 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 75.5%

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

      7. clear-num 75.5%

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

      8. unpow2 75.5%

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

      9. associate-*r/ 75.5%

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

      10. tan-quot 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-un-lft-identity 75.4%

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

      4. associate-/r/ 75.4%

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

      5. /-rgt-identity 75.4%

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

      6. times-frac 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 86.8%

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

      3. associate-/l* 86.8%

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

      4. associate-/r/ 86.7%

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

   10. Simplified 86.7%

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

   if 7.80000000000000038e-59 < t < 4.39999999999999999e91

   1. Initial program 78.0%

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

      1. associate-/r* 78.6%

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

      2. sqr-neg 78.6%

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

   3. Simplified 87.8%

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

      1. frac-times 90.2%

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

      2. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. unpow2 90.2%

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

      2. associate-*r/ 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 4.39999999999999999e91 < t

   1. Initial program 61.1%

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

      1. unpow3 61.1%

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

      2. times-frac 80.6%

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

      3. pow1 80.6%

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

      4. pow1 80.6%

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

      5. pow-sqr 80.6%

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

      6. metadata-eval 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. unpow2 60.9%

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

      2. clear-num 60.9%

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

      3. un-div-inv 60.9%

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

   5. Applied egg-rr 80.6%

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

      1. unpow2 80.6%

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

      2. add-sqr-sqrt 48.0%

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

      3. frac-times 57.5%

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

      4. associate-*l* 57.5%

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

      5. clear-num 57.5%

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

      6. associate-*l/ 57.5%

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

      7. *-un-lft-identity 57.5%

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

      8. frac-times 45.2%

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

      9. unpow2 45.2%

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

      10. *-commutative 45.2%

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

   7. Applied egg-rr 45.2%

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

      1. associate-/l/ 45.2%

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

      2. associate-/r* 36.0%

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

      3. associate-/l* 33.5%

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

      4. *-commutative 33.5%

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

      5. associate-*r/ 36.0%

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

      6. times-frac 48.0%

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

      7. unpow2 48.0%

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

      8. associate-*r/ 52.8%

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

      9. associate-*l/ 57.5%

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

      10. unpow2 57.5%

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

   9. Simplified 57.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot {\left(\frac{t}{\sqrt{\ell}}\right)}^{2}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) + 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot {\left(\frac{t}{\sqrt{\ell}}\right)}^{2}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \end{array} \]

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot t_1}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sin k) l)))
   (if (<= t -9.5e+103)
     (/
      2.0
      (*
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
       (* (* (tan k) (* t (sin k))) (pow (/ t l) 2.0))))
     (if (<= t -7.5e-37)
       (/
        (* l (* (/ l (sin k)) (/ 2.0 (* (tan k) (pow t 3.0)))))
        (+ 2.0 (/ (/ k t) (/ t k))))
       (if (<= t 1.15e-58)
         (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) t_1)))
         (if (<= t 1.1e+93)
           (/
            (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) t_1))
            (+ 2.0 (/ (* k (/ k t)) t)))
           (/
            2.0
            (*
             (* (tan k) (* (sin k) (* (/ t l) (* t (/ t l)))))
             (+ 1.0 (+ 1.0 (/ k (* t (/ t k)))))))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * ((tan(k) * (t * sin(k))) * pow((t / l), 2.0)));
	} else if (t <= -7.5e-37) {
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 1.15e-58) {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * t_1));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) / l
    if (t <= (-9.5d+103)) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * ((tan(k) * (t * sin(k))) * ((t / l) ** 2.0d0)))
    else if (t <= (-7.5d-37)) then
        tmp = (l * ((l / sin(k)) * (2.0d0 / (tan(k) * (t ** 3.0d0))))) / (2.0d0 + ((k / t) / (t / k)))
    else if (t <= 1.15d-58) then
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * t_1))
    else if (t <= 1.1d+93) then
        tmp = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * t_1)) / (2.0d0 + ((k * (k / t)) / t))
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0d0 + (1.0d0 + (k / (t * (t / k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * ((Math.tan(k) * (t * Math.sin(k))) * Math.pow((t / l), 2.0)));
	} else if (t <= -7.5e-37) {
		tmp = (l * ((l / Math.sin(k)) * (2.0 / (Math.tan(k) * Math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 1.15e-58) {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * t_1));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) / l
	tmp = 0
	if t <= -9.5e+103:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * ((math.tan(k) * (t * math.sin(k))) * math.pow((t / l), 2.0)))
	elif t <= -7.5e-37:
		tmp = (l * ((l / math.sin(k)) * (2.0 / (math.tan(k) * math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)))
	elif t <= 1.15e-58:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * t_1))
	elif t <= 1.1e+93:
		tmp = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) / l)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(Float64(tan(k) * Float64(t * sin(k))) * (Float64(t / l) ^ 2.0))));
	elseif (t <= -7.5e-37)
		tmp = Float64(Float64(l * Float64(Float64(l / sin(k)) * Float64(2.0 / Float64(tan(k) * (t ^ 3.0))))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	elseif (t <= 1.15e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * t_1)));
	elseif (t <= 1.1e+93)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * t_1)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + Float64(k / Float64(t * Float64(t / k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) / l;
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * ((tan(k) * (t * sin(k))) * ((t / l) ^ 2.0)));
	elseif (t <= -7.5e-37)
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * (t ^ 3.0))))) / (2.0 + ((k / t) / (t / k)));
	elseif (t <= 1.15e-58)
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * t_1));
	elseif (t <= 1.1e+93)
		tmp = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, -9.5e+103], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-37], N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-58], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e+93], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.49999999999999922e103

   1. Initial program 53.7%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. unpow2 71.8%

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

      2. add-sqr-sqrt 52.4%

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

      3. times-frac 67.5%

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

   5. Applied egg-rr 67.5%

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

   7. Applied egg-rr 94.1%

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

      1. distribute-rgt1-in 94.1%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -9.49999999999999922e103 < t < -7.5000000000000004e-37

   1. Initial program 75.3%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

   3. Simplified 91.8%

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

      1. unpow2 91.8%

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

      2. clear-num 91.8%

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

      3. un-div-inv 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.1%

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

      3. associate-/r* 96.3%

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

      4. *-commutative 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. associate-/r/ 96.2%

