Toniolo and Linder, Equation (10-)

Percentage Accurate: 33.8% → 78.0%

Time: 29.6s

Alternatives: 16

Speedup: 2.0×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 33.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 4.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \log \left({\left(e^{t_1}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\frac{t_1 \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 4.5e-160)
     (/
      2.0
      (* (/ (* t (pow k 2.0)) (cos k)) (log (pow (exp t_1) (pow l -2.0)))))
     (if (<= k 2.8e+97)
       (/ 2.0 (/ (* t_1 (/ t (/ (cos k) (pow k 2.0)))) (pow l 2.0)))
       (/
        2.0
        (pow
         (*
          (/ t (pow (cbrt l) 2.0))
          (cbrt (* (sin k) (* (tan k) (pow (/ k t) 2.0)))))
         3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 4.5e-160) {
		tmp = 2.0 / (((t * pow(k, 2.0)) / cos(k)) * log(pow(exp(t_1), pow(l, -2.0))));
	} else if (k <= 2.8e+97) {
		tmp = 2.0 / ((t_1 * (t / (cos(k) / pow(k, 2.0)))) / pow(l, 2.0));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((sin(k) * (tan(k) * pow((k / t), 2.0))))), 3.0);
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 4.5e-160) {
		tmp = 2.0 / (((t * Math.pow(k, 2.0)) / Math.cos(k)) * Math.log(Math.pow(Math.exp(t_1), Math.pow(l, -2.0))));
	} else if (k <= 2.8e+97) {
		tmp = 2.0 / ((t_1 * (t / (Math.cos(k) / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0))))), 3.0);
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 4.5e-160)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (k ^ 2.0)) / cos(k)) * log((exp(t_1) ^ (l ^ -2.0)))));
	elseif (k <= 2.8e+97)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(t / Float64(cos(k) / (k ^ 2.0)))) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0))))) ^ 3.0));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 4.5e-160], N[(2.0 / N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Log[N[Power[N[Exp[t$95$1], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e+97], N[(2.0 / N[(N[(t$95$1 * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.50000000000000026e-160

   1. Initial program 40.2%

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

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 48.4%

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

      2. times-frac 59.6%

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

      3. pow2 59.6%

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

   5. Applied egg-rr 59.6%

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

   6. Taylor expanded in t around 0 75.8%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 75.9%

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

      4. *-commutative 75.9%

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

   8. Simplified 75.9%

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

      1. add-log-exp 61.6%

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

      2. div-inv 61.6%

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

      3. exp-prod 68.3%

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

      4. pow-flip 68.3%

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

      5. metadata-eval 68.3%

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

   10. Applied egg-rr 68.3%

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

   if 4.50000000000000026e-160 < k < 2.7999999999999999e97

   1. Initial program 27.9%

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

   3. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 37.1%

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

      2. times-frac 44.5%

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

      3. pow2 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in t around 0 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

      3. times-frac 81.9%

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

      4. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. associate-*r/ 82.2%

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

      2. associate-/l* 82.3%

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

   10. Applied egg-rr 82.3%

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

   if 2.7999999999999999e97 < k

   1. Initial program 38.8%

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

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 46.7%

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

      2. times-frac 56.3%

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

      3. pow2 56.3%

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

   5. Applied egg-rr 56.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 56.3%

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

      2. clear-num 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 56.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 56.3%

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

   7. Applied egg-rr 56.3%

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

   9. Applied egg-rr 71.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \log \left({\left(e^{{\sin k}^{2}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternative 2: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-160} \lor \neg \left(k \leq 3.2 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= k 4.5e-160) (not (<= k 3.2e+97)))
   (/
    2.0
    (pow
     (*
      (/ t (pow (cbrt l) 2.0))
      (cbrt (* (sin k) (* (tan k) (pow (/ k t) 2.0)))))
     3.0))
   (/
    2.0
    (/ (* (pow (sin k) 2.0) (/ t (/ (cos k) (pow k 2.0)))) (pow l 2.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((k <= 4.5e-160) || !(k <= 3.2e+97)) {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((sin(k) * (tan(k) * pow((k / t), 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / (cos(k) / pow(k, 2.0)))) / pow(l, 2.0));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 4.5e-160) || !(k <= 3.2e+97)) {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (Math.cos(k) / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((k <= 4.5e-160) || !(k <= 3.2e+97))
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(cos(k) / (k ^ 2.0)))) / (l ^ 2.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[k, 4.5e-160], N[Not[LessEqual[k, 3.2e+97]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000026e-160 or 3.20000000000000016e97 < k

   1. Initial program 39.8%

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

   3. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 47.9%

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

      2. times-frac 58.8%

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

      3. pow2 58.8%

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

   5. Applied egg-rr 58.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 58.8%

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

      2. clear-num 58.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 58.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 58.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 58.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 58.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 58.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 58.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 58.8%

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

   7. Applied egg-rr 58.8%

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

   9. Applied egg-rr 75.1%

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

   if 4.50000000000000026e-160 < k < 3.20000000000000016e97

   1. Initial program 27.9%

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

   3. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 37.1%

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

      2. times-frac 44.5%

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

      3. pow2 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in t around 0 82.3%

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

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

      3. times-frac 81.9%

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

      4. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. associate-*r/ 82.2%

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

      2. associate-/l* 82.3%

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

   10. Applied egg-rr 82.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-160} \lor \neg \left(k \leq 3.2 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \end{array} \]

Alternative 3: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.32e+154)
   (/ 2.0 (/ (* (pow (sin k) 2.0) (/ t (/ (cos k) (pow k 2.0)))) (pow l 2.0)))
   (/
    2.0
    (*
     (* (sin k) (* (tan k) (pow (/ k t) 2.0)))
     (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.32e+154) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / (cos(k) / pow(k, 2.0)))) / pow(l, 2.0));
	} else {
		tmp = 2.0 / ((sin(k) * (tan(k) * pow((k / t), 2.0))) * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.32e+154) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (Math.cos(k) / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0))) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.32e+154)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(cos(k) / (k ^ 2.0)))) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0))) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.32e+154], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.31999999999999998e154

