Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.0% → 79.7%

Time: 34.9s

Alternatives: 23

Speedup: 2.0×

• 34.1% 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: 34.0% 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: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left(t_1 \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{t}{t_1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\log \left({\left(e^{{k}^{2}}\right)}^{\left({\ell}^{-2}\right)}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= l 2.7e-162)
     (* 2.0 (pow (* t_1 (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
     (if (<= l 1.4e+157)
       (/
        2.0
        (* (pow k 2.0) (* (pow l -2.0) (* (pow (sin k) 2.0) (/ t (cos k))))))
       (if (<= l 2.8e+227)
         (/
          2.0
          (pow
           (* (cbrt (* (* (sin k) (tan k)) (pow (/ k t) 2.0))) (/ t t_1))
           3.0))
         (/
          2.0
          (*
           (log (pow (exp (pow k 2.0)) (pow l -2.0)))
           (/ (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k)))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if (l <= 2.7e-162) {
		tmp = 2.0 * pow((t_1 * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else if (l <= 1.4e+157) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(sin(k), 2.0) * (t / cos(k)))));
	} else if (l <= 2.8e+227) {
		tmp = 2.0 / pow((cbrt(((sin(k) * tan(k)) * pow((k / t), 2.0))) * (t / t_1)), 3.0);
	} else {
		tmp = 2.0 / (log(pow(exp(pow(k, 2.0)), pow(l, -2.0))) * ((t * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (l <= 2.7e-162) {
		tmp = 2.0 * Math.pow((t_1 * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else if (l <= 1.4e+157) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(Math.sin(k), 2.0) * (t / Math.cos(k)))));
	} else if (l <= 2.8e+227) {
		tmp = 2.0 / Math.pow((Math.cbrt(((Math.sin(k) * Math.tan(k)) * Math.pow((k / t), 2.0))) * (t / t_1)), 3.0);
	} else {
		tmp = 2.0 / (Math.log(Math.pow(Math.exp(Math.pow(k, 2.0)), Math.pow(l, -2.0))) * ((t * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (l <= 2.7e-162)
		tmp = Float64(2.0 * (Float64(t_1 * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	elseif (l <= 1.4e+157)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((sin(k) ^ 2.0) * Float64(t / cos(k))))));
	elseif (l <= 2.8e+227)
		tmp = Float64(2.0 / (Float64(cbrt(Float64(Float64(sin(k) * tan(k)) * (Float64(k / t) ^ 2.0))) * Float64(t / t_1)) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(log((exp((k ^ 2.0)) ^ (l ^ -2.0))) * Float64(Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 2.7e-162], N[(2.0 * N[Power[N[(t$95$1 * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.4e+157], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e+227], N[(2.0 / N[Power[N[(N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Log[N[Power[N[Exp[N[Power[k, 2.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 2.69999999999999984e-162

   1. Initial program 34.3%

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

      1. associate-/r* 34.4%

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

      2. associate-*l/ 34.4%

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

      3. associate--l+ 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in k around 0 57.1%

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

      1. add-cube-cbrt 57.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 57.1%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 57.1%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 57.1%

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

      1. pow1/3 38.9%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 38.9%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 34.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 34.0%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 34.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 38.2%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 38.2%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 38.2%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 66.8%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 66.8%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 66.8%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 2.69999999999999984e-162 < l < 1.4000000000000001e157

   1. Initial program 37.9%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in t around 0 85.3%

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

      1. times-frac 89.8%

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

   6. Simplified 89.8%

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

      1. expm1-log1p-u 64.0%

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

      2. expm1-udef 34.1%

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

      3. div-inv 34.1%

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

      4. pow-flip 35.4%

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

      5. metadata-eval 35.4%

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

      6. associate-/l* 35.4%

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

   8. Applied egg-rr 35.4%

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

      1. expm1-def 64.3%

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

      2. expm1-log1p 90.1%

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

      3. associate-*l* 90.1%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

   10. Simplified 91.0%

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

   if 1.4000000000000001e157 < l < 2.79999999999999984e227

   1. Initial program 16.0%

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

   3. Simplified 16.0%

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

      1. unpow3 16.0%

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

      2. times-frac 40.3%

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

      3. pow2 40.3%

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

   5. Applied egg-rr 40.3%

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

      1. add-cube-cbrt 39.9%

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

      2. pow3 39.8%

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

   7. Applied egg-rr 78.4%

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

      1. associate-*r* 77.9%

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

      2. *-commutative 77.9%

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

   9. Simplified 77.9%

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

   if 2.79999999999999984e227 < l

   1. Initial program 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in t around 0 71.4%

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

      1. times-frac 71.7%

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

   6. Simplified 71.7%

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

      1. unpow2 71.7%

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

      2. sin-mult 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. div-sub 71.7%

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

      2. +-inverses 71.7%

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

      3. cos-0 71.7%

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

      4. metadata-eval 71.7%

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

      5. count-2 71.7%

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

      6. *-commutative 71.7%

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

   10. Simplified 71.7%

       \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\cos k}} \]
   11. Step-by-step derivation

       1. add-log-exp 71.7%

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

       2. div-inv 71.7%

          \[\leadsto \frac{2}{\log \left(e^{\color{blue}{{k}^{2} \cdot \frac{1}{{\ell}^{2}}}}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\cos k}} \]

       3. exp-prod 72.4%

          \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{{k}^{2}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\cos k}} \]

       4. pow-flip 72.4%

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

       5. metadata-eval 72.4%

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

   12. Applied egg-rr 72.4%

       \[\leadsto \frac{2}{\color{blue}{\log \left({\left(e^{{k}^{2}}\right)}^{\left({\ell}^{-2}\right)}\right)} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{\cos k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\log \left({\left(e^{{k}^{2}}\right)}^{\left({\ell}^{-2}\right)}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 2: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left(t_1 \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{t}{t_1}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= (* l l) 0.0)
     (* 2.0 (pow (* t_1 (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
     (if (<= (* l l) 2e+296)
       (/
        2.0
        (* (pow k 2.0) (* (pow l -2.0) (* (pow (sin k) 2.0) (/ t (cos k))))))
       (/
        2.0
        (pow
         (* (cbrt (* (* (sin k) (tan k)) (pow (/ k t) 2.0))) (/ t t_1))
         3.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * pow((t_1 * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else if ((l * l) <= 2e+296) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(sin(k), 2.0) * (t / cos(k)))));
	} else {
		tmp = 2.0 / pow((cbrt(((sin(k) * tan(k)) * pow((k / t), 2.0))) * (t / t_1)), 3.0);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * Math.pow((t_1 * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else if ((l * l) <= 2e+296) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(Math.sin(k), 2.0) * (t / Math.cos(k)))));
	} else {
		tmp = 2.0 / Math.pow((Math.cbrt(((Math.sin(k) * Math.tan(k)) * Math.pow((k / t), 2.0))) * (t / t_1)), 3.0);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * (Float64(t_1 * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	elseif (Float64(l * l) <= 2e+296)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((sin(k) ^ 2.0) * Float64(t / cos(k))))));
	else
		tmp = Float64(2.0 / (Float64(cbrt(Float64(Float64(sin(k) * tan(k)) * (Float64(k / t) ^ 2.0))) * Float64(t / t_1)) ^ 3.0));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[Power[N[(t$95$1 * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+296], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 16.1%

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

      1. associate-/r* 16.1%

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

      2. associate-*l/ 16.1%

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

      3. associate--l+ 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. add-cube-cbrt 49.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 49.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 49.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 49.0%

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

      1. pow1/3 49.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 49.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.7%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 50.6%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 76.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 0.0 < (*.f64 l l) < 1.99999999999999996e296

   1. Initial program 42.9%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in t around 0 85.4%

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

      1. times-frac 87.9%

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

   6. Simplified 87.9%

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

      1. expm1-log1p-u 61.2%

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

      2. expm1-udef 29.2%

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

      3. div-inv 29.2%

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

      4. pow-flip 29.2%

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

      5. metadata-eval 29.2%

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

      6. associate-/l* 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. expm1-def 60.7%