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

      2. associate-/l* 85.7%

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

      3. associate-/r/ 96.2%

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

      4. associate-/l/ 96.4%

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

   9. Simplified 96.4%

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

   if -7.5000000000000004e-37 < t < 1.1499999999999999e-58

   1. Initial program 38.0%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.1%

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

   4. Simplified 72.1%

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

      1. associate-*l/ 75.5%

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

      2. unpow2 75.5%

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

      3. clear-num 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 75.5%

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

      7. clear-num 75.5%

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

      8. unpow2 75.5%

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

      9. associate-*r/ 75.5%

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

      10. tan-quot 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-un-lft-identity 75.4%

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

      4. associate-/r/ 75.4%

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

      5. /-rgt-identity 75.4%

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

      6. times-frac 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 86.8%

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

      3. associate-/l* 86.8%

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

      4. associate-/r/ 86.7%

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

   10. Simplified 86.7%

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

   if 1.1499999999999999e-58 < t < 1.10000000000000011e93

   1. Initial program 78.0%

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

      1. associate-/r* 78.6%

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

      2. sqr-neg 78.6%

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

   3. Simplified 87.8%

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

      1. frac-times 90.2%

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

      2. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. unpow2 90.2%

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

      2. associate-*r/ 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 1.10000000000000011e93 < t

   1. Initial program 61.1%

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

      1. unpow3 61.1%

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

      2. times-frac 80.6%

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

      3. pow1 80.6%

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

      4. pow1 80.6%

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

      5. pow-sqr 80.6%

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

      6. metadata-eval 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. div-inv 58.9%

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

      2. unpow2 58.9%

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

      3. associate-*l* 64.8%

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

      4. div-inv 64.8%

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

   5. Applied egg-rr 94.8%

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

      1. unpow2 94.8%

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

      2. clear-num 94.8%

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

      3. frac-times 94.9%

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

      4. *-un-lft-identity 94.9%

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

   7. Applied egg-rr 94.9%

      \[\leadsto \frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{\frac{t}{k} \cdot t}}\right) + 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ t_3 := \frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot t_1}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sin k) l))
        (t_2
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (* (/ t l) (* t (/ t l)))))
           (+ 1.0 (+ 1.0 (/ k (* t (/ t k))))))))
        (t_3
         (/
          (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) t_1))
          (+ 2.0 (/ (* k (/ k t)) t)))))
   (if (<= t -9.5e+103)
     t_2
     (if (<= t -1.05e-58)
       t_3
       (if (<= t -2.5e-303)
         (/ 2.0 (* (* (/ k l) (/ k l)) (/ (* t (pow (sin k) 2.0)) (cos k))))
         (if (<= t 2.95e-58)
           (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) t_1)))
           (if (<= t 8.8e+92) t_3 t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) / l;
	double t_2 = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	double t_3 = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	double tmp;
	if (t <= -9.5e+103) {
		tmp = t_2;
	} else if (t <= -1.05e-58) {
		tmp = t_3;
	} else if (t <= -2.5e-303) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t * pow(sin(k), 2.0)) / cos(k)));
	} else if (t <= 2.95e-58) {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * t_1));
	} else if (t <= 8.8e+92) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = sin(k) / l
    t_2 = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0d0 + (1.0d0 + (k / (t * (t / k))))))
    t_3 = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * t_1)) / (2.0d0 + ((k * (k / t)) / t))
    if (t <= (-9.5d+103)) then
        tmp = t_2
    else if (t <= (-1.05d-58)) then
        tmp = t_3
    else if (t <= (-2.5d-303)) then
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((t * (sin(k) ** 2.0d0)) / cos(k)))
    else if (t <= 2.95d-58) then
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * t_1))
    else if (t <= 8.8d+92) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) / l;
	double t_2 = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	double t_3 = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	double tmp;
	if (t <= -9.5e+103) {
		tmp = t_2;
	} else if (t <= -1.05e-58) {
		tmp = t_3;
	} else if (t <= -2.5e-303) {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else if (t <= 2.95e-58) {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * t_1));
	} else if (t <= 8.8e+92) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) / l
	t_2 = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))))
	t_3 = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t))
	tmp = 0
	if t <= -9.5e+103:
		tmp = t_2
	elif t <= -1.05e-58:
		tmp = t_3
	elif t <= -2.5e-303:
		tmp = 2.0 / (((k / l) * (k / l)) * ((t * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	elif t <= 2.95e-58:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * t_1))
	elif t <= 8.8e+92:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) / l)
	t_2 = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + Float64(k / Float64(t * Float64(t / k)))))))
	t_3 = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * t_1)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)))
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = t_2;
	elseif (t <= -1.05e-58)
		tmp = t_3;
	elseif (t <= -2.5e-303)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(t * (sin(k) ^ 2.0)) / cos(k))));
	elseif (t <= 2.95e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * t_1)));
	elseif (t <= 8.8e+92)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) / l;
	t_2 = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	t_3 = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * t_1)) / (2.0 + ((k * (k / t)) / t));
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = t_2;
	elseif (t <= -1.05e-58)
		tmp = t_3;
	elseif (t <= -2.5e-303)
		tmp = 2.0 / (((k / l) * (k / l)) * ((t * (sin(k) ^ 2.0)) / cos(k)));
	elseif (t <= 2.95e-58)
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * t_1));
	elseif (t <= 8.8e+92)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e+103], t$95$2, If[LessEqual[t, -1.05e-58], t$95$3, If[LessEqual[t, -2.5e-303], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.95e-58], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e+92], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.49999999999999922e103 or 8.79999999999999969e92 < t

   1. Initial program 57.5%

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

      1. unpow3 57.5%

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

      2. times-frac 76.3%

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

      3. pow1 76.3%

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

      4. pow1 76.3%

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

      5. pow-sqr 76.3%

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

      6. metadata-eval 76.3%

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

   3. Applied egg-rr 76.3%

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

      1. div-inv 61.3%

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

      2. unpow2 61.3%

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

      3. associate-*l* 69.9%

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

      4. div-inv 69.9%

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

   5. Applied egg-rr 93.3%

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

      1. unpow2 93.3%

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

      2. clear-num 93.3%

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

      3. frac-times 93.3%

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

      4. *-un-lft-identity 93.3%

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

   7. Applied egg-rr 93.3%

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

   if -9.49999999999999922e103 < t < -1.04999999999999994e-58 or 2.95e-58 < t < 8.79999999999999969e92