   1. Initial program 36.9%

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

   3. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 45.3%

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

      2. times-frac 55.8%

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

      3. pow2 55.8%

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

   5. Applied egg-rr 55.8%

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

   6. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. times-frac 77.2%

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

      4. *-commutative 77.2%

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

   8. Simplified 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.3%

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

   10. Applied egg-rr 77.3%

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

   if 1.31999999999999998e154 < k

   1. Initial program 39.7%

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

   3. Simplified 47.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 47.6%

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

      2. times-frac 55.7%

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

      3. pow2 55.7%

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

   5. Applied egg-rr 55.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 55.7%

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

      2. clear-num 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 55.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 55.7%

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

   7. Applied egg-rr 55.7%

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

   9. Applied egg-rr 74.0%

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

       1. *-commutative 74.0%

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

       2. cube-prod 67.9%

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

       3. associate-*r* 67.9%

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

       4. *-commutative 67.9%

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

       5. rem-cube-cbrt 68.0%

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

       6. *-commutative 68.0%

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

       7. associate-*r* 68.0%

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

   11. Simplified 68.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 4: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t \cdot {k}^{2}}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (pow l 2.0) (pow (sin k) 2.0)) (/ (cos k) (* t (pow k 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / pow(sin(k), 2.0)) * (cos(k) / (t * pow(k, 2.0))));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l ** 2.0d0) / (sin(k) ** 2.0d0)) * (cos(k) / (t * (k ** 2.0d0))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / (t * Math.pow(k, 2.0))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)) * (math.cos(k) / (t * math.pow(k, 2.0))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / (sin(k) ^ 2.0)) * Float64(cos(k) / Float64(t * (k ^ 2.0)))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) / (sin(k) ^ 2.0)) * (cos(k) / (t * (k ^ 2.0))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. associate-*l/ 37.7%

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

   3. associate--l+ 37.7%

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

3. Simplified 37.7%

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

   1. +-commutative 37.7%

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

   2. associate-+l- 46.0%

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

   3. metadata-eval 46.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}} \]

   4. --rgt-identity 46.0%

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

   5. unpow2 46.0%

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

   6. clear-num 46.0%

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

   7. clear-num 46.0%

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

   8. frac-times 45.6%

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

   9. metadata-eval 45.6%

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

5. Applied egg-rr 45.6%

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

6. Taylor expanded in t around 0 74.9%

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

   1. associate-*r* 74.9%

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

   2. *-commutative 74.9%

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

   3. times-frac 74.8%

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

   4. *-commutative 74.8%

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

8. Simplified 74.8%

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

9. Final simplification 74.8%

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

Alternative 5: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
}

Fortran

NOTE: k should be positive before calling this function
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 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * (t * (sin(k) ** 2.0d0))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * (t * math.pow(math.sin(k), 2.0))))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * (t * (sin(k) ^ 2.0))));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

3. Simplified 45.6%

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

4. Taylor expanded in t around 0 74.9%

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

5. Final simplification 74.9%

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

Alternative 6: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{\cos k}{{k}^{2}}}}{{\ell}^{2}}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (* (pow (sin k) 2.0) (/ t (/ (cos k) (pow k 2.0)))) (pow l 2.0))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / ((pow(sin(k), 2.0) * (t / (cos(k) / pow(k, 2.0)))) / pow(l, 2.0));
}

Fortran

NOTE: k should be positive before calling this function
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 / (((sin(k) ** 2.0d0) * (t / (cos(k) / (k ** 2.0d0)))) / (l ** 2.0d0))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (Math.cos(k) / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((math.pow(math.sin(k), 2.0) * (t / (math.cos(k) / math.pow(k, 2.0)))) / math.pow(l, 2.0))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(cos(k) / (k ^ 2.0)))) / (l ^ 2.0)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / (((sin(k) ^ 2.0) * (t / (cos(k) / (k ^ 2.0)))) / (l ^ 2.0));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

3. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Step-by-step derivation

   1. unpow3 45.6%

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

   2. times-frac 55.8%

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

   3. pow2 55.8%

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

5. Applied egg-rr 55.8%

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

6. Taylor expanded in t around 0 74.9%

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

   1. associate-*r* 74.9%

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

   2. *-commutative 74.9%

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

   3. times-frac 74.9%

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

   4. *-commutative 74.9%

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

8. Simplified 74.9%

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

   1. associate-*r/ 74.9%

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

   2. associate-/l* 74.9%

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

10. Applied egg-rr 74.9%

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

11. Final simplification 74.9%

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

Alternative 7: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := t \cdot {k}^{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t_1}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow k 2.0))))
   (if (<= t -1.5e+151)
     (* 2.0 (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) t_1)))
     (if (<= t -3.1e-140)
       (/
        2.0
        (*
         (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))
         (* (tan k) (* k (/ (/ k t) t)))))
       (if (<= t 2.4e-116)
         (/ 2.0 (* (/ t_1 (cos k)) (/ (pow k 2.0) (pow l 2.0))))
         (if (<= t 6.5e+83)
           (/
            (/ 2.0 (* (tan k) (* (/ (pow t 3.0) l) (/ (sin k) l))))
            (/ 1.0 (* (/ t k) (/ t k))))
           (/
            2.0
            (/
             (* (pow (sin k) 2.0) (/ t (/ 1.0 (pow k 2.0))))
             (pow l 2.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t * pow(k, 2.0);
	double tmp;
	if (t <= -1.5e+151) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * t_1));
	} else if (t <= -3.1e-140) {
		tmp = 2.0 / ((sin(k) * ((pow(t, 2.0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))));
	} else if (t <= 2.4e-116) {
		tmp = 2.0 / ((t_1 / cos(k)) * (pow(k, 2.0) / pow(l, 2.0)));
	} else if (t <= 6.5e+83) {
		tmp = (2.0 / (tan(k) * ((pow(t, 3.0) / l) * (sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / (1.0 / pow(k, 2.0)))) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 = t * (k ** 2.0d0)
    if (t <= (-1.5d+151)) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * t_1))
    else if (t <= (-3.1d-140)) then
        tmp = 2.0d0 / ((sin(k) * (((t ** 2.0d0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))))
    else if (t <= 2.4d-116) then
        tmp = 2.0d0 / ((t_1 / cos(k)) * ((k ** 2.0d0) / (l ** 2.0d0)))
    else if (t <= 6.5d+83) then
        tmp = (2.0d0 / (tan(k) * (((t ** 3.0d0) / l) * (sin(k) / l)))) / (1.0d0 / ((t / k) * (t / k)))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t / (1.0d0 / (k ** 2.0d0)))) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(k, 2.0);
	double tmp;
	if (t <= -1.5e+151) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * t_1));
	} else if (t <= -3.1e-140) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))) * (Math.tan(k) * (k * ((k / t) / t))));
	} else if (t <= 2.4e-116) {
		tmp = 2.0 / ((t_1 / Math.cos(k)) * (Math.pow(k, 2.0) / Math.pow(l, 2.0)));
	} else if (t <= 6.5e+83) {
		tmp = (2.0 / (Math.tan(k) * ((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (1.0 / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t * math.pow(k, 2.0)
	tmp = 0
	if t <= -1.5e+151:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * t_1))
	elif t <= -3.1e-140:
		tmp = 2.0 / ((math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))) * (math.tan(k) * (k * ((k / t) / t))))
	elif t <= 2.4e-116:
		tmp = 2.0 / ((t_1 / math.cos(k)) * (math.pow(k, 2.0) / math.pow(l, 2.0)))
	elif t <= 6.5e+83:
		tmp = (2.0 / (math.tan(k) * ((math.pow(t, 3.0) / l) * (math.sin(k) / l)))) / (1.0 / ((t / k) * (t / k)))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t / (1.0 / math.pow(k, 2.0)))) / math.pow(l, 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t * (k ^ 2.0))
	tmp = 0.0
	if (t <= -1.5e+151)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * t_1)));
	elseif (t <= -3.1e-140)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))) * Float64(tan(k) * Float64(k * Float64(Float64(k / t) / t)))));
	elseif (t <= 2.4e-116)
		tmp = Float64(2.0 / Float64(Float64(t_1 / cos(k)) * Float64((k ^ 2.0) / (l ^ 2.0))));
	elseif (t <= 6.5e+83)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)))) / Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(1.0 / (k ^ 2.0)))) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t * (k ^ 2.0);
	tmp = 0.0;
	if (t <= -1.5e+151)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * t_1));
	elseif (t <= -3.1e-140)
		tmp = 2.0 / ((sin(k) * (((t ^ 2.0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))));
	elseif (t <= 2.4e-116)
		tmp = 2.0 / ((t_1 / cos(k)) * ((k ^ 2.0) / (l ^ 2.0)));
	elseif (t <= 6.5e+83)
		tmp = (2.0 / (tan(k) * (((t ^ 3.0) / l) * (sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t / (1.0 / (k ^ 2.0)))) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+151], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.1e-140], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-116], N[(2.0 / N[(N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+83], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.5e151

   1. Initial program 13.2%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 89.6%

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

   5. Taylor expanded in k around 0 87.3%

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

   if -1.5e151 < t < -3.0999999999999999e-140

   1. Initial program 52.5%

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

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 58.3%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 77.0%

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

      2. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 77.0%

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

   7. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -3.0999999999999999e-140 < t < 2.39999999999999993e-116

   1. Initial program 35.5%

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

   3. Simplified 35.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 35.5%

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

      2. times-frac 42.2%

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

      3. pow2 42.2%

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

   5. Applied egg-rr 42.2%

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

   6. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. times-frac 78.1%

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

      4. *-commutative 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in k around 0 64.5%

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

   if 2.39999999999999993e-116 < t < 6.5000000000000003e83

   1. Initial program 55.5%

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

      1. associate-/r* 55.4%

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

      2. associate-*l/ 55.4%

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

      3. associate--l+ 55.4%

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

   3. Simplified 55.4%

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

      1. +-commutative 55.4%

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

      2. associate-+l- 71.0%

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

      3. metadata-eval 71.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}} \]

      4. --rgt-identity 71.0%

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

      5. unpow2 71.0%

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

      6. clear-num 71.0%

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

      7. clear-num 71.0%

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

      8. frac-times 71.1%

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

      9. metadata-eval 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. times-frac 77.8%

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

   7. Applied egg-rr 77.8%

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

   if 6.5000000000000003e83 < t

   1. Initial program 21.1%

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

   3. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 31.9%

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

      2. times-frac 45.4%

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

      3. pow2 45.4%

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

   5. Applied egg-rr 45.4%

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

   6. Taylor expanded in t around 0 74.7%

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

      1. associate-*r* 74.7%

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

      2. *-commutative 74.7%

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

      3. times-frac 74.9%

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

      4. *-commutative 74.9%

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

   8. Simplified 74.9%

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

      1. associate-*r/ 74.7%

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

      2. associate-/l* 74.6%

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

   10. Applied egg-rr 74.6%

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

   11. Taylor expanded in k around 0 72.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \end{array} \]

Alternative 8: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.6e-5)
   (/ 2.0 (/ (* (pow (sin k) 2.0) (/ t (/ 1.0 (pow k 2.0)))) (pow l 2.0)))
   (*
    2.0
    (/
     (* (cos k) (pow l 2.0))
     (* (pow k 2.0) (* t (- 0.5 (/ (cos (* k 2.0)) 2.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-5) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / (1.0 / pow(k, 2.0)))) / pow(l, 2.0));
	} else {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * (0.5 - (cos((k * 2.0)) / 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.6d-5) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t / (1.0d0 / (k ** 2.0d0)))) / (l ** 2.0d0))
    else
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * (t * (0.5d0 - (cos((k * 2.0d0)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-5) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (1.0 / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (t * (0.5 - (Math.cos((k * 2.0)) / 2.0)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.6e-5:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t / (1.0 / math.pow(k, 2.0)))) / math.pow(l, 2.0))
	else:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * (t * (0.5 - (math.cos((k * 2.0)) / 2.0)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.6e-5)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(1.0 / (k ^ 2.0)))) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k * 2.0)) / 2.0))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.6e-5)
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t / (1.0 / (k ^ 2.0)))) / (l ^ 2.0));
	else
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * (t * (0.5 - (cos((k * 2.0)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.6e-5], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.59999999999999992e-5