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

      2. expm1-log1p 87.3%

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

      3. associate-*l* 88.6%

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

      4. associate-/r/ 89.0%

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

      5. *-commutative 89.0%

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

   10. Simplified 89.0%

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

   if 1.99999999999999996e296 < (*.f64 l l)

   1. Initial program 31.4%

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

   3. Simplified 31.4%

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

      1. unpow3 31.4%

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

      2. times-frac 46.3%

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

      3. pow2 46.3%

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

   5. Applied egg-rr 46.3%

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

      1. add-cube-cbrt 46.1%

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

      2. pow3 46.1%

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

   7. Applied egg-rr 67.1%

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

      1. associate-*r* 67.0%

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

      2. *-commutative 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 3: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* 2.0 (pow (* (pow (cbrt l) 2.0) (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
   (*
    2.0
    (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 16.1%

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

      1. associate-/r* 16.1%

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

      2. associate-*l/ 16.1%

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

      3. associate--l+ 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. add-cube-cbrt 49.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 49.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 49.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 49.0%

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

      1. pow1/3 49.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 49.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.7%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 50.6%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 76.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 39.7%

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

      1. associate-/r* 39.7%

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

      2. associate-*l/ 39.7%

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

      3. associate--l+ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around 0 76.7%

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

      1. times-frac 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* 2.0 (pow (* (pow (cbrt l) 2.0) (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
   (*
    2.0
    (* (/ (pow l 2.0) (pow k 2.0)) (/ (/ (cos k) t) (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * ((cos(k) / t) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 16.1%

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

      1. associate-/r* 16.1%

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

      2. associate-*l/ 16.1%

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

      3. associate--l+ 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. add-cube-cbrt 49.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 49.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 49.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 49.0%

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

      1. pow1/3 49.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 49.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.7%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 50.6%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 76.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 39.7%

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

      1. associate-/r* 39.7%

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

      2. associate-*l/ 39.7%

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

      3. associate--l+ 39.7%

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

   3. Simplified 39.7%

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

      1. unpow2 46.5%

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

      2. clear-num 46.5%

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

      3. un-div-inv 46.5%

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

   5. Applied egg-rr 39.8%

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

      1. associate-+r- 39.7%

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

      2. add-exp-log 39.5%

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

      3. log1p-udef 39.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 46.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 47.0%

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

      6. associate-/r/ 41.9%

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

   7. Applied egg-rr 41.9%

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

   8. Taylor expanded in t around 0 76.7%

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

      1. times-frac 78.2%

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

      2. associate-/r* 78.3%

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

   10. Simplified 78.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* 2.0 (pow (* (pow (cbrt l) 2.0) (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
   (*
    2.0
    (/ (/ (* (cos k) (pow l 2.0)) (pow k 2.0)) (* t (pow (sin k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * (((cos(k) * pow(l, 2.0)) / pow(k, 2.0)) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 * (((Math.cos(k) * Math.pow(l, 2.0)) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * (l ^ 2.0)) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 16.1%

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

      1. associate-/r* 16.1%

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

      2. associate-*l/ 16.1%

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

      3. associate--l+ 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. add-cube-cbrt 49.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 49.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 49.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 49.0%

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

      1. pow1/3 49.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 49.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.7%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 50.6%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 76.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 0.0 < (*.f64 l l)

   1. Initial program 39.7%

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

      1. associate-/r* 39.7%

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

      2. associate-*l/ 39.7%

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

      3. associate--l+ 39.7%

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

   3. Simplified 39.7%

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

      1. unpow2 46.5%

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

      2. clear-num 46.5%

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

      3. un-div-inv 46.5%

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

   5. Applied egg-rr 39.8%

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

   6. Taylor expanded in t around 0 76.7%

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

      1. associate-/r* 78.3%

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

   8. Simplified 78.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 6: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e-298)
   (* 2.0 (pow (* (pow (cbrt l) 2.0) (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
   (/
    2.0
    (* (/ (pow k 2.0) (pow l 2.0)) (/ (* t (pow (sin k) 2.0)) (cos k))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-298) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * ((t * pow(sin(k), 2.0)) / cos(k)));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-298) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * ((t * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-298)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * (sin(k) ^ 2.0)) / cos(k))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-298], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999982e-298

   1. Initial program 16.9%

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

      1. associate-/r* 16.9%

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

      2. associate-*l/ 16.9%

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

      3. associate--l+ 16.9%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in k around 0 50.1%

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

      1. add-cube-cbrt 50.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 50.1%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 50.1%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 50.1%

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

      1. pow1/3 50.1%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 50.1%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.5%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.5%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.5%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 49.8%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 49.8%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 49.8%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 75.8%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 75.8%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 75.8%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 1.99999999999999982e-298 < (*.f64 l l)

   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 47.6%

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

   4. Taylor expanded in t around 0 76.3%

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

      1. times-frac 78.7%

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

   6. Simplified 78.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-298}:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 7: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* 2.0 (pow (* (pow (cbrt l) 2.0) (cbrt (/ (/ 1.0 t) (pow k 4.0)))) 3.0))
   (/
    2.0
    (* (pow k 2.0) (* (pow l -2.0) (* (pow (sin k) 2.0) (/ t (cos k))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) * cbrt(((1.0 / t) / pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(sin(k), 2.0) * (t / cos(k)))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(((1.0 / t) / Math.pow(k, 4.0)))), 3.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(Math.sin(k), 2.0) * (t / Math.cos(k)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) * cbrt(Float64(Float64(1.0 / t) / (k ^ 4.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((sin(k) ^ 2.0) * Float64(t / cos(k))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 16.1%

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

      1. associate-/r* 16.1%

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

      2. associate-*l/ 16.1%

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

      3. associate--l+ 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in k around 0 49.0%

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

      1. add-cube-cbrt 49.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 49.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 49.0%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 49.0%

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

      1. pow1/3 49.0%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{{\ell}^{2}}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. div-inv 49.0%

         \[\leadsto 2 \cdot {\left({\color{blue}{\left({\ell}^{2} \cdot \frac{1}{t \cdot {k}^{4}}\right)}}^{0.3333333333333333}\right)}^{3} \]

      3. unpow-prod-down 39.7%

         \[\leadsto 2 \cdot {\color{blue}{\left({\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]

      4. pow1/3 39.7%

         \[\leadsto 2 \cdot {\left(\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      5. pow2 39.7%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      6. cbrt-prod 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

      7. pow2 50.6%

         \[\leadsto 2 \cdot {\left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}^{3} \]

   8. Applied egg-rr 50.6%

      \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left(\frac{1}{t \cdot {k}^{4}}\right)}^{0.3333333333333333}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow1/3 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{\sqrt[3]{\frac{1}{t \cdot {k}^{4}}}}\right)}^{3} \]

      2. associate-/r* 76.2%

         \[\leadsto 2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{1}{t}}{{k}^{4}}}}\right)}^{3} \]

   10. Simplified 76.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}}^{3} \]

   if 0.0 < (*.f64 l l)

   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 47.4%

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

   4. Taylor expanded in t around 0 76.6%

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

      1. times-frac 78.5%

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

   6. Simplified 78.5%

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

      1. expm1-log1p-u 59.2%

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

      2. expm1-udef 24.3%

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

      3. div-inv 24.3%

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

      4. pow-flip 24.8%

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

      5. metadata-eval 24.8%

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

      6. associate-/l* 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. expm1-def 59.4%

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

      2. expm1-log1p 78.7%

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

      3. associate-*l* 79.6%

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

      4. associate-/r/ 80.0%

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

      5. *-commutative 80.0%

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

   10. Simplified 80.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{1}{t}}{{k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \end{array} \]