   1. Initial program 76.2%

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

      1. associate-/r* 76.5%

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

      2. sqr-neg 76.5%

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

   3. Simplified 88.4%

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

      1. frac-times 92.9%

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

      2. *-commutative 92.9%

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

   5. Applied egg-rr 92.9%

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

      1. unpow2 92.9%

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

      2. associate-*r/ 91.5%

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

   7. Applied egg-rr 91.5%

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

   if -1.04999999999999994e-58 < t < -2.4999999999999999e-303

   1. Initial program 52.1%

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

   2. Taylor expanded in t around 0 75.2%

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

      1. times-frac 75.2%

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

   4. Simplified 75.2%

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

      1. unpow2 75.2%

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

      2. unpow2 75.2%

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

      3. times-frac 90.0%

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

   6. Applied egg-rr 90.0%

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

   if -2.4999999999999999e-303 < t < 2.95e-58

   1. Initial program 26.5%

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

   2. Taylor expanded in t around 0 76.3%

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

      1. times-frac 70.4%

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

   4. Simplified 70.4%

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

      1. associate-*l/ 76.4%

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

      2. unpow2 76.4%

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

      3. clear-num 76.4%

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

      4. unpow2 76.4%

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

      5. associate-/l* 76.4%

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

      6. associate-*r/ 76.4%

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

      7. clear-num 76.4%

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

      8. unpow2 76.4%

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

      9. associate-*r/ 76.4%

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

      10. tan-quot 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-/l* 76.3%

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

      2. unpow2 76.3%

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

      3. *-un-lft-identity 76.3%

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

      4. associate-/r/ 76.3%

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

      5. /-rgt-identity 76.3%

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

      6. times-frac 87.5%

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

      7. *-commutative 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. associate-/l* 84.9%

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

      2. associate-/r/ 88.8%

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

      3. associate-/l* 88.8%

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

      4. associate-/r/ 88.8%

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

   10. Simplified 88.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ t_2 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot t_2}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (* (/ t l) (* t (/ t l)))))
           (+ 1.0 (+ 1.0 (/ k (* t (/ t k))))))))
        (t_2 (/ (sin k) l)))
   (if (<= t -9.5e+103)
     t_1
     (if (<= t -5e-40)
       (/
        (* l (* (/ l (sin k)) (/ 2.0 (* (tan k) (pow t 3.0)))))
        (+ 2.0 (/ (/ k t) (/ t k))))
       (if (<= t 2.5e-58)
         (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) t_2)))
         (if (<= t 1.1e+93)
           (/
            (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) t_2))
            (+ 2.0 (/ (* k (/ k t)) t)))
           t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	double t_2 = sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = t_1;
	} else if (t <= -5e-40) {
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 2.5e-58) {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * t_2));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0d0 + (1.0d0 + (k / (t * (t / k))))))
    t_2 = sin(k) / l
    if (t <= (-9.5d+103)) then
        tmp = t_1
    else if (t <= (-5d-40)) then
        tmp = (l * ((l / sin(k)) * (2.0d0 / (tan(k) * (t ** 3.0d0))))) / (2.0d0 + ((k / t) / (t / k)))
    else if (t <= 2.5d-58) then
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * t_2))
    else if (t <= 1.1d+93) then
        tmp = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * t_2)) / (2.0d0 + ((k * (k / t)) / t))
    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.tan(k) * (Math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	double t_2 = Math.sin(k) / l;
	double tmp;
	if (t <= -9.5e+103) {
		tmp = t_1;
	} else if (t <= -5e-40) {
		tmp = (l * ((l / Math.sin(k)) * (2.0 / (Math.tan(k) * Math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)));
	} else if (t <= 2.5e-58) {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * t_2));
	} else if (t <= 1.1e+93) {
		tmp = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))))
	t_2 = math.sin(k) / l
	tmp = 0
	if t <= -9.5e+103:
		tmp = t_1
	elif t <= -5e-40:
		tmp = (l * ((l / math.sin(k)) * (2.0 / (math.tan(k) * math.pow(t, 3.0))))) / (2.0 + ((k / t) / (t / k)))
	elif t <= 2.5e-58:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * t_2))
	elif t <= 1.1e+93:
		tmp = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + Float64(k / Float64(t * Float64(t / k)))))))
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = t_1;
	elseif (t <= -5e-40)
		tmp = Float64(Float64(l * Float64(Float64(l / sin(k)) * Float64(2.0 / Float64(tan(k) * (t ^ 3.0))))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	elseif (t <= 2.5e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * t_2)));
	elseif (t <= 1.1e+93)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * t_2)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	t_2 = sin(k) / l;
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = t_1;
	elseif (t <= -5e-40)
		tmp = (l * ((l / sin(k)) * (2.0 / (tan(k) * (t ^ 3.0))))) / (2.0 + ((k / t) / (t / k)));
	elseif (t <= 2.5e-58)
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * t_2));
	elseif (t <= 1.1e+93)
		tmp = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * t_2)) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, -9.5e+103], t$95$1, If[LessEqual[t, -5e-40], N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-58], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e+93], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.49999999999999922e103 or 1.10000000000000011e93 < t