   1. Initial program 38.6%

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

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 46.8%

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

      2. times-frac 57.5%

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

      3. pow2 57.5%

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

   5. Applied egg-rr 57.5%

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

   6. Taylor expanded in t around 0 77.3%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. times-frac 77.3%

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

      4. *-commutative 77.3%

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

   8. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-/l* 77.3%

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

   10. Applied egg-rr 77.3%

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

   11. Taylor expanded in k around 0 71.5%

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

   if 5.59999999999999992e-5 < k

   1. Initial program 34.0%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in t around 0 68.5%

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

      1. unpow2 68.5%

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

      2. sin-mult 68.3%

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

   6. Applied egg-rr 68.3%

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

      1. div-sub 68.3%

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

      2. +-inverses 68.3%

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

      3. cos-0 68.3%

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

      4. metadata-eval 68.3%

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

      5. count-2 68.3%

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

   8. Simplified 68.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}\\ \end{array} \]

Alternative 9: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\frac{\cos k}{{k}^{2}}} \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.6e-5)
   (/ 2.0 (/ (* (pow (sin k) 2.0) (/ t (/ 1.0 (pow k 2.0)))) (pow l 2.0)))
   (/
    2.0
    (/
     (* (/ t (/ (cos k) (pow k 2.0))) (- 0.5 (/ (cos (* k 2.0)) 2.0)))
     (pow l 2.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-5) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / (1.0 / pow(k, 2.0)))) / pow(l, 2.0));
	} else {
		tmp = 2.0 / (((t / (cos(k) / pow(k, 2.0))) * (0.5 - (cos((k * 2.0)) / 2.0))) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.6d-5) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t / (1.0d0 / (k ** 2.0d0)))) / (l ** 2.0d0))
    else
        tmp = 2.0d0 / (((t / (cos(k) / (k ** 2.0d0))) * (0.5d0 - (cos((k * 2.0d0)) / 2.0d0))) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-5) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / (1.0 / Math.pow(k, 2.0)))) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / (((t / (Math.cos(k) / Math.pow(k, 2.0))) * (0.5 - (Math.cos((k * 2.0)) / 2.0))) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.6e-5:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t / (1.0 / math.pow(k, 2.0)))) / math.pow(l, 2.0))
	else:
		tmp = 2.0 / (((t / (math.cos(k) / math.pow(k, 2.0))) * (0.5 - (math.cos((k * 2.0)) / 2.0))) / math.pow(l, 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.6e-5)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / Float64(1.0 / (k ^ 2.0)))) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / Float64(cos(k) / (k ^ 2.0))) * Float64(0.5 - Float64(cos(Float64(k * 2.0)) / 2.0))) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.6e-5)
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t / (1.0 / (k ^ 2.0)))) / (l ^ 2.0));
	else
		tmp = 2.0 / (((t / (cos(k) / (k ^ 2.0))) * (0.5 - (cos((k * 2.0)) / 2.0))) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.6e-5], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.59999999999999992e-5

   1. Initial program 38.6%

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

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 46.8%

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

      2. times-frac 57.5%

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

      3. pow2 57.5%

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

   5. Applied egg-rr 57.5%

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

   6. Taylor expanded in t around 0 77.3%

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

      1. associate-*r* 77.3%

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

      2. *-commutative 77.3%

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

      3. times-frac 77.3%

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

      4. *-commutative 77.3%

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

   8. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-/l* 77.3%

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

   10. Applied egg-rr 77.3%

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

   11. Taylor expanded in k around 0 71.5%

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

   if 5.59999999999999992e-5 < k

   1. Initial program 34.0%

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

   3. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 42.7%

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

      2. times-frac 51.1%

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

      3. pow2 51.1%

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

   5. Applied egg-rr 51.1%

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

   6. Taylor expanded in t around 0 68.5%

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

      1. associate-*r* 68.6%

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

      2. *-commutative 68.6%

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

      3. times-frac 68.5%

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

      4. *-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. associate-*r/ 68.5%

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

      2. associate-/l* 68.6%

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

   10. Applied egg-rr 68.6%

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

       1. unpow2 68.5%

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

       2. sin-mult 68.3%

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

   12. Applied egg-rr 68.3%

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

       1. div-sub 68.3%

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

       2. +-inverses 68.3%

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

       3. cos-0 68.3%

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

       4. metadata-eval 68.3%

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

       5. count-2 68.3%

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

   14. Simplified 68.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{\frac{1}{{k}^{2}}}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\frac{\cos k}{{k}^{2}}} \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2}}}\\ \end{array} \]

Alternative 10: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ t_2 := \sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-122} \lor \neg \left(t \leq 1.42 \cdot 10^{+196}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (/ (* (pow (sin k) 2.0) (* t (pow k 2.0))) (pow l 2.0))))
        (t_2 (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
   (if (<= t -3.2e+159)
     t_1
     (if (<= t -2e-143)
       (/ 2.0 (* t_2 (* (tan k) (* k (/ (/ k t) t)))))
       (if (or (<= t 3.2e-122) (not (<= t 1.42e+196)))
         t_1
         (/ 2.0 (* t_2 (* (tan k) (/ k (* t (/ t k)))))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = 2.0 / ((pow(sin(k), 2.0) * (t * pow(k, 2.0))) / pow(l, 2.0));
	double t_2 = sin(k) * ((pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -3.2e+159) {
		tmp = t_1;
	} else if (t <= -2e-143) {
		tmp = 2.0 / (t_2 * (tan(k) * (k * ((k / t) / t))));
	} else if ((t <= 3.2e-122) || !(t <= 1.42e+196)) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * (k / (t * (t / k)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 / (((sin(k) ** 2.0d0) * (t * (k ** 2.0d0))) / (l ** 2.0d0))
    t_2 = sin(k) * (((t ** 2.0d0) / l) * (t / l))
    if (t <= (-3.2d+159)) then
        tmp = t_1
    else if (t <= (-2d-143)) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (k * ((k / t) / t))))
    else if ((t <= 3.2d-122) .or. (.not. (t <= 1.42d+196))) then
        tmp = t_1
    else
        tmp = 2.0d0 / (t_2 * (tan(k) * (k / (t * (t / k)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0))) / Math.pow(l, 2.0));
	double t_2 = Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -3.2e+159) {
		tmp = t_1;
	} else if (t <= -2e-143) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (k * ((k / t) / t))));
	} else if ((t <= 3.2e-122) || !(t <= 1.42e+196)) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (k / (t * (t / k)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = 2.0 / ((math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0))) / math.pow(l, 2.0))
	t_2 = math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))
	tmp = 0
	if t <= -3.2e+159:
		tmp = t_1
	elif t <= -2e-143:
		tmp = 2.0 / (t_2 * (math.tan(k) * (k * ((k / t) / t))))
	elif (t <= 3.2e-122) or not (t <= 1.42e+196):
		tmp = t_1
	else:
		tmp = 2.0 / (t_2 * (math.tan(k) * (k / (t * (t / k)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0))) / (l ^ 2.0)))
	t_2 = Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))
	tmp = 0.0
	if (t <= -3.2e+159)
		tmp = t_1;
	elseif (t <= -2e-143)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(k * Float64(Float64(k / t) / t)))));
	elseif ((t <= 3.2e-122) || !(t <= 1.42e+196))
		tmp = t_1;
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(k / Float64(t * Float64(t / k))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (((sin(k) ^ 2.0) * (t * (k ^ 2.0))) / (l ^ 2.0));
	t_2 = sin(k) * (((t ^ 2.0) / l) * (t / l));
	tmp = 0.0;
	if (t <= -3.2e+159)
		tmp = t_1;
	elseif (t <= -2e-143)
		tmp = 2.0 / (t_2 * (tan(k) * (k * ((k / t) / t))));
	elseif ((t <= 3.2e-122) || ~((t <= 1.42e+196)))
		tmp = t_1;
	else
		tmp = 2.0 / (t_2 * (tan(k) * (k / (t * (t / k)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.2e+159], t$95$1, If[LessEqual[t, -2e-143], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 3.2e-122], N[Not[LessEqual[t, 1.42e+196]], $MachinePrecision]], t$95$1, N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.19999999999999985e159 or -1.9999999999999999e-143 < t < 3.2000000000000002e-122 or 1.42000000000000002e196 < t