Alternative 8: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-151}:\\ \;\;\;\;2 \cdot {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\frac{t}{\cos k} \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.85e-151)
   (* 2.0 (pow (/ (pow (cbrt l) 2.0) (cbrt (* t (pow k 4.0)))) 3.0))
   (if (<= k 9.5e-5)
     (/ 2.0 (* (pow k 2.0) (/ (pow k 2.0) (/ (pow l 2.0) t))))
     (/
      2.0
      (*
       (pow k 2.0)
       (* (pow l -2.0) (* (/ t (cos k)) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-151) {
		tmp = 2.0 * pow((pow(cbrt(l), 2.0) / cbrt((t * pow(k, 4.0)))), 3.0);
	} else if (k <= 9.5e-5) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) / (pow(l, 2.0) / t)));
	} else {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * ((t / cos(k)) * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-151) {
		tmp = 2.0 * Math.pow((Math.pow(Math.cbrt(l), 2.0) / Math.cbrt((t * Math.pow(k, 4.0)))), 3.0);
	} else if (k <= 9.5e-5) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) / (Math.pow(l, 2.0) / t)));
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * ((t / Math.cos(k)) * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.85e-151)
		tmp = Float64(2.0 * (Float64((cbrt(l) ^ 2.0) / cbrt(Float64(t * (k ^ 4.0)))) ^ 3.0));
	elseif (k <= 9.5e-5)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) / Float64((l ^ 2.0) / t))));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(Float64(t / cos(k)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.85e-151], N[(2.0 * N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.5e-5], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.85e-151

   1. Initial program 39.7%

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

      1. associate-/r* 39.7%

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

      2. associate-*l/ 39.7%

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

      3. associate--l+ 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in k around 0 59.1%

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

      1. add-cube-cbrt 59.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. pow3 59.1%

         \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)}^{3}} \]

      3. *-commutative 59.1%

         \[\leadsto 2 \cdot {\left(\sqrt[3]{\frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}}}\right)}^{3} \]

   6. Applied egg-rr 59.1%

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

      1. cbrt-div 59.1%

         \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}}^{3} \]

      2. pow2 59.1%

         \[\leadsto 2 \cdot {\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3} \]

      3. cbrt-prod 68.0%

         \[\leadsto 2 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3} \]

      4. pow2 68.0%

         \[\leadsto 2 \cdot {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3} \]

   8. Applied egg-rr 68.0%

      \[\leadsto 2 \cdot {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}}^{3} \]

   if 1.85e-151 < k < 9.5000000000000005e-5

   1. Initial program 26.7%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in t around 0 71.7%

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

      1. times-frac 75.4%

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

   6. Simplified 75.4%

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

      1. expm1-log1p-u 64.0%

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

      2. expm1-udef 20.4%

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

      3. div-inv 20.4%

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

      4. pow-flip 20.4%

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

      5. metadata-eval 20.4%

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

      6. associate-/l* 20.4%

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

   8. Applied egg-rr 20.4%

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

      1. expm1-def 63.9%

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

      2. expm1-log1p 75.3%

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

      3. associate-*l* 75.3%

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

      4. associate-/r/ 75.4%

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

      5. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   11. Taylor expanded in k around 0 74.3%

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

       1. associate-/l* 77.6%

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

   13. Simplified 77.6%

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

   if 9.5000000000000005e-5 < k

   1. Initial program 26.8%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around 0 69.2%

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

      1. times-frac 75.6%

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

   6. Simplified 75.6%

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

      1. expm1-log1p-u 37.3%

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

      2. expm1-udef 27.9%

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

      3. div-inv 27.9%

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

      4. pow-flip 29.2%

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

      5. metadata-eval 29.2%

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

      6. associate-/l* 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. expm1-def 38.6%

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

      2. expm1-log1p 76.9%

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

      3. associate-*l* 76.8%

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

      4. associate-/r/ 76.8%

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

      5. *-commutative 76.8%

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

   10. Simplified 76.8%

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

       1. unpow2 75.6%

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

       2. sin-mult 75.4%

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

   12. Applied egg-rr 76.6%

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

       1. div-sub 75.4%

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

       2. +-inverses 75.4%

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

       3. cos-0 75.4%

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

       4. metadata-eval 75.4%

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

       5. count-2 75.4%

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

       6. *-commutative 75.4%

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

   14. Simplified 76.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-151}:\\ \;\;\;\;2 \cdot {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\frac{t}{\cos k} \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 9: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot t_1\right)\right)}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \sqrt{t_1}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(\sin k \cdot t_3\right)\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t_4\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot t_3\right) \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{{\left(t_4 \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cos k)))
        (t_2 (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* (pow k 2.0) t_1)))))
        (t_3 (pow (/ k t) 2.0))
        (t_4 (sqrt t_1)))
   (if (<= t -5.5e+102)
     t_2
     (if (<= t -7.2e-112)
       (/ 2.0 (/ (* (* (tan k) (* (sin k) t_3)) (/ (pow t 3.0) l)) l))
       (if (<= t -2e-310)
         t_2
         (if (<= t 1.05e-77)
           (/ 2.0 (pow (* (* (sin k) (/ k l)) t_4) 2.0))
           (if (<= t 5.6e+102)
             (/ 2.0 (/ (* (* (tan k) t_3) (/ (* (sin k) (pow t 3.0)) l)) l))
             (if (<= t 2.8e+191)
               (/ 2.0 (pow (* t_4 (/ (* k (sin k)) l)) 2.0))
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t / cos(k);
	double t_2 = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(k, 2.0) * t_1)));
	double t_3 = pow((k / t), 2.0);
	double t_4 = sqrt(t_1);
	double tmp;
	if (t <= -5.5e+102) {
		tmp = t_2;
	} else if (t <= -7.2e-112) {
		tmp = 2.0 / (((tan(k) * (sin(k) * t_3)) * (pow(t, 3.0) / l)) / l);
	} else if (t <= -2e-310) {
		tmp = t_2;
	} else if (t <= 1.05e-77) {
		tmp = 2.0 / pow(((sin(k) * (k / l)) * t_4), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = 2.0 / (((tan(k) * t_3) * ((sin(k) * pow(t, 3.0)) / l)) / l);
	} else if (t <= 2.8e+191) {
		tmp = 2.0 / pow((t_4 * ((k * sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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) :: t_4
    real(8) :: tmp
    t_1 = t / cos(k)
    t_2 = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * ((k ** 2.0d0) * t_1)))
    t_3 = (k / t) ** 2.0d0
    t_4 = sqrt(t_1)
    if (t <= (-5.5d+102)) then
        tmp = t_2
    else if (t <= (-7.2d-112)) then
        tmp = 2.0d0 / (((tan(k) * (sin(k) * t_3)) * ((t ** 3.0d0) / l)) / l)
    else if (t <= (-2d-310)) then
        tmp = t_2
    else if (t <= 1.05d-77) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) * t_4) ** 2.0d0)
    else if (t <= 5.6d+102) then
        tmp = 2.0d0 / (((tan(k) * t_3) * ((sin(k) * (t ** 3.0d0)) / l)) / l)
    else if (t <= 2.8d+191) then
        tmp = 2.0d0 / ((t_4 * ((k * sin(k)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t / Math.cos(k);
	double t_2 = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(k, 2.0) * t_1)));
	double t_3 = Math.pow((k / t), 2.0);
	double t_4 = Math.sqrt(t_1);
	double tmp;
	if (t <= -5.5e+102) {
		tmp = t_2;
	} else if (t <= -7.2e-112) {
		tmp = 2.0 / (((Math.tan(k) * (Math.sin(k) * t_3)) * (Math.pow(t, 3.0) / l)) / l);
	} else if (t <= -2e-310) {
		tmp = t_2;
	} else if (t <= 1.05e-77) {
		tmp = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * t_4), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = 2.0 / (((Math.tan(k) * t_3) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l);
	} else if (t <= 2.8e+191) {
		tmp = 2.0 / Math.pow((t_4 * ((k * Math.sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t / math.cos(k)
	t_2 = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (math.pow(k, 2.0) * t_1)))
	t_3 = math.pow((k / t), 2.0)
	t_4 = math.sqrt(t_1)
	tmp = 0
	if t <= -5.5e+102:
		tmp = t_2
	elif t <= -7.2e-112:
		tmp = 2.0 / (((math.tan(k) * (math.sin(k) * t_3)) * (math.pow(t, 3.0) / l)) / l)
	elif t <= -2e-310:
		tmp = t_2
	elif t <= 1.05e-77:
		tmp = 2.0 / math.pow(((math.sin(k) * (k / l)) * t_4), 2.0)
	elif t <= 5.6e+102:
		tmp = 2.0 / (((math.tan(k) * t_3) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)
	elif t <= 2.8e+191:
		tmp = 2.0 / math.pow((t_4 * ((k * math.sin(k)) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t / cos(k))
	t_2 = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((k ^ 2.0) * t_1))))
	t_3 = Float64(k / t) ^ 2.0
	t_4 = sqrt(t_1)
	tmp = 0.0
	if (t <= -5.5e+102)
		tmp = t_2;
	elseif (t <= -7.2e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(sin(k) * t_3)) * Float64((t ^ 3.0) / l)) / l));
	elseif (t <= -2e-310)
		tmp = t_2;
	elseif (t <= 1.05e-77)
		tmp = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * t_4) ^ 2.0));
	elseif (t <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * t_3) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l));
	elseif (t <= 2.8e+191)
		tmp = Float64(2.0 / (Float64(t_4 * Float64(Float64(k * sin(k)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t / cos(k);
	t_2 = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * ((k ^ 2.0) * t_1)));
	t_3 = (k / t) ^ 2.0;
	t_4 = sqrt(t_1);
	tmp = 0.0;
	if (t <= -5.5e+102)
		tmp = t_2;
	elseif (t <= -7.2e-112)
		tmp = 2.0 / (((tan(k) * (sin(k) * t_3)) * ((t ^ 3.0) / l)) / l);
	elseif (t <= -2e-310)
		tmp = t_2;
	elseif (t <= 1.05e-77)
		tmp = 2.0 / (((sin(k) * (k / l)) * t_4) ^ 2.0);
	elseif (t <= 5.6e+102)
		tmp = 2.0 / (((tan(k) * t_3) * ((sin(k) * (t ^ 3.0)) / l)) / l);
	elseif (t <= 2.8e+191)
		tmp = 2.0 / ((t_4 * ((k * sin(k)) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[t, -5.5e+102], t$95$2, If[LessEqual[t, -7.2e-112], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2e-310], t$95$2, If[LessEqual[t, 1.05e-77], N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+191], N[(2.0 / N[Power[N[(t$95$4 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.49999999999999981e102 or -7.2000000000000002e-112 < t < -1.999999999999994e-310