   1. Initial program 57.5%

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

      1. unpow3 57.5%

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

      2. times-frac 76.3%

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

      3. pow1 76.3%

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

      4. pow1 76.3%

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

      5. pow-sqr 76.3%

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

      6. metadata-eval 76.3%

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

   3. Applied egg-rr 76.3%

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

      1. div-inv 61.3%

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

      2. unpow2 61.3%

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

      3. associate-*l* 69.9%

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

      4. div-inv 69.9%

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

   5. Applied egg-rr 93.3%

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

      1. unpow2 93.3%

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

      2. clear-num 93.3%

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

      3. frac-times 93.3%

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

      4. *-un-lft-identity 93.3%

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

   7. Applied egg-rr 93.3%

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

   if -9.49999999999999922e103 < t < -4.99999999999999965e-40

   1. Initial program 75.3%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

   3. Simplified 91.8%

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

      1. unpow2 91.8%

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

      2. clear-num 91.8%

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

      3. un-div-inv 91.8%

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

   5. Applied egg-rr 91.8%

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

      1. associate-*r/ 96.2%

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

      2. div-inv 96.1%

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

      3. associate-/r* 96.3%

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

      4. *-commutative 96.3%

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

   7. Applied egg-rr 96.3%

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

      1. associate-/r/ 96.2%

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

      2. associate-/l* 85.7%

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

      3. associate-/r/ 96.2%

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

      4. associate-/l/ 96.4%

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

   9. Simplified 96.4%

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

   if -4.99999999999999965e-40 < t < 2.49999999999999989e-58

   1. Initial program 38.0%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.1%

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

   4. Simplified 72.1%

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

      1. associate-*l/ 75.5%

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

      2. unpow2 75.5%

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

      3. clear-num 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 75.5%

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

      7. clear-num 75.5%

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

      8. unpow2 75.5%

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

      9. associate-*r/ 75.5%

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

      10. tan-quot 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-un-lft-identity 75.4%

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

      4. associate-/r/ 75.4%

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

      5. /-rgt-identity 75.4%

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

      6. times-frac 84.4%

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

      7. *-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 86.8%

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

      3. associate-/l* 86.8%

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

      4. associate-/r/ 86.7%

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

   10. Simplified 86.7%

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

   if 2.49999999999999989e-58 < t < 1.10000000000000011e93

   1. Initial program 78.0%

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

      1. associate-/r* 78.6%

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

      2. sqr-neg 78.6%

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

   3. Simplified 87.8%

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

      1. frac-times 90.2%

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

      2. *-commutative 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. unpow2 90.2%

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

      2. associate-*r/ 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{\tan k \cdot {t}^{3}}\right)}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \end{array} \]

Alternative 6: 84.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-39} \lor \neg \left(t \leq 6.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.2e-39) (not (<= t 6.5e-17)))
   (/
    2.0
    (*
     (* (tan k) (* (sin k) (* (/ t l) (* t (/ t l)))))
     (+ 1.0 (+ 1.0 (/ k (* t (/ t k)))))))
   (/ 2.0 (* (* (/ t l) (pow k 2.0)) (* (tan k) (/ (sin k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-39) || !(t <= 6.5e-17)) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	} else {
		tmp = 2.0 / (((t / l) * pow(k, 2.0)) * (tan(k) * (sin(k) / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-4.2d-39)) .or. (.not. (t <= 6.5d-17))) then
        tmp = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0d0 + (1.0d0 + (k / (t * (t / k))))))
    else
        tmp = 2.0d0 / (((t / l) * (k ** 2.0d0)) * (tan(k) * (sin(k) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-39) || !(t <= 6.5e-17)) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	} else {
		tmp = 2.0 / (((t / l) * Math.pow(k, 2.0)) * (Math.tan(k) * (Math.sin(k) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -4.2e-39) or not (t <= 6.5e-17):
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))))
	else:
		tmp = 2.0 / (((t / l) * math.pow(k, 2.0)) * (math.tan(k) * (math.sin(k) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.2e-39) || !(t <= 6.5e-17))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + Float64(k / Float64(t * Float64(t / k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * (k ^ 2.0)) * Float64(tan(k) * Float64(sin(k) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -4.2e-39) || ~((t <= 6.5e-17)))
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	else
		tmp = 2.0 / (((t / l) * (k ^ 2.0)) * (tan(k) * (sin(k) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -4.2e-39], N[Not[LessEqual[t, 6.5e-17]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.19999999999999987e-39 or 6.4999999999999996e-17 < t

   1. Initial program 65.9%

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

      1. unpow3 65.9%

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

      2. times-frac 77.2%

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

      3. pow1 77.2%

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

      4. pow1 77.2%

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

      5. pow-sqr 77.2%

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

      6. metadata-eval 77.2%

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

   3. Applied egg-rr 77.2%

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

      1. div-inv 60.8%

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

      2. unpow2 60.8%

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

      3. associate-*l* 65.6%

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

      4. div-inv 65.6%

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

   5. Applied egg-rr 86.8%

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

      1. unpow2 86.8%

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

      2. clear-num 86.9%

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

      3. frac-times 86.9%

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

      4. *-un-lft-identity 86.9%

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

   7. Applied egg-rr 86.9%

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

   if -4.19999999999999987e-39 < t < 6.4999999999999996e-17

   1. Initial program 40.6%

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

   2. Taylor expanded in t around 0 75.5%

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

      1. times-frac 72.4%

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

   4. Simplified 72.4%

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

      1. associate-*l/ 75.5%

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

      2. unpow2 75.5%

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

      3. clear-num 75.5%

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

      4. unpow2 75.5%

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

      5. associate-/l* 75.5%

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

      6. associate-*r/ 75.5%

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

      7. clear-num 75.5%

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

      8. unpow2 75.5%

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

      9. associate-*r/ 75.5%

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

      10. tan-quot 75.5%

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

   6. Applied egg-rr 75.5%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-un-lft-identity 75.4%

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

      4. associate-/r/ 75.4%

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

      5. /-rgt-identity 75.4%

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

      6. times-frac 84.6%

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

      7. *-commutative 84.6%

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

   8. Applied egg-rr 84.6%

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

      1. associate-/l* 84.9%

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

      2. associate-/r/ 86.9%

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

      3. associate-/l* 86.9%

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

      4. associate-/r/ 86.8%

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

   10. Simplified 86.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-39} \lor \neg \left(t \leq 6.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)}\\ \end{array} \]

Alternative 7: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-124} \lor \neg \left(t \leq 8 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.8e-124) (not (<= t 8e-94)))
   (/
    2.0
    (*
     (* (tan k) (* (sin k) (* (/ t l) (* t (/ t l)))))
     (+ 1.0 (+ 1.0 (/ k (* t (/ t k)))))))
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.8e-124) || !(t <= 8e-94)) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-6.8d-124)) .or. (.not. (t <= 8d-94))) then
        tmp = 2.0d0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0d0 + (1.0d0 + (k / (t * (t / k))))))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.8e-124) || !(t <= 8e-94)) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.8e-124) or not (t <= 8e-94):
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.8e-124) || !(t <= 8e-94))
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64(t / l) * Float64(t * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + Float64(k / Float64(t * Float64(t / k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.8e-124) || ~((t <= 8e-94)))
		tmp = 2.0 / ((tan(k) * (sin(k) * ((t / l) * (t * (t / l))))) * (1.0 + (1.0 + (k / (t * (t / k))))));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.8e-124], N[Not[LessEqual[t, 8e-94]], $MachinePrecision]], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.8000000000000001e-124 or 7.9999999999999996e-94 < t