   1. Initial program 26.4%

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

   3. Simplified 32.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 32.7%

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

      2. times-frac 36.0%

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

      3. pow2 36.0%

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

   5. Applied egg-rr 36.0%

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

   6. Taylor expanded in t around 0 80.7%

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

      1. associate-*r* 80.7%

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

      2. *-commutative 80.7%

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

      3. times-frac 80.6%

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

      4. *-commutative 80.6%

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

   8. Simplified 80.6%

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

      1. associate-*r/ 80.7%

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

      2. associate-/l* 80.7%

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

   10. Applied egg-rr 80.7%

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

   11. Taylor expanded in k around 0 71.8%

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

   if -3.19999999999999985e159 < t < -1.9999999999999999e-143

   1. Initial program 51.1%

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

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 56.8%

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

      2. times-frac 75.0%

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

      3. pow2 75.0%

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

   5. Applied egg-rr 75.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 75.0%

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

      2. clear-num 75.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 75.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 75.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 75.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 75.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 75.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 75.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 75.0%

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

   7. Applied egg-rr 75.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 75.1%

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

   9. Applied egg-rr 75.1%

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

   if 3.2000000000000002e-122 < t < 1.42000000000000002e196

   1. Initial program 44.7%

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

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 61.7%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. div-inv 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\color{blue}{\left(k \cdot \frac{1}{t}\right)}}^{2} + 0\right)\right)} \]

      2. unpow-prod-down 67.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

   7. Applied egg-rr 67.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. pow-prod-down 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(k \cdot \frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

      2. div-inv 77.0%

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

      3. pow2 77.0%

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

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. div-inv 77.1%

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

      6. associate-/l/ 77.1%

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

   9. Applied egg-rr 77.1%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{\frac{t}{k} \cdot t}} + 0\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-122} \lor \neg \left(t \leq 1.42 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \end{array} \]

Alternative 11: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := t \cdot {k}^{2}\\ t_2 := 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t_1}\\ t_3 := \sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+196}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot t_1}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow k 2.0)))
        (t_2 (* 2.0 (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) t_1))))
        (t_3 (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
   (if (<= t -3.1e+152)
     t_2
     (if (<= t -5.5e-145)
       (/ 2.0 (* t_3 (* (tan k) (* k (/ (/ k t) t)))))
       (if (<= t 1.9e-123)
         t_2
         (if (<= t 1.42e+196)
           (/ 2.0 (* t_3 (* (tan k) (/ k (* t (/ t k))))))
           (/ 2.0 (/ (* (pow (sin k) 2.0) t_1) (pow l 2.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t * pow(k, 2.0);
	double t_2 = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * t_1));
	double t_3 = sin(k) * ((pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -3.1e+152) {
		tmp = t_2;
	} else if (t <= -5.5e-145) {
		tmp = 2.0 / (t_3 * (tan(k) * (k * ((k / t) / t))));
	} else if (t <= 1.9e-123) {
		tmp = t_2;
	} else if (t <= 1.42e+196) {
		tmp = 2.0 / (t_3 * (tan(k) * (k / (t * (t / k)))));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * t_1) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 * (k ** 2.0d0)
    t_2 = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * t_1))
    t_3 = sin(k) * (((t ** 2.0d0) / l) * (t / l))
    if (t <= (-3.1d+152)) then
        tmp = t_2
    else if (t <= (-5.5d-145)) then
        tmp = 2.0d0 / (t_3 * (tan(k) * (k * ((k / t) / t))))
    else if (t <= 1.9d-123) then
        tmp = t_2
    else if (t <= 1.42d+196) then
        tmp = 2.0d0 / (t_3 * (tan(k) * (k / (t * (t / k)))))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * t_1) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(k, 2.0);
	double t_2 = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * t_1));
	double t_3 = Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -3.1e+152) {
		tmp = t_2;
	} else if (t <= -5.5e-145) {
		tmp = 2.0 / (t_3 * (Math.tan(k) * (k * ((k / t) / t))));
	} else if (t <= 1.9e-123) {
		tmp = t_2;
	} else if (t <= 1.42e+196) {
		tmp = 2.0 / (t_3 * (Math.tan(k) * (k / (t * (t / k)))));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * t_1) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t * math.pow(k, 2.0)
	t_2 = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * t_1))
	t_3 = math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))
	tmp = 0
	if t <= -3.1e+152:
		tmp = t_2
	elif t <= -5.5e-145:
		tmp = 2.0 / (t_3 * (math.tan(k) * (k * ((k / t) / t))))
	elif t <= 1.9e-123:
		tmp = t_2
	elif t <= 1.42e+196:
		tmp = 2.0 / (t_3 * (math.tan(k) * (k / (t * (t / k)))))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * t_1) / math.pow(l, 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t * (k ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * t_1)))
	t_3 = Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))
	tmp = 0.0
	if (t <= -3.1e+152)
		tmp = t_2;
	elseif (t <= -5.5e-145)
		tmp = Float64(2.0 / Float64(t_3 * Float64(tan(k) * Float64(k * Float64(Float64(k / t) / t)))));
	elseif (t <= 1.9e-123)
		tmp = t_2;
	elseif (t <= 1.42e+196)
		tmp = Float64(2.0 / Float64(t_3 * Float64(tan(k) * Float64(k / Float64(t * Float64(t / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t_1) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t * (k ^ 2.0);
	t_2 = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * t_1));
	t_3 = sin(k) * (((t ^ 2.0) / l) * (t / l));
	tmp = 0.0;
	if (t <= -3.1e+152)
		tmp = t_2;
	elseif (t <= -5.5e-145)
		tmp = 2.0 / (t_3 * (tan(k) * (k * ((k / t) / t))));
	elseif (t <= 1.9e-123)
		tmp = t_2;
	elseif (t <= 1.42e+196)
		tmp = 2.0 / (t_3 * (tan(k) * (k / (t * (t / k)))));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * t_1) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.1e+152], t$95$2, If[LessEqual[t, -5.5e-145], N[(2.0 / N[(t$95$3 * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-123], t$95$2, If[LessEqual[t, 1.42e+196], N[(2.0 / N[(t$95$3 * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.1e152 or -5.50000000000000015e-145 < t < 1.89999999999999998e-123