   1. Initial program 17.5%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in t around 0 73.2%

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

      1. times-frac 73.0%

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

   6. Simplified 73.0%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-udef 10.5%

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

      3. div-inv 10.5%

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

      4. pow-flip 11.8%

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

      5. metadata-eval 11.8%

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

      6. associate-/l* 11.8%

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

   8. Applied egg-rr 11.8%

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

      1. expm1-def 57.0%

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

      2. expm1-log1p 73.6%

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

      3. associate-*l* 74.8%

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

      4. associate-/r/ 75.7%

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

      5. *-commutative 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in k around 0 65.2%

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

   if -5.49999999999999981e102 < t < -7.2000000000000002e-112

   1. Initial program 66.1%

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

   3. Simplified 71.9%

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

      1. associate-*l* 71.9%

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

      2. +-rgt-identity 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r* 71.9%

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

      5. *-commutative 71.9%

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

      6. *-commutative 71.9%

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

      7. associate-/r* 77.2%

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

      8. associate-*r/ 85.7%

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

   5. Applied egg-rr 85.7%

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

   if -1.999999999999994e-310 < t < 1.05000000000000008e-77

   1. Initial program 26.0%

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

   3. Simplified 28.5%

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

      1. associate-*l* 26.0%

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

      2. +-rgt-identity 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. add-sqr-sqrt 26.1%

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

      7. pow2 26.1%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

   if 1.05000000000000008e-77 < t < 5.60000000000000037e102

   1. Initial program 68.4%

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

   3. Simplified 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*l/ 71.1%

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

      3. associate-/r* 76.7%

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

      4. associate-*r/ 82.9%

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

      5. +-rgt-identity 82.9%

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

   5. Applied egg-rr 82.9%

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

   if 5.60000000000000037e102 < t < 2.7999999999999999e191

   1. Initial program 15.0%

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

   3. Simplified 30.5%

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

      1. associate-*l* 30.5%

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

      2. +-rgt-identity 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r* 30.5%

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

      5. *-commutative 30.5%

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

      6. add-sqr-sqrt 29.2%

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

      7. pow2 29.2%

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

   5. Applied egg-rr 51.2%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 2.7999999999999999e191 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 10: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ t_2 := \frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+197}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
          (* k (/ (/ k t) t))))
        (t_2 (/ 2.0 (pow (* (* (sin k) (/ k l)) (sqrt (/ t (cos k)))) 2.0))))
   (if (<= t -5.6e+102)
     (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* t (pow k 2.0)))))
     (if (<= t -7.2e-112)
       t_1
       (if (<= t -2e-310)
         (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
         (if (<= t 3.5e-76)
           t_2
           (if (<= t 2.6e+102)
             t_1
             (if (<= t 1.02e+197)
               t_2
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double t_2 = 2.0 / pow(((sin(k) * (k / l)) * sqrt((t / cos(k)))), 2.0);
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (t * pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = t_1;
	} else if (t <= -2e-310) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else if (t <= 3.5e-76) {
		tmp = t_2;
	} else if (t <= 2.6e+102) {
		tmp = t_1;
	} else if (t <= 1.02e+197) {
		tmp = t_2;
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    t_2 = 2.0d0 / (((sin(k) * (k / l)) * sqrt((t / cos(k)))) ** 2.0d0)
    if (t <= (-5.6d+102)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * (t * (k ** 2.0d0))))
    else if (t <= (-7.2d-112)) then
        tmp = t_1
    else if (t <= (-2d-310)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else if (t <= 3.5d-76) then
        tmp = t_2
    else if (t <= 2.6d+102) then
        tmp = t_1
    else if (t <= 1.02d+197) then
        tmp = t_2
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double t_2 = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * Math.sqrt((t / Math.cos(k)))), 2.0);
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (t * Math.pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = t_1;
	} else if (t <= -2e-310) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else if (t <= 3.5e-76) {
		tmp = t_2;
	} else if (t <= 2.6e+102) {
		tmp = t_1;
	} else if (t <= 1.02e+197) {
		tmp = t_2;
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	t_2 = 2.0 / math.pow(((math.sin(k) * (k / l)) * math.sqrt((t / math.cos(k)))), 2.0)
	tmp = 0
	if t <= -5.6e+102:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (t * math.pow(k, 2.0))))
	elif t <= -7.2e-112:
		tmp = t_1
	elif t <= -2e-310:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	elif t <= 3.5e-76:
		tmp = t_2
	elif t <= 2.6e+102:
		tmp = t_1
	elif t <= 1.02e+197:
		tmp = t_2
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)))
	t_2 = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * sqrt(Float64(t / cos(k)))) ^ 2.0))
	tmp = 0.0
	if (t <= -5.6e+102)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(t * (k ^ 2.0)))));
	elseif (t <= -7.2e-112)
		tmp = t_1;
	elseif (t <= -2e-310)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	elseif (t <= 3.5e-76)
		tmp = t_2;
	elseif (t <= 2.6e+102)
		tmp = t_1;
	elseif (t <= 1.02e+197)
		tmp = t_2;
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	t_2 = 2.0 / (((sin(k) * (k / l)) * sqrt((t / cos(k)))) ^ 2.0);
	tmp = 0.0;
	if (t <= -5.6e+102)
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * (t * (k ^ 2.0))));
	elseif (t <= -7.2e-112)
		tmp = t_1;
	elseif (t <= -2e-310)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	elseif (t <= 3.5e-76)
		tmp = t_2;
	elseif (t <= 2.6e+102)
		tmp = t_1;
	elseif (t <= 1.02e+197)
		tmp = t_2;
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+102], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.2e-112], t$95$1, If[LessEqual[t, -2e-310], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-76], t$95$2, If[LessEqual[t, 2.6e+102], t$95$1, If[LessEqual[t, 1.02e+197], t$95$2, N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.60000000000000037e102