   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. unpow3 65.6%

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

      2. times-frac 75.5%

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

      3. pow1 75.5%

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

      4. pow1 75.5%

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

      5. pow-sqr 75.5%

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

      6. metadata-eval 75.5%

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

   3. Applied egg-rr 75.5%

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

      1. div-inv 58.0%

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

      2. unpow2 58.0%

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

      3. associate-*l* 62.0%

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

      4. div-inv 62.0%

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

   5. Applied egg-rr 83.4%

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

      1. unpow2 83.4%

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

      2. clear-num 83.4%

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

      3. frac-times 83.4%

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

      4. *-un-lft-identity 83.4%

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

   7. Applied egg-rr 83.4%

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

   if -6.8000000000000001e-124 < t < 7.9999999999999996e-94

   1. Initial program 32.6%

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

   2. Taylor expanded in t around 0 74.8%

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

      1. times-frac 71.7%

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

   4. Simplified 71.7%

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

   5. Taylor expanded in k around 0 62.0%

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

      1. unpow2 62.0%

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

      2. times-frac 69.8%

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

   7. Applied egg-rr 69.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-124} \lor \neg \left(t \leq 8 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 8: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+104}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-43} \lor \neg \left(t \leq 5.6 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{2}{k \cdot {t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.85e+104)
   (* 2.0 (/ 1.0 (* t (* (pow (/ t l) 2.0) (* 2.0 (pow k 2.0))))))
   (if (or (<= t -3e-43) (not (<= t 5.6e-83)))
     (/
      (* (/ 2.0 (* k (pow t 3.0))) (/ l (/ (sin k) l)))
      (+ 2.0 (/ (/ k t) (/ t k))))
     (/ 2.0 (* (/ t l) (/ (pow k 4.0) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.85e+104) {
		tmp = 2.0 * (1.0 / (t * (pow((t / l), 2.0) * (2.0 * pow(k, 2.0)))));
	} else if ((t <= -3e-43) || !(t <= 5.6e-83)) {
		tmp = ((2.0 / (k * pow(t, 3.0))) * (l / (sin(k) / l))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.85d+104)) then
        tmp = 2.0d0 * (1.0d0 / (t * (((t / l) ** 2.0d0) * (2.0d0 * (k ** 2.0d0)))))
    else if ((t <= (-3d-43)) .or. (.not. (t <= 5.6d-83))) then
        tmp = ((2.0d0 / (k * (t ** 3.0d0))) * (l / (sin(k) / l))) / (2.0d0 + ((k / t) / (t / k)))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.85e+104) {
		tmp = 2.0 * (1.0 / (t * (Math.pow((t / l), 2.0) * (2.0 * Math.pow(k, 2.0)))));
	} else if ((t <= -3e-43) || !(t <= 5.6e-83)) {
		tmp = ((2.0 / (k * Math.pow(t, 3.0))) * (l / (Math.sin(k) / l))) / (2.0 + ((k / t) / (t / k)));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.85e+104:
		tmp = 2.0 * (1.0 / (t * (math.pow((t / l), 2.0) * (2.0 * math.pow(k, 2.0)))))
	elif (t <= -3e-43) or not (t <= 5.6e-83):
		tmp = ((2.0 / (k * math.pow(t, 3.0))) * (l / (math.sin(k) / l))) / (2.0 + ((k / t) / (t / k)))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.85e+104)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t * Float64((Float64(t / l) ^ 2.0) * Float64(2.0 * (k ^ 2.0))))));
	elseif ((t <= -3e-43) || !(t <= 5.6e-83))
		tmp = Float64(Float64(Float64(2.0 / Float64(k * (t ^ 3.0))) * Float64(l / Float64(sin(k) / l))) / Float64(2.0 + Float64(Float64(k / t) / Float64(t / k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.85e+104)
		tmp = 2.0 * (1.0 / (t * (((t / l) ^ 2.0) * (2.0 * (k ^ 2.0)))));
	elseif ((t <= -3e-43) || ~((t <= 5.6e-83)))
		tmp = ((2.0 / (k * (t ^ 3.0))) * (l / (sin(k) / l))) / (2.0 + ((k / t) / (t / k)));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.85e+104], N[(2.0 * N[(1.0 / N[(t * N[(N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -3e-43], N[Not[LessEqual[t, 5.6e-83]], $MachinePrecision]], N[(N[(N[(2.0 / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8499999999999999e104

   1. Initial program 53.7%

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

      1. associate-*l* 48.2%

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

      2. associate-*l* 48.2%

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

      3. +-commutative 48.2%

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

      4. unpow2 48.2%

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

      5. times-frac 37.3%

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

      6. sqr-neg 37.3%

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

      7. times-frac 48.2%

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

      8. unpow2 48.2%

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

      9. +-commutative 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around 0 48.2%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   6. Applied egg-rr 63.8%

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

      1. div-inv 63.8%

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

      2. *-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l* 75.2%

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

      6. associate-*l* 77.6%

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

      7. pow1 77.6%

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

      8. pow1 77.6%

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

      9. pow-sqr 77.6%

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

      10. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

   if -1.8499999999999999e104 < t < -3.00000000000000003e-43 or 5.6000000000000002e-83 < t