   1. Initial program 28.3%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in t around 0 81.2%

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

   5. Taylor expanded in k around 0 71.2%

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

   if -3.1e152 < t < -5.50000000000000015e-145

   1. Initial program 52.5%

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

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 58.3%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 77.0%

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

      2. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 77.0%

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

   7. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if 1.89999999999999998e-123 < t < 1.42000000000000002e196

   1. Initial program 44.7%

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

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 61.7%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. div-inv 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\color{blue}{\left(k \cdot \frac{1}{t}\right)}}^{2} + 0\right)\right)} \]

      2. unpow-prod-down 67.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

   7. Applied egg-rr 67.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. pow-prod-down 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(k \cdot \frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

      2. div-inv 77.0%

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

      3. pow2 77.0%

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

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. div-inv 77.1%

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

      6. associate-/l/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if 1.42000000000000002e196 < t

   1. Initial program 11.1%

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

   3. Simplified 11.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 11.7%

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

      2. times-frac 12.2%

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

      3. pow2 12.2%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in t around 0 74.4%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. times-frac 74.6%

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

      4. *-commutative 74.6%

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

   8. Simplified 74.6%

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

      1. associate-*r/ 74.4%

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

      2. associate-/l* 74.4%

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

   10. Applied egg-rr 74.4%

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

   11. Taylor expanded in k around 0 70.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-123}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+196}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \end{array} \]

Alternative 12: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := t \cdot {k}^{2}\\ t_2 := \sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t_1}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+196}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot t_1}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow k 2.0)))
        (t_2 (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
   (if (<= t -5.2e+153)
     (* 2.0 (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) t_1)))
     (if (<= t -4.2e-141)
       (/ 2.0 (* t_2 (* (tan k) (* k (/ (/ k t) t)))))
       (if (<= t 9e-120)
         (/ 2.0 (* (/ t_1 (cos k)) (/ (pow k 2.0) (pow l 2.0))))
         (if (<= t 1.42e+196)
           (/ 2.0 (* t_2 (* (tan k) (/ k (* t (/ t k))))))
           (/ 2.0 (/ (* (pow (sin k) 2.0) t_1) (pow l 2.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t * pow(k, 2.0);
	double t_2 = sin(k) * ((pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -5.2e+153) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * t_1));
	} else if (t <= -4.2e-141) {
		tmp = 2.0 / (t_2 * (tan(k) * (k * ((k / t) / t))));
	} else if (t <= 9e-120) {
		tmp = 2.0 / ((t_1 / cos(k)) * (pow(k, 2.0) / pow(l, 2.0)));
	} else if (t <= 1.42e+196) {
		tmp = 2.0 / (t_2 * (tan(k) * (k / (t * (t / k)))));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * t_1) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 = t * (k ** 2.0d0)
    t_2 = sin(k) * (((t ** 2.0d0) / l) * (t / l))
    if (t <= (-5.2d+153)) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * t_1))
    else if (t <= (-4.2d-141)) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (k * ((k / t) / t))))
    else if (t <= 9d-120) then
        tmp = 2.0d0 / ((t_1 / cos(k)) * ((k ** 2.0d0) / (l ** 2.0d0)))
    else if (t <= 1.42d+196) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (k / (t * (t / k)))))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * t_1) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(k, 2.0);
	double t_2 = Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l));
	double tmp;
	if (t <= -5.2e+153) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * t_1));
	} else if (t <= -4.2e-141) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (k * ((k / t) / t))));
	} else if (t <= 9e-120) {
		tmp = 2.0 / ((t_1 / Math.cos(k)) * (Math.pow(k, 2.0) / Math.pow(l, 2.0)));
	} else if (t <= 1.42e+196) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (k / (t * (t / k)))));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * t_1) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t * math.pow(k, 2.0)
	t_2 = math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))
	tmp = 0
	if t <= -5.2e+153:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * t_1))
	elif t <= -4.2e-141:
		tmp = 2.0 / (t_2 * (math.tan(k) * (k * ((k / t) / t))))
	elif t <= 9e-120:
		tmp = 2.0 / ((t_1 / math.cos(k)) * (math.pow(k, 2.0) / math.pow(l, 2.0)))
	elif t <= 1.42e+196:
		tmp = 2.0 / (t_2 * (math.tan(k) * (k / (t * (t / k)))))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * t_1) / math.pow(l, 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t * (k ^ 2.0))
	t_2 = Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))
	tmp = 0.0
	if (t <= -5.2e+153)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * t_1)));
	elseif (t <= -4.2e-141)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(k * Float64(Float64(k / t) / t)))));
	elseif (t <= 9e-120)
		tmp = Float64(2.0 / Float64(Float64(t_1 / cos(k)) * Float64((k ^ 2.0) / (l ^ 2.0))));
	elseif (t <= 1.42e+196)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(k / Float64(t * Float64(t / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t_1) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t * (k ^ 2.0);
	t_2 = sin(k) * (((t ^ 2.0) / l) * (t / l));
	tmp = 0.0;
	if (t <= -5.2e+153)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * t_1));
	elseif (t <= -4.2e-141)
		tmp = 2.0 / (t_2 * (tan(k) * (k * ((k / t) / t))));
	elseif (t <= 9e-120)
		tmp = 2.0 / ((t_1 / cos(k)) * ((k ^ 2.0) / (l ^ 2.0)));
	elseif (t <= 1.42e+196)
		tmp = 2.0 / (t_2 * (tan(k) * (k / (t * (t / k)))));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * t_1) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e+153], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.2e-141], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-120], N[(2.0 / N[(N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.42e+196], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.1999999999999998e153