   1. Initial program 8.3%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in t around 0 72.9%

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

      1. times-frac 77.6%

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

   6. Simplified 77.6%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 5.7%

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

      3. div-inv 5.7%

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

      4. pow-flip 8.5%

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

      5. metadata-eval 8.5%

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

      6. associate-/l* 8.5%

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

   8. Applied egg-rr 8.5%

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

      1. expm1-def 53.5%

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

      2. expm1-log1p 78.7%

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

      3. associate-*l* 76.3%

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

      4. associate-/r/ 78.4%

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

      5. *-commutative 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 75.2%

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

   if -5.60000000000000037e102 < t < -7.2000000000000002e-112 or 3.49999999999999997e-76 < t < 2.60000000000000006e102

   1. Initial program 67.0%

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

      1. associate-/r* 66.9%

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

      2. associate-*l/ 66.9%

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

      3. associate--l+ 67.0%

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

   3. Simplified 67.0%

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

      1. unpow2 70.6%

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

      2. clear-num 70.6%

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

      3. un-div-inv 70.6%

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

   5. Applied egg-rr 67.0%

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

      1. associate-+r- 67.0%

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

      2. add-exp-log 66.5%

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

      3. log1p-udef 66.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 71.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.6%

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

      6. associate-/r/ 71.6%

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

   7. Applied egg-rr 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*l/ 71.6%

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

      4. associate-/r* 79.2%

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

      5. associate-*r/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -7.2000000000000002e-112 < t < -1.999999999999994e-310

   1. Initial program 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in t around 0 73.5%

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

      1. times-frac 69.3%

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

   6. Simplified 69.3%

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

      1. expm1-log1p-u 59.9%

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

      2. expm1-udef 14.5%

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

      3. div-inv 14.5%

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

      4. pow-flip 14.5%

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

      5. metadata-eval 14.5%

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

      6. associate-/l* 14.5%

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

   8. Applied egg-rr 14.5%

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

      1. expm1-def 59.9%

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

      2. expm1-log1p 69.3%

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

      3. associate-*l* 73.5%

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

      4. associate-/r/ 73.5%

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

      5. *-commutative 73.5%

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

   10. Simplified 73.5%

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

   11. Taylor expanded in k around 0 49.7%

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

       1. associate-/l* 53.8%

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

       2. associate-/r/ 53.9%

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

   13. Simplified 53.9%

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

   if -1.999999999999994e-310 < t < 3.49999999999999997e-76 or 2.60000000000000006e102 < t < 1.02000000000000008e197

   1. Initial program 22.9%

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

   3. Simplified 29.1%

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

      1. associate-*l* 27.3%

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

      2. +-rgt-identity 27.3%

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

      3. *-commutative 27.3%

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

      4. associate-*r* 27.3%

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

      5. *-commutative 27.3%

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

      6. add-sqr-sqrt 27.0%

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

      7. pow2 27.0%

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

   5. Applied egg-rr 49.8%

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

   6. Taylor expanded in k around inf 75.0%

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

      1. associate-/l* 74.8%

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

      2. associate-/r/ 75.0%

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

   8. Simplified 75.0%

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

   if 1.02000000000000008e197 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 11: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{t}{\cos k}}\\ t_2 := \frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t_1\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{{\left(t_1 \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sqrt (/ t (cos k))))
        (t_2
         (/
          (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
          (* k (/ (/ k t) t)))))
   (if (<= t -5.6e+102)
     (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* t (pow k 2.0)))))
     (if (<= t -7.2e-112)
       t_2
       (if (<= t -2e-309)
         (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
         (if (<= t 3e-71)
           (/ 2.0 (pow (* (* (sin k) (/ k l)) t_1) 2.0))
           (if (<= t 2.6e+102)
             t_2
             (if (<= t 4e+194)
               (/ 2.0 (pow (* t_1 (/ (* k (sin k)) l)) 2.0))
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = sqrt((t / cos(k)));
	double t_2 = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (t * pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = t_2;
	} else if (t <= -2e-309) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else if (t <= 3e-71) {
		tmp = 2.0 / pow(((sin(k) * (k / l)) * t_1), 2.0);
	} else if (t <= 2.6e+102) {
		tmp = t_2;
	} else if (t <= 4e+194) {
		tmp = 2.0 / pow((t_1 * ((k * sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 = sqrt((t / cos(k)))
    t_2 = (2.0d0 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    if (t <= (-5.6d+102)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * (t * (k ** 2.0d0))))
    else if (t <= (-7.2d-112)) then
        tmp = t_2
    else if (t <= (-2d-309)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else if (t <= 3d-71) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) * t_1) ** 2.0d0)
    else if (t <= 2.6d+102) then
        tmp = t_2
    else if (t <= 4d+194) then
        tmp = 2.0d0 / ((t_1 * ((k * sin(k)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt((t / Math.cos(k)));
	double t_2 = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (t * Math.pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = t_2;
	} else if (t <= -2e-309) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else if (t <= 3e-71) {
		tmp = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * t_1), 2.0);
	} else if (t <= 2.6e+102) {
		tmp = t_2;
	} else if (t <= 4e+194) {
		tmp = 2.0 / Math.pow((t_1 * ((k * Math.sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.sqrt((t / math.cos(k)))
	t_2 = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	tmp = 0
	if t <= -5.6e+102:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (t * math.pow(k, 2.0))))
	elif t <= -7.2e-112:
		tmp = t_2
	elif t <= -2e-309:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	elif t <= 3e-71:
		tmp = 2.0 / math.pow(((math.sin(k) * (k / l)) * t_1), 2.0)
	elif t <= 2.6e+102:
		tmp = t_2
	elif t <= 4e+194:
		tmp = 2.0 / math.pow((t_1 * ((k * math.sin(k)) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = sqrt(Float64(t / cos(k)))
	t_2 = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)))
	tmp = 0.0
	if (t <= -5.6e+102)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(t * (k ^ 2.0)))));
	elseif (t <= -7.2e-112)
		tmp = t_2;
	elseif (t <= -2e-309)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	elseif (t <= 3e-71)
		tmp = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * t_1) ^ 2.0));
	elseif (t <= 2.6e+102)
		tmp = t_2;
	elseif (t <= 4e+194)
		tmp = Float64(2.0 / (Float64(t_1 * Float64(Float64(k * sin(k)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = sqrt((t / cos(k)));
	t_2 = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	tmp = 0.0;
	if (t <= -5.6e+102)
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * (t * (k ^ 2.0))));
	elseif (t <= -7.2e-112)
		tmp = t_2;
	elseif (t <= -2e-309)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	elseif (t <= 3e-71)
		tmp = 2.0 / (((sin(k) * (k / l)) * t_1) ^ 2.0);
	elseif (t <= 2.6e+102)
		tmp = t_2;
	elseif (t <= 4e+194)
		tmp = 2.0 / ((t_1 * ((k * sin(k)) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+102], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.2e-112], t$95$2, If[LessEqual[t, -2e-309], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-71], N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e+102], t$95$2, If[LessEqual[t, 4e+194], N[(2.0 / N[Power[N[(t$95$1 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.60000000000000037e102