   1. Initial program 68.3%

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

      1. associate-/r* 68.4%

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

      2. sqr-neg 68.4%

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

   3. Simplified 75.6%

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

      1. unpow2 75.6%

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

      2. clear-num 75.6%

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

      3. un-div-inv 75.6%

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

   5. Applied egg-rr 75.6%

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

   6. Taylor expanded in k around 0 65.9%

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

      1. *-commutative 65.9%

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

   8. Simplified 65.9%

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

   if -3.00000000000000003e-43 < t < 5.6000000000000002e-83

   1. Initial program 39.1%

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

   2. Taylor expanded in t around 0 76.8%

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

      1. times-frac 74.1%

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

   4. Simplified 74.1%

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

   5. Taylor expanded in k around 0 61.5%

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

      1. unpow2 61.5%

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

      2. times-frac 69.0%

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

   7. Applied egg-rr 69.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+104}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-43} \lor \neg \left(t \leq 5.6 \cdot 10^{-83}\right):\\ \;\;\;\;\frac{\frac{2}{k \cdot {t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 9: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-58} \lor \neg \left(t \leq 2.8 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -9.5e+103)
   (* 2.0 (/ 1.0 (* t (* (pow (/ t l) 2.0) (* 2.0 (pow k 2.0))))))
   (if (or (<= t -2.8e-58) (not (<= t 2.8e-58)))
     (/
      (/ (* l (/ 2.0 (tan k))) (* (pow t 3.0) (/ k l)))
      (+ 2.0 (/ (* k (/ k t)) t)))
     (/ 2.0 (* (/ t l) (/ (pow k 4.0) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 * (1.0 / (t * (pow((t / l), 2.0) * (2.0 * pow(k, 2.0)))));
	} else if ((t <= -2.8e-58) || !(t <= 2.8e-58)) {
		tmp = ((l * (2.0 / tan(k))) / (pow(t, 3.0) * (k / l))) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-9.5d+103)) then
        tmp = 2.0d0 * (1.0d0 / (t * (((t / l) ** 2.0d0) * (2.0d0 * (k ** 2.0d0)))))
    else if ((t <= (-2.8d-58)) .or. (.not. (t <= 2.8d-58))) then
        tmp = ((l * (2.0d0 / tan(k))) / ((t ** 3.0d0) * (k / l))) / (2.0d0 + ((k * (k / t)) / t))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 * (1.0 / (t * (Math.pow((t / l), 2.0) * (2.0 * Math.pow(k, 2.0)))));
	} else if ((t <= -2.8e-58) || !(t <= 2.8e-58)) {
		tmp = ((l * (2.0 / Math.tan(k))) / (Math.pow(t, 3.0) * (k / l))) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -9.5e+103:
		tmp = 2.0 * (1.0 / (t * (math.pow((t / l), 2.0) * (2.0 * math.pow(k, 2.0)))))
	elif (t <= -2.8e-58) or not (t <= 2.8e-58):
		tmp = ((l * (2.0 / math.tan(k))) / (math.pow(t, 3.0) * (k / l))) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t * Float64((Float64(t / l) ^ 2.0) * Float64(2.0 * (k ^ 2.0))))));
	elseif ((t <= -2.8e-58) || !(t <= 2.8e-58))
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64((t ^ 3.0) * Float64(k / l))) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = 2.0 * (1.0 / (t * (((t / l) ^ 2.0) * (2.0 * (k ^ 2.0)))));
	elseif ((t <= -2.8e-58) || ~((t <= 2.8e-58)))
		tmp = ((l * (2.0 / tan(k))) / ((t ^ 3.0) * (k / l))) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -9.5e+103], N[(2.0 * N[(1.0 / N[(t * N[(N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.8e-58], N[Not[LessEqual[t, 2.8e-58]], $MachinePrecision]], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.49999999999999922e103

   1. Initial program 53.7%

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

      1. associate-*l* 48.2%

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

      2. associate-*l* 48.2%

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

      3. +-commutative 48.2%

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

      4. unpow2 48.2%

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

      5. times-frac 37.3%

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

      6. sqr-neg 37.3%

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

      7. times-frac 48.2%

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

      8. unpow2 48.2%

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

      9. +-commutative 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around 0 48.2%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   6. Applied egg-rr 63.8%

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

      1. div-inv 63.8%

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

      2. *-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l* 75.2%

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

      6. associate-*l* 77.6%

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

      7. pow1 77.6%

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

      8. pow1 77.6%

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

      9. pow-sqr 77.6%

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

      10. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

   if -9.49999999999999922e103 < t < -2.8000000000000001e-58 or 2.8000000000000001e-58 < t