   1. Initial program 13.2%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 89.6%

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

   5. Taylor expanded in k around 0 87.3%

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

   if -5.1999999999999998e153 < t < -4.1999999999999999e-141

   1. Initial program 52.5%

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

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 58.3%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 77.0%

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

      2. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 77.0%

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

   7. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -4.1999999999999999e-141 < t < 9e-120

   1. Initial program 36.0%

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

   3. Simplified 36.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 36.0%

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

      2. times-frac 41.4%

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

      3. pow2 41.4%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in t around 0 76.9%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

      3. times-frac 77.8%

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

      4. *-commutative 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in k around 0 64.1%

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

   if 9e-120 < t < 1.42000000000000002e196

   1. Initial program 44.7%

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

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 61.7%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. div-inv 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\color{blue}{\left(k \cdot \frac{1}{t}\right)}}^{2} + 0\right)\right)} \]

      2. unpow-prod-down 67.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

   7. Applied egg-rr 67.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{k}^{2} \cdot {\left(\frac{1}{t}\right)}^{2}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. pow-prod-down 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(k \cdot \frac{1}{t}\right)}^{2}} + 0\right)\right)} \]

      2. div-inv 77.0%

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

      3. pow2 77.0%

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

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. div-inv 77.1%

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

      6. associate-/l/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if 1.42000000000000002e196 < t

   1. Initial program 11.1%

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

   3. Simplified 11.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 11.7%

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

      2. times-frac 12.2%

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

      3. pow2 12.2%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in t around 0 74.4%

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

      1. associate-*r* 74.4%

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

      2. *-commutative 74.4%

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

      3. times-frac 74.6%

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

      4. *-commutative 74.6%

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

   8. Simplified 74.6%

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

      1. associate-*r/ 74.4%

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

      2. associate-/l* 74.4%

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

   10. Applied egg-rr 74.4%

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

   11. Taylor expanded in k around 0 70.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+196}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \end{array} \]

Alternative 13: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-116}:\\ \;\;\;\;{k}^{-4} \cdot \left({\ell}^{2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -8e+149)
   (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t (pow k 4.0))))
   (if (<= t -2.1e-146)
     (/
      2.0
      (*
       (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))
       (* (tan k) (* k (/ (/ k t) t)))))
     (if (<= t 2.35e-116)
       (* (pow k -4.0) (* (pow l 2.0) (/ 2.0 t)))
       (if (<= t 4.5e+89)
         (/
          (/ 2.0 (* (tan k) (* (/ (pow t 3.0) l) (/ (sin k) l))))
          (/ 1.0 (* (/ t k) (/ t k))))
         (* (/ 2.0 t) (/ (pow l 2.0) (pow k 4.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e+149) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t * pow(k, 4.0)));
	} else if (t <= -2.1e-146) {
		tmp = 2.0 / ((sin(k) * ((pow(t, 2.0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))));
	} else if (t <= 2.35e-116) {
		tmp = pow(k, -4.0) * (pow(l, 2.0) * (2.0 / t));
	} else if (t <= 4.5e+89) {
		tmp = (2.0 / (tan(k) * ((pow(t, 3.0) / l) * (sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	} else {
		tmp = (2.0 / t) * (pow(l, 2.0) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-8d+149)) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t * (k ** 4.0d0)))
    else if (t <= (-2.1d-146)) then
        tmp = 2.0d0 / ((sin(k) * (((t ** 2.0d0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))))
    else if (t <= 2.35d-116) then
        tmp = (k ** (-4.0d0)) * ((l ** 2.0d0) * (2.0d0 / t))
    else if (t <= 4.5d+89) then
        tmp = (2.0d0 / (tan(k) * (((t ** 3.0d0) / l) * (sin(k) / l)))) / (1.0d0 / ((t / k) * (t / k)))
    else
        tmp = (2.0d0 / t) * ((l ** 2.0d0) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e+149) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t * Math.pow(k, 4.0)));
	} else if (t <= -2.1e-146) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))) * (Math.tan(k) * (k * ((k / t) / t))));
	} else if (t <= 2.35e-116) {
		tmp = Math.pow(k, -4.0) * (Math.pow(l, 2.0) * (2.0 / t));
	} else if (t <= 4.5e+89) {
		tmp = (2.0 / (Math.tan(k) * ((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	} else {
		tmp = (2.0 / t) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -8e+149:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t * math.pow(k, 4.0)))
	elif t <= -2.1e-146:
		tmp = 2.0 / ((math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))) * (math.tan(k) * (k * ((k / t) / t))))
	elif t <= 2.35e-116:
		tmp = math.pow(k, -4.0) * (math.pow(l, 2.0) * (2.0 / t))
	elif t <= 4.5e+89:
		tmp = (2.0 / (math.tan(k) * ((math.pow(t, 3.0) / l) * (math.sin(k) / l)))) / (1.0 / ((t / k) * (t / k)))
	else:
		tmp = (2.0 / t) * (math.pow(l, 2.0) / math.pow(k, 4.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -8e+149)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t * (k ^ 4.0))));
	elseif (t <= -2.1e-146)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))) * Float64(tan(k) * Float64(k * Float64(Float64(k / t) / t)))));
	elseif (t <= 2.35e-116)
		tmp = Float64((k ^ -4.0) * Float64((l ^ 2.0) * Float64(2.0 / t)));
	elseif (t <= 4.5e+89)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)))) / Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))));
	else
		tmp = Float64(Float64(2.0 / t) * Float64((l ^ 2.0) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8e+149)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t * (k ^ 4.0)));
	elseif (t <= -2.1e-146)
		tmp = 2.0 / ((sin(k) * (((t ^ 2.0) / l) * (t / l))) * (tan(k) * (k * ((k / t) / t))));
	elseif (t <= 2.35e-116)
		tmp = (k ^ -4.0) * ((l ^ 2.0) * (2.0 / t));
	elseif (t <= 4.5e+89)
		tmp = (2.0 / (tan(k) * (((t ^ 3.0) / l) * (sin(k) / l)))) / (1.0 / ((t / k) * (t / k)));
	else
		tmp = (2.0 / t) * ((l ^ 2.0) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -8e+149], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.1e-146], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.35e-116], N[(N[Power[k, -4.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e+89], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.00000000000000039e149