   1. Initial program 8.3%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in t around 0 72.9%

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

      1. times-frac 77.6%

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

   6. Simplified 77.6%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 5.7%

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

      3. div-inv 5.7%

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

      4. pow-flip 8.5%

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

      5. metadata-eval 8.5%

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

      6. associate-/l* 8.5%

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

   8. Applied egg-rr 8.5%

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

      1. expm1-def 53.5%

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

      2. expm1-log1p 78.7%

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

      3. associate-*l* 76.3%

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

      4. associate-/r/ 78.4%

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

      5. *-commutative 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 75.2%

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

   if -5.60000000000000037e102 < t < -7.2000000000000002e-112 or 3.0000000000000001e-71 < t < 2.60000000000000006e102

   1. Initial program 67.0%

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

      1. associate-/r* 66.9%

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

      2. associate-*l/ 66.9%

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

      3. associate--l+ 67.0%

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

   3. Simplified 67.0%

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

      1. unpow2 70.6%

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

      2. clear-num 70.6%

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

      3. un-div-inv 70.6%

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

   5. Applied egg-rr 67.0%

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

      1. associate-+r- 67.0%

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

      2. add-exp-log 66.5%

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

      3. log1p-udef 66.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 71.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.6%

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

      6. associate-/r/ 71.6%

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

   7. Applied egg-rr 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*l/ 71.6%

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

      4. associate-/r* 79.2%

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

      5. associate-*r/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -7.2000000000000002e-112 < t < -1.9999999999999988e-309

   1. Initial program 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in t around 0 73.5%

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

      1. times-frac 69.3%

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

   6. Simplified 69.3%

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

      1. expm1-log1p-u 59.9%

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

      2. expm1-udef 14.5%

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

      3. div-inv 14.5%

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

      4. pow-flip 14.5%

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

      5. metadata-eval 14.5%

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

      6. associate-/l* 14.5%

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

   8. Applied egg-rr 14.5%

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

      1. expm1-def 59.9%

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

      2. expm1-log1p 69.3%

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

      3. associate-*l* 73.5%

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

      4. associate-/r/ 73.5%

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

      5. *-commutative 73.5%

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

   10. Simplified 73.5%

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

   11. Taylor expanded in k around 0 49.7%

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

       1. associate-/l* 53.8%

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

       2. associate-/r/ 53.9%

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

   13. Simplified 53.9%

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

   if -1.9999999999999988e-309 < t < 3.0000000000000001e-71

   1. Initial program 26.0%

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

   3. Simplified 28.5%

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

      1. associate-*l* 26.0%

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

      2. +-rgt-identity 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. add-sqr-sqrt 26.1%

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

      7. pow2 26.1%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

   if 2.60000000000000006e102 < t < 3.99999999999999978e194

   1. Initial program 15.0%

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

   3. Simplified 30.5%

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

      1. associate-*l* 30.5%

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

      2. +-rgt-identity 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r* 30.5%

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

      5. *-commutative 30.5%

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

      6. add-sqr-sqrt 29.2%

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

      7. pow2 29.2%

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

   5. Applied egg-rr 51.2%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 3.99999999999999978e194 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 12: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot {k}^{2}\\ t_2 := \sqrt{\frac{t}{\cos k}}\\ t_3 := \frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot t_1\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t_1}{\cos k}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t_2\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow k 2.0)))
        (t_2 (sqrt (/ t (cos k))))
        (t_3
         (/
          (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
          (* k (/ (/ k t) t)))))
   (if (<= t -5.6e+102)
     (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) t_1)))
     (if (<= t -7.2e-112)
       t_3
       (if (<= t -2e-310)
         (/ 2.0 (* (/ (pow k 2.0) (pow l 2.0)) (/ t_1 (cos k))))
         (if (<= t 1.1e-76)
           (/ 2.0 (pow (* (* (sin k) (/ k l)) t_2) 2.0))
           (if (<= t 5.6e+102)
             t_3
             (if (<= t 3.1e+190)
               (/ 2.0 (pow (* t_2 (/ (* k (sin k)) l)) 2.0))
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t * pow(k, 2.0);
	double t_2 = sqrt((t / cos(k)));
	double t_3 = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * t_1));
	} else if (t <= -7.2e-112) {
		tmp = t_3;
	} else if (t <= -2e-310) {
		tmp = 2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * (t_1 / cos(k)));
	} else if (t <= 1.1e-76) {
		tmp = 2.0 / pow(((sin(k) * (k / l)) * t_2), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = t_3;
	} else if (t <= 3.1e+190) {
		tmp = 2.0 / pow((t_2 * ((k * sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 = sqrt((t / cos(k)))
    t_3 = (2.0d0 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    if (t <= (-5.6d+102)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * t_1))
    else if (t <= (-7.2d-112)) then
        tmp = t_3
    else if (t <= (-2d-310)) then
        tmp = 2.0d0 / (((k ** 2.0d0) / (l ** 2.0d0)) * (t_1 / cos(k)))
    else if (t <= 1.1d-76) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) * t_2) ** 2.0d0)
    else if (t <= 5.6d+102) then
        tmp = t_3
    else if (t <= 3.1d+190) then
        tmp = 2.0d0 / ((t_2 * ((k * sin(k)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(k, 2.0);
	double t_2 = Math.sqrt((t / Math.cos(k)));
	double t_3 = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double tmp;
	if (t <= -5.6e+102) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * t_1));
	} else if (t <= -7.2e-112) {
		tmp = t_3;
	} else if (t <= -2e-310) {
		tmp = 2.0 / ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * (t_1 / Math.cos(k)));
	} else if (t <= 1.1e-76) {
		tmp = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * t_2), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = t_3;
	} else if (t <= 3.1e+190) {
		tmp = 2.0 / Math.pow((t_2 * ((k * Math.sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t * math.pow(k, 2.0)
	t_2 = math.sqrt((t / math.cos(k)))
	t_3 = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	tmp = 0
	if t <= -5.6e+102:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * t_1))
	elif t <= -7.2e-112:
		tmp = t_3
	elif t <= -2e-310:
		tmp = 2.0 / ((math.pow(k, 2.0) / math.pow(l, 2.0)) * (t_1 / math.cos(k)))
	elif t <= 1.1e-76:
		tmp = 2.0 / math.pow(((math.sin(k) * (k / l)) * t_2), 2.0)
	elif t <= 5.6e+102:
		tmp = t_3
	elif t <= 3.1e+190:
		tmp = 2.0 / math.pow((t_2 * ((k * math.sin(k)) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t * (k ^ 2.0))
	t_2 = sqrt(Float64(t / cos(k)))
	t_3 = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)))
	tmp = 0.0
	if (t <= -5.6e+102)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * t_1)));
	elseif (t <= -7.2e-112)
		tmp = t_3;
	elseif (t <= -2e-310)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(t_1 / cos(k))));
	elseif (t <= 1.1e-76)
		tmp = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * t_2) ^ 2.0));
	elseif (t <= 5.6e+102)
		tmp = t_3;
	elseif (t <= 3.1e+190)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(Float64(k * sin(k)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t * (k ^ 2.0);
	t_2 = sqrt((t / cos(k)));
	t_3 = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	tmp = 0.0;
	if (t <= -5.6e+102)
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * t_1));
	elseif (t <= -7.2e-112)
		tmp = t_3;
	elseif (t <= -2e-310)
		tmp = 2.0 / (((k ^ 2.0) / (l ^ 2.0)) * (t_1 / cos(k)));
	elseif (t <= 1.1e-76)
		tmp = 2.0 / (((sin(k) * (k / l)) * t_2) ^ 2.0);
	elseif (t <= 5.6e+102)
		tmp = t_3;
	elseif (t <= 3.1e+190)
		tmp = 2.0 / ((t_2 * ((k * sin(k)) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l 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[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+102], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.2e-112], t$95$3, If[LessEqual[t, -2e-310], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-76], N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], t$95$3, If[LessEqual[t, 3.1e+190], N[(2.0 / N[Power[N[(t$95$2 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.60000000000000037e102