   1. Initial program 71.3%

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

      1. associate-/r* 71.5%

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

      2. sqr-neg 71.5%

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

   3. Simplified 79.0%

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

      1. frac-times 82.1%

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

      2. *-commutative 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*r/ 81.2%

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

   7. Applied egg-rr 81.2%

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

   8. Taylor expanded in k around 0 71.2%

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

      1. associate-/l* 66.9%

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

      2. associate-/r/ 71.2%

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

   10. Simplified 71.2%

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

   if -2.8000000000000001e-58 < t < 2.8000000000000001e-58

   1. Initial program 37.5%

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

   2. Taylor expanded in t around 0 75.2%

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

      1. times-frac 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

      2. times-frac 66.9%

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

   7. Applied egg-rr 66.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-58} \lor \neg \left(t \leq 2.8 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{{t}^{3} \cdot \frac{k}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 10: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-58} \lor \neg \left(t \leq 7.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{\frac{k \cdot {t}^{3}}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -9.5e+103)
   (* 2.0 (/ 1.0 (* t (* (pow (/ t l) 2.0) (* 2.0 (pow k 2.0))))))
   (if (or (<= t -2.8e-58) (not (<= t 7.7e-59)))
     (/
      (/ (* l (/ 2.0 (tan k))) (/ (* k (pow t 3.0)) l))
      (+ 2.0 (/ (* k (/ k t)) t)))
     (/ 2.0 (* (/ t l) (/ (pow k 4.0) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 * (1.0 / (t * (pow((t / l), 2.0) * (2.0 * pow(k, 2.0)))));
	} else if ((t <= -2.8e-58) || !(t <= 7.7e-59)) {
		tmp = ((l * (2.0 / tan(k))) / ((k * pow(t, 3.0)) / l)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-9.5d+103)) then
        tmp = 2.0d0 * (1.0d0 / (t * (((t / l) ** 2.0d0) * (2.0d0 * (k ** 2.0d0)))))
    else if ((t <= (-2.8d-58)) .or. (.not. (t <= 7.7d-59))) then
        tmp = ((l * (2.0d0 / tan(k))) / ((k * (t ** 3.0d0)) / l)) / (2.0d0 + ((k * (k / t)) / t))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.5e+103) {
		tmp = 2.0 * (1.0 / (t * (Math.pow((t / l), 2.0) * (2.0 * Math.pow(k, 2.0)))));
	} else if ((t <= -2.8e-58) || !(t <= 7.7e-59)) {
		tmp = ((l * (2.0 / Math.tan(k))) / ((k * Math.pow(t, 3.0)) / l)) / (2.0 + ((k * (k / t)) / t));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -9.5e+103:
		tmp = 2.0 * (1.0 / (t * (math.pow((t / l), 2.0) * (2.0 * math.pow(k, 2.0)))))
	elif (t <= -2.8e-58) or not (t <= 7.7e-59):
		tmp = ((l * (2.0 / math.tan(k))) / ((k * math.pow(t, 3.0)) / l)) / (2.0 + ((k * (k / t)) / t))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -9.5e+103)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t * Float64((Float64(t / l) ^ 2.0) * Float64(2.0 * (k ^ 2.0))))));
	elseif ((t <= -2.8e-58) || !(t <= 7.7e-59))
		tmp = Float64(Float64(Float64(l * Float64(2.0 / tan(k))) / Float64(Float64(k * (t ^ 3.0)) / l)) / Float64(2.0 + Float64(Float64(k * Float64(k / t)) / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -9.5e+103)
		tmp = 2.0 * (1.0 / (t * (((t / l) ^ 2.0) * (2.0 * (k ^ 2.0)))));
	elseif ((t <= -2.8e-58) || ~((t <= 7.7e-59)))
		tmp = ((l * (2.0 / tan(k))) / ((k * (t ^ 3.0)) / l)) / (2.0 + ((k * (k / t)) / t));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -9.5e+103], N[(2.0 * N[(1.0 / N[(t * N[(N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.8e-58], N[Not[LessEqual[t, 7.7e-59]], $MachinePrecision]], N[(N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k * N[(k / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.49999999999999922e103

   1. Initial program 53.7%

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

      1. associate-*l* 48.2%

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

      2. associate-*l* 48.2%

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

      3. +-commutative 48.2%

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

      4. unpow2 48.2%

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

      5. times-frac 37.3%

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

      6. sqr-neg 37.3%

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

      7. times-frac 48.2%

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

      8. unpow2 48.2%

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

      9. +-commutative 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around 0 48.2%

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

      1. unpow3 53.7%

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

      2. times-frac 71.8%

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

      3. pow1 71.8%

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

      4. pow1 71.8%

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

      5. pow-sqr 71.8%

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

      6. metadata-eval 71.8%

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

   6. Applied egg-rr 63.8%

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

      1. div-inv 63.8%

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

      2. *-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. associate-*r/ 75.2%

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

      5. associate-*l* 75.2%

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

      6. associate-*l* 77.6%

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

      7. pow1 77.6%

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

      8. pow1 77.6%

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

      9. pow-sqr 77.6%

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

      10. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

   if -9.49999999999999922e103 < t < -2.8000000000000001e-58 or 7.7e-59 < t

   1. Initial program 71.3%

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

      1. associate-/r* 71.5%

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

      2. sqr-neg 71.5%

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

   3. Simplified 79.0%

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

      1. frac-times 82.1%

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

      2. *-commutative 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*r/ 81.2%

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

   7. Applied egg-rr 81.2%

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

   8. Taylor expanded in k around 0 71.2%

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

   if -2.8000000000000001e-58 < t < 7.7e-59

   1. Initial program 37.5%

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

   2. Taylor expanded in t around 0 75.2%

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

      1. times-frac 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

      2. times-frac 66.9%

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

   7. Applied egg-rr 66.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-58} \lor \neg \left(t \leq 7.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k}}{\frac{k \cdot {t}^{3}}{\ell}}}{2 + \frac{k \cdot \frac{k}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 11: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot {k}^{2}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow k 2.0))))
   (if (<= t -9.2e-33)
     (* 2.0 (/ 1.0 (* t (* (pow (/ t l) 2.0) t_1))))
     (if (<= t 1.9e-43)
       (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))
       (/ 2.0 (* (* (/ t l) (* t (/ t l))) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 * pow(k, 2.0);
	double tmp;
	if (t <= -9.2e-33) {
		tmp = 2.0 * (1.0 / (t * (pow((t / l), 2.0) * t_1)));
	} else if (t <= 1.9e-43) {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	} else {
		tmp = 2.0 / (((t / l) * (t * (t / l))) * 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 * (k ** 2.0d0)
    if (t <= (-9.2d-33)) then
        tmp = 2.0d0 * (1.0d0 / (t * (((t / l) ** 2.0d0) * t_1)))
    else if (t <= 1.9d-43) then
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    else
        tmp = 2.0d0 / (((t / l) * (t * (t / l))) * 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.pow(k, 2.0);
	double tmp;
	if (t <= -9.2e-33) {
		tmp = 2.0 * (1.0 / (t * (Math.pow((t / l), 2.0) * t_1)));
	} else if (t <= 1.9e-43) {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	} else {
		tmp = 2.0 / (((t / l) * (t * (t / l))) * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 * math.pow(k, 2.0)
	tmp = 0
	if t <= -9.2e-33:
		tmp = 2.0 * (1.0 / (t * (math.pow((t / l), 2.0) * t_1)))
	elif t <= 1.9e-43:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	else:
		tmp = 2.0 / (((t / l) * (t * (t / l))) * t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 * (k ^ 2.0))
	tmp = 0.0
	if (t <= -9.2e-33)
		tmp = Float64(2.0 * Float64(1.0 / Float64(t * Float64((Float64(t / l) ^ 2.0) * t_1))));
	elseif (t <= 1.9e-43)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(t * Float64(t / l))) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 * (k ^ 2.0);
	tmp = 0.0;
	if (t <= -9.2e-33)
		tmp = 2.0 * (1.0 / (t * (((t / l) ^ 2.0) * t_1)));
	elseif (t <= 1.9e-43)
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	else
		tmp = 2.0 / (((t / l) * (t * (t / l))) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.2e-33], N[(2.0 * N[(1.0 / N[(t * N[(N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-43], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.19999999999999942e-33