   1. Initial program 13.2%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in t around 0 89.6%

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

   5. Taylor expanded in k around 0 79.9%

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

   if -8.00000000000000039e149 < t < -2.0999999999999999e-146

   1. Initial program 52.5%

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

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow3 58.3%

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

      2. times-frac 77.0%

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

      3. pow2 77.0%

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

   5. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 77.0%

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

      2. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k}\right)}^{-1}} \cdot \frac{k}{t} + 0\right)\right)} \]

      4. clear-num 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      5. inv-pow 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{t}{k}\right)}^{-1} \cdot \color{blue}{{\left(\frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      6. unpow-prod-down 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{{\left(\frac{t}{k} \cdot \frac{t}{k}\right)}^{-1}} + 0\right)\right)} \]

      7. inv-pow 77.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      8. associate-/r* 77.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{1}{\frac{t}{k}}}{\frac{t}{k}}} + 0\right)\right)} \]

      9. clear-num 77.0%

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

   7. Applied egg-rr 77.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} + 0\right)\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -2.0999999999999999e-146 < t < 2.34999999999999997e-116

   1. Initial program 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in k around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. *-commutative 59.5%

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

      3. times-frac 60.5%

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

   6. Simplified 60.5%

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

      1. expm1-log1p-u 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\right)\right)} \]

      2. expm1-udef 30.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\right)} - 1} \]

      3. *-commutative 30.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}}\right)} - 1 \]

      4. div-inv 30.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{4}}\right)} \cdot \frac{2}{t}\right)} - 1 \]

      5. pow-flip 31.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \frac{2}{t}\right)} - 1 \]

      6. metadata-eval 31.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right) \cdot \frac{2}{t}\right)} - 1 \]

   8. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-4}\right) \cdot \frac{2}{t}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 31.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\ell}^{2} \cdot {k}^{-4}\right) \cdot \frac{2}{t}\right)\right)} \]

      2. expm1-log1p 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l* 62.9%

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

   10. Simplified 62.9%

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

   if 2.34999999999999997e-116 < t < 4.5e89

   1. Initial program 55.5%

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

      1. associate-/r* 55.4%

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

      2. associate-*l/ 55.4%

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

      3. associate--l+ 55.4%

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

   3. Simplified 55.4%

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

      1. +-commutative 55.4%

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

      2. associate-+l- 71.0%

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

      3. metadata-eval 71.0%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}} \]

      4. --rgt-identity 71.0%

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

      5. unpow2 71.0%

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

      6. clear-num 71.0%

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

      7. clear-num 71.0%

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

      8. frac-times 71.1%

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

      9. metadata-eval 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. times-frac 77.8%

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

   7. Applied egg-rr 77.8%

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

   if 4.5e89 < t

   1. Initial program 21.1%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in k around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. times-frac 69.9%

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

   6. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\frac{k}{t}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-116}:\\ \;\;\;\;{k}^{-4} \cdot \left({\ell}^{2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \end{array} \]

Alternative 14: 60.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((cos(k) * pow(l, 2.0)) / (t * pow(k, 4.0)));
}

Fortran

NOTE: k should be positive before calling this function
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 * ((cos(k) * (l ** 2.0d0)) / (t * (k ** 4.0d0)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t * Math.pow(k, 4.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t * math.pow(k, 4.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t * (k ^ 4.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t * (k ^ 4.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

3. Simplified 45.6%

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

4. Taylor expanded in t around 0 74.9%

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

5. Taylor expanded in k around 0 65.3%

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

6. Final simplification 65.3%

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

Alternative 15: 59.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (pow l 2.0) (* t (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k, 4.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

3. Simplified 45.6%

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

4. Taylor expanded in k around 0 64.0%

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

5. Final simplification 64.0%

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

Alternative 16: 59.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2 \cdot \left({\ell}^{2} \cdot {k}^{-4}\right)}{t} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (/ (* 2.0 (* (pow l 2.0) (pow k -4.0))) t))

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 * (pow(l, 2.0) * pow(k, -4.0))) / t;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 * ((l ** 2.0d0) * (k ** (-4.0d0)))) / t
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (2.0 * (Math.pow(l, 2.0) * Math.pow(k, -4.0))) / t;
}

Python

k = abs(k)
def code(t, l, k):
	return (2.0 * (math.pow(l, 2.0) * math.pow(k, -4.0))) / t

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(2.0 * Float64((l ^ 2.0) * (k ^ -4.0))) / t)
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (2.0 * ((l ^ 2.0) * (k ^ -4.0))) / t;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

3. Simplified 45.6%

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

4. Taylor expanded in k around 0 64.0%

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

   1. associate-*r/ 64.0%

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

   2. *-commutative 64.0%

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

   3. times-frac 63.9%

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

6. Simplified 63.9%

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

   1. associate-*l/ 63.9%

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

   2. div-inv 63.9%

      \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{4}}\right)}}{t} \]

   3. pow-flip 64.3%

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

   4. metadata-eval 64.3%

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

8. Applied egg-rr 64.3%

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

9. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2023311 
(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))))