   1. Initial program 8.3%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in t around 0 72.9%

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

      1. times-frac 77.6%

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

   6. Simplified 77.6%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 5.7%

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

      3. div-inv 5.7%

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

      4. pow-flip 8.5%

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

      5. metadata-eval 8.5%

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

      6. associate-/l* 8.5%

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

   8. Applied egg-rr 8.5%

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

      1. expm1-def 53.5%

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

      2. expm1-log1p 78.7%

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

      3. associate-*l* 76.3%

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

      4. associate-/r/ 78.4%

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

      5. *-commutative 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 75.2%

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

   if -5.60000000000000037e102 < t < -7.2000000000000002e-112 or 1.1e-76 < t < 5.60000000000000037e102

   1. Initial program 67.0%

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

      1. associate-/r* 66.9%

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

      2. associate-*l/ 66.9%

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

      3. associate--l+ 67.0%

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

   3. Simplified 67.0%

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

      1. unpow2 70.6%

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

      2. clear-num 70.6%

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

      3. un-div-inv 70.6%

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

   5. Applied egg-rr 67.0%

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

      1. associate-+r- 67.0%

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

      2. add-exp-log 66.5%

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

      3. log1p-udef 66.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 71.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.6%

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

      6. associate-/r/ 71.6%

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

   7. Applied egg-rr 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*l/ 71.6%

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

      4. associate-/r* 79.2%

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

      5. associate-*r/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -7.2000000000000002e-112 < t < -1.999999999999994e-310

   1. Initial program 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in t around 0 73.5%

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

      1. times-frac 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in k around 0 56.8%

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

   if -1.999999999999994e-310 < t < 1.1e-76

   1. Initial program 26.0%

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

   3. Simplified 28.5%

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

      1. associate-*l* 26.0%

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

      2. +-rgt-identity 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. add-sqr-sqrt 26.1%

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

      7. pow2 26.1%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

   if 5.60000000000000037e102 < t < 3.1000000000000001e190

   1. Initial program 15.0%

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

   3. Simplified 30.5%

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

      1. associate-*l* 30.5%

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

      2. +-rgt-identity 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r* 30.5%

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

      5. *-commutative 30.5%

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

      6. add-sqr-sqrt 29.2%

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

      7. pow2 29.2%

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

   5. Applied egg-rr 51.2%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 3.1000000000000001e190 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 13: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ t_3 := \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot t_1\right)\right)}\\ t_4 := \sqrt{t_1}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+101}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t_4\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{{\left(t_4 \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cos k)))
        (t_2
         (/
          (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
          (* k (/ (/ k t) t))))
        (t_3 (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* (pow k 2.0) t_1)))))
        (t_4 (sqrt t_1)))
   (if (<= t -1.05e+101)
     t_3
     (if (<= t -7.2e-112)
       t_2
       (if (<= t -2e-310)
         t_3
         (if (<= t 1.35e-67)
           (/ 2.0 (pow (* (* (sin k) (/ k l)) t_4) 2.0))
           (if (<= t 5.6e+102)
             t_2
             (if (<= t 4.8e+191)
               (/ 2.0 (pow (* t_4 (/ (* k (sin k)) l)) 2.0))
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t / cos(k);
	double t_2 = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double t_3 = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(k, 2.0) * t_1)));
	double t_4 = sqrt(t_1);
	double tmp;
	if (t <= -1.05e+101) {
		tmp = t_3;
	} else if (t <= -7.2e-112) {
		tmp = t_2;
	} else if (t <= -2e-310) {
		tmp = t_3;
	} else if (t <= 1.35e-67) {
		tmp = 2.0 / pow(((sin(k) * (k / l)) * t_4), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = t_2;
	} else if (t <= 4.8e+191) {
		tmp = 2.0 / pow((t_4 * ((k * sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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) :: t_4
    real(8) :: tmp
    t_1 = t / cos(k)
    t_2 = (2.0d0 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    t_3 = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * ((k ** 2.0d0) * t_1)))
    t_4 = sqrt(t_1)
    if (t <= (-1.05d+101)) then
        tmp = t_3
    else if (t <= (-7.2d-112)) then
        tmp = t_2
    else if (t <= (-2d-310)) then
        tmp = t_3
    else if (t <= 1.35d-67) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) * t_4) ** 2.0d0)
    else if (t <= 5.6d+102) then
        tmp = t_2
    else if (t <= 4.8d+191) then
        tmp = 2.0d0 / ((t_4 * ((k * sin(k)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t / Math.cos(k);
	double t_2 = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	double t_3 = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(k, 2.0) * t_1)));
	double t_4 = Math.sqrt(t_1);
	double tmp;
	if (t <= -1.05e+101) {
		tmp = t_3;
	} else if (t <= -7.2e-112) {
		tmp = t_2;
	} else if (t <= -2e-310) {
		tmp = t_3;
	} else if (t <= 1.35e-67) {
		tmp = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * t_4), 2.0);
	} else if (t <= 5.6e+102) {
		tmp = t_2;
	} else if (t <= 4.8e+191) {
		tmp = 2.0 / Math.pow((t_4 * ((k * Math.sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t / math.cos(k)
	t_2 = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	t_3 = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (math.pow(k, 2.0) * t_1)))
	t_4 = math.sqrt(t_1)
	tmp = 0
	if t <= -1.05e+101:
		tmp = t_3
	elif t <= -7.2e-112:
		tmp = t_2
	elif t <= -2e-310:
		tmp = t_3
	elif t <= 1.35e-67:
		tmp = 2.0 / math.pow(((math.sin(k) * (k / l)) * t_4), 2.0)
	elif t <= 5.6e+102:
		tmp = t_2
	elif t <= 4.8e+191:
		tmp = 2.0 / math.pow((t_4 * ((k * math.sin(k)) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t / cos(k))
	t_2 = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)))
	t_3 = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((k ^ 2.0) * t_1))))
	t_4 = sqrt(t_1)
	tmp = 0.0
	if (t <= -1.05e+101)
		tmp = t_3;
	elseif (t <= -7.2e-112)
		tmp = t_2;
	elseif (t <= -2e-310)
		tmp = t_3;
	elseif (t <= 1.35e-67)
		tmp = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * t_4) ^ 2.0));
	elseif (t <= 5.6e+102)
		tmp = t_2;
	elseif (t <= 4.8e+191)
		tmp = Float64(2.0 / (Float64(t_4 * Float64(Float64(k * sin(k)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t / cos(k);
	t_2 = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	t_3 = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * ((k ^ 2.0) * t_1)));
	t_4 = sqrt(t_1);
	tmp = 0.0;
	if (t <= -1.05e+101)
		tmp = t_3;
	elseif (t <= -7.2e-112)
		tmp = t_2;
	elseif (t <= -2e-310)
		tmp = t_3;
	elseif (t <= 1.35e-67)
		tmp = 2.0 / (((sin(k) * (k / l)) * t_4) ^ 2.0);
	elseif (t <= 5.6e+102)
		tmp = t_2;
	elseif (t <= 4.8e+191)
		tmp = 2.0 / ((t_4 * ((k * sin(k)) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[t, -1.05e+101], t$95$3, If[LessEqual[t, -7.2e-112], t$95$2, If[LessEqual[t, -2e-310], t$95$3, If[LessEqual[t, 1.35e-67], N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+102], t$95$2, If[LessEqual[t, 4.8e+191], N[(2.0 / N[Power[N[(t$95$4 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.05e101 or -7.2000000000000002e-112 < t < -1.999999999999994e-310

   1. Initial program 17.5%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in t around 0 73.2%

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

      1. times-frac 73.0%

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

   6. Simplified 73.0%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-udef 10.5%

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

      3. div-inv 10.5%

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

      4. pow-flip 11.8%

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

      5. metadata-eval 11.8%

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

      6. associate-/l* 11.8%

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

   8. Applied egg-rr 11.8%

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

      1. expm1-def 57.0%

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

      2. expm1-log1p 73.6%

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

      3. associate-*l* 74.8%

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

      4. associate-/r/ 75.7%

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

      5. *-commutative 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in k around 0 65.2%