   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* 58.6%

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

      2. associate-*l* 58.6%

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

      3. +-commutative 58.6%

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

      4. unpow2 58.6%

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

      5. times-frac 50.5%

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

      6. sqr-neg 50.5%

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

      7. times-frac 58.6%

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

      8. unpow2 58.6%

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

      9. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in k around 0 52.2%

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

      1. unpow3 63.6%

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

      2. times-frac 75.7%

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

      3. pow1 75.7%

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

      4. pow1 75.7%

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

      5. pow-sqr 75.7%

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

      6. metadata-eval 75.7%

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

   6. Applied egg-rr 61.5%

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

      1. div-inv 61.5%

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

      2. *-commutative 61.5%

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

      3. unpow2 61.5%

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

      4. associate-*r/ 68.2%

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

      5. associate-*l* 68.2%

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

      6. associate-*l* 69.6%

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

      7. pow1 69.6%

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

      8. pow1 69.6%

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

      9. pow-sqr 69.6%

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

      10. metadata-eval 69.6%

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

   8. Applied egg-rr 69.6%

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

   if -9.19999999999999942e-33 < t < 1.89999999999999985e-43

   1. Initial program 39.4%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.2%

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

   4. Simplified 72.2%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. unpow2 58.9%

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

      2. times-frac 65.8%

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

   7. Applied egg-rr 65.8%

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

   if 1.89999999999999985e-43 < t

   1. Initial program 68.4%

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

      1. associate-*l* 63.0%

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

      2. associate-*l* 63.0%

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

      3. +-commutative 63.0%

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

      4. unpow2 63.0%

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

      5. times-frac 52.7%

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

      6. sqr-neg 52.7%

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

      7. times-frac 63.0%

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

      8. unpow2 63.0%

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

      9. +-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. unpow3 68.4%

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

      2. times-frac 78.5%

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

      3. pow1 78.5%

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

      4. pow1 78.5%

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

      5. pow-sqr 78.5%

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

      6. metadata-eval 78.5%

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

   6. Applied egg-rr 61.0%

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

      1. div-inv 61.0%

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

      2. unpow2 61.0%

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

      3. associate-*l* 64.1%

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

      4. div-inv 64.1%

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

   8. Applied egg-rr 64.1%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{1}{t \cdot \left({\left(\frac{t}{\ell}\right)}^{2} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 12: 65.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-27} \lor \neg \left(t \leq 9.5 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.3e-27) (not (<= t 9.5e-43)))
   (/ 2.0 (* (* (/ t l) (* t (/ t l))) (* 2.0 (pow k 2.0))))
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-27) || !(t <= 9.5e-43)) {
		tmp = 2.0 / (((t / l) * (t * (t / l))) * (2.0 * pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.3d-27)) .or. (.not. (t <= 9.5d-43))) then
        tmp = 2.0d0 / (((t / l) * (t * (t / l))) * (2.0d0 * (k ** 2.0d0)))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.3e-27) || !(t <= 9.5e-43)) {
		tmp = 2.0 / (((t / l) * (t * (t / l))) * (2.0 * Math.pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.3e-27) or not (t <= 9.5e-43):
		tmp = 2.0 / (((t / l) * (t * (t / l))) * (2.0 * math.pow(k, 2.0)))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.3e-27) || !(t <= 9.5e-43))
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(t * Float64(t / l))) * Float64(2.0 * (k ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.3e-27) || ~((t <= 9.5e-43)))
		tmp = 2.0 / (((t / l) * (t * (t / l))) * (2.0 * (k ^ 2.0)));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.3e-27], N[Not[LessEqual[t, 9.5e-43]], $MachinePrecision]], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.29999999999999998e-27 or 9.50000000000000044e-43 < t

   1. Initial program 66.2%

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

      1. associate-*l* 61.0%

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

      2. associate-*l* 61.0%

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

      3. +-commutative 61.0%

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

      4. unpow2 61.0%

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

      5. times-frac 51.7%

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

      6. sqr-neg 51.7%

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

      7. times-frac 61.0%

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

      8. unpow2 61.0%

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

      9. +-commutative 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 54.8%

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

      1. unpow3 66.2%

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

      2. times-frac 77.2%

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

      3. pow1 77.2%

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

      4. pow1 77.2%

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

      5. pow-sqr 77.2%

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

      6. metadata-eval 77.2%

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

   6. Applied egg-rr 61.3%

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

      1. div-inv 61.2%

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

      2. unpow2 61.2%

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

      3. associate-*l* 66.0%

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

      4. div-inv 66.0%

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

   8. Applied egg-rr 66.0%

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

   if -3.29999999999999998e-27 < t < 9.50000000000000044e-43

   1. Initial program 39.4%

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

   2. Taylor expanded in t around 0 75.4%

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

      1. times-frac 72.2%

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

   4. Simplified 72.2%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. unpow2 58.9%

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

      2. times-frac 65.8%

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

   7. Applied egg-rr 65.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-27} \lor \neg \left(t \leq 9.5 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 13: 55.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* l (/ 2.0 (* t (pow k 4.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(l * N[(l * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in t around 0 61.0%

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

   1. times-frac 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in k around 0 50.5%

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

   1. associate-/r/ 50.5%

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

   2. unpow2 50.5%

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

   3. associate-*r* 53.5%

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

   4. *-commutative 53.5%

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

7. Applied egg-rr 53.5%

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

8. Final simplification 53.5%

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

Alternative 14: 55.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (/ 2.0 (/ (* t (pow k 4.0)) l))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(l * N[(2.0 / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in t around 0 61.0%

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

   1. times-frac 59.7%

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

4. Simplified 59.7%

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

5. Taylor expanded in k around 0 50.5%

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

   1. associate-/r/ 50.5%

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

   2. unpow2 50.5%

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

   3. associate-*r* 53.5%

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

   4. *-commutative 53.5%

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

7. Applied egg-rr 53.5%

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

   1. associate-*l/ 53.5%

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

   2. associate-/l* 53.5%

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

9. Applied egg-rr 53.5%

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

10. Final simplification 53.5%

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

Alternative 15: 56.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* (/ t l) (/ (pow k 4.0) l))))

C

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

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 / l) * ((k ** 4.0d0) / l))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 61.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. times-frac 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

4. Simplified 59.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

5. Taylor expanded in k around 0 50.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
6. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   2. times-frac 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

7. Applied egg-rr 53.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}} \]

8. Final simplification 53.7%

   \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}} \]

Reproduce

herbie shell --seed 2023301 
(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))))