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

   if -1.05e101 < t < -7.2000000000000002e-112 or 1.35000000000000008e-67 < t < 5.60000000000000037e102

   1. Initial program 67.0%

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

      1. associate-/r* 66.9%

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

      2. associate-*l/ 66.9%

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

      3. associate--l+ 67.0%

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

   3. Simplified 67.0%

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

      1. unpow2 70.6%

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

      2. clear-num 70.6%

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

      3. un-div-inv 70.6%

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

   5. Applied egg-rr 67.0%

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

      1. associate-+r- 67.0%

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

      2. add-exp-log 66.5%

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

      3. log1p-udef 66.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 71.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.6%

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

      6. associate-/r/ 71.6%

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

   7. Applied egg-rr 71.6%

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

      1. associate-*l/ 71.7%

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

      2. *-commutative 71.7%

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

      3. associate-*l/ 71.6%

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

      4. associate-/r* 79.2%

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

      5. associate-*r/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -1.999999999999994e-310 < t < 1.35000000000000008e-67

   1. Initial program 26.0%

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

   3. Simplified 28.5%

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

      1. associate-*l* 26.0%

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

      2. +-rgt-identity 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. add-sqr-sqrt 26.1%

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

      7. pow2 26.1%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

   if 5.60000000000000037e102 < t < 4.79999999999999972e191

   1. Initial program 15.0%

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

   3. Simplified 30.5%

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

      1. associate-*l* 30.5%

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

      2. +-rgt-identity 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r* 30.5%

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

      5. *-commutative 30.5%

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

      6. add-sqr-sqrt 29.2%

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

      7. pow2 29.2%

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

   5. Applied egg-rr 51.2%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 4.79999999999999972e191 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 14: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot t_1\right)\right)}\\ t_3 := \sqrt{t_1}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t_3\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{{\left(t_3 \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cos k)))
        (t_2 (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* (pow k 2.0) t_1)))))
        (t_3 (sqrt t_1)))
   (if (<= t -5.2e+102)
     t_2
     (if (<= t -7.2e-112)
       (/
        2.0
        (/ (* (* (tan k) (* (sin k) (pow (/ k t) 2.0))) (/ (pow t 3.0) l)) l))
       (if (<= t 9.8e-307)
         t_2
         (if (<= t 2.75e-73)
           (/ 2.0 (pow (* (* (sin k) (/ k l)) t_3) 2.0))
           (if (<= t 9.5e+99)
             (/
              (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
              (* k (/ (/ k t) t)))
             (if (<= t 3e+193)
               (/ 2.0 (pow (* t_3 (/ (* k (sin k)) l)) 2.0))
               (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t / cos(k);
	double t_2 = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (pow(k, 2.0) * t_1)));
	double t_3 = sqrt(t_1);
	double tmp;
	if (t <= -5.2e+102) {
		tmp = t_2;
	} else if (t <= -7.2e-112) {
		tmp = 2.0 / (((tan(k) * (sin(k) * pow((k / t), 2.0))) * (pow(t, 3.0) / l)) / l);
	} else if (t <= 9.8e-307) {
		tmp = t_2;
	} else if (t <= 2.75e-73) {
		tmp = 2.0 / pow(((sin(k) * (k / l)) * t_3), 2.0);
	} else if (t <= 9.5e+99) {
		tmp = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	} else if (t <= 3e+193) {
		tmp = 2.0 / pow((t_3 * ((k * sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 / cos(k)
    t_2 = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * ((k ** 2.0d0) * t_1)))
    t_3 = sqrt(t_1)
    if (t <= (-5.2d+102)) then
        tmp = t_2
    else if (t <= (-7.2d-112)) then
        tmp = 2.0d0 / (((tan(k) * (sin(k) * ((k / t) ** 2.0d0))) * ((t ** 3.0d0) / l)) / l)
    else if (t <= 9.8d-307) then
        tmp = t_2
    else if (t <= 2.75d-73) then
        tmp = 2.0d0 / (((sin(k) * (k / l)) * t_3) ** 2.0d0)
    else if (t <= 9.5d+99) then
        tmp = (2.0d0 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    else if (t <= 3d+193) then
        tmp = 2.0d0 / ((t_3 * ((k * sin(k)) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t / Math.cos(k);
	double t_2 = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (Math.pow(k, 2.0) * t_1)));
	double t_3 = Math.sqrt(t_1);
	double tmp;
	if (t <= -5.2e+102) {
		tmp = t_2;
	} else if (t <= -7.2e-112) {
		tmp = 2.0 / (((Math.tan(k) * (Math.sin(k) * Math.pow((k / t), 2.0))) * (Math.pow(t, 3.0) / l)) / l);
	} else if (t <= 9.8e-307) {
		tmp = t_2;
	} else if (t <= 2.75e-73) {
		tmp = 2.0 / Math.pow(((Math.sin(k) * (k / l)) * t_3), 2.0);
	} else if (t <= 9.5e+99) {
		tmp = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	} else if (t <= 3e+193) {
		tmp = 2.0 / Math.pow((t_3 * ((k * Math.sin(k)) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t / math.cos(k)
	t_2 = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (math.pow(k, 2.0) * t_1)))
	t_3 = math.sqrt(t_1)
	tmp = 0
	if t <= -5.2e+102:
		tmp = t_2
	elif t <= -7.2e-112:
		tmp = 2.0 / (((math.tan(k) * (math.sin(k) * math.pow((k / t), 2.0))) * (math.pow(t, 3.0) / l)) / l)
	elif t <= 9.8e-307:
		tmp = t_2
	elif t <= 2.75e-73:
		tmp = 2.0 / math.pow(((math.sin(k) * (k / l)) * t_3), 2.0)
	elif t <= 9.5e+99:
		tmp = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	elif t <= 3e+193:
		tmp = 2.0 / math.pow((t_3 * ((k * math.sin(k)) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t / cos(k))
	t_2 = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64((k ^ 2.0) * t_1))))
	t_3 = sqrt(t_1)
	tmp = 0.0
	if (t <= -5.2e+102)
		tmp = t_2;
	elseif (t <= -7.2e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(sin(k) * (Float64(k / t) ^ 2.0))) * Float64((t ^ 3.0) / l)) / l));
	elseif (t <= 9.8e-307)
		tmp = t_2;
	elseif (t <= 2.75e-73)
		tmp = Float64(2.0 / (Float64(Float64(sin(k) * Float64(k / l)) * t_3) ^ 2.0));
	elseif (t <= 9.5e+99)
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)));
	elseif (t <= 3e+193)
		tmp = Float64(2.0 / (Float64(t_3 * Float64(Float64(k * sin(k)) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t / cos(k);
	t_2 = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * ((k ^ 2.0) * t_1)));
	t_3 = sqrt(t_1);
	tmp = 0.0;
	if (t <= -5.2e+102)
		tmp = t_2;
	elseif (t <= -7.2e-112)
		tmp = 2.0 / (((tan(k) * (sin(k) * ((k / t) ^ 2.0))) * ((t ^ 3.0) / l)) / l);
	elseif (t <= 9.8e-307)
		tmp = t_2;
	elseif (t <= 2.75e-73)
		tmp = 2.0 / (((sin(k) * (k / l)) * t_3) ^ 2.0);
	elseif (t <= 9.5e+99)
		tmp = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	elseif (t <= 3e+193)
		tmp = 2.0 / ((t_3 * ((k * sin(k)) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[t, -5.2e+102], t$95$2, If[LessEqual[t, -7.2e-112], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.8e-307], t$95$2, If[LessEqual[t, 2.75e-73], N[(2.0 / N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+99], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+193], N[(2.0 / N[Power[N[(t$95$3 * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -5.20000000000000013e102 or -7.2000000000000002e-112 < t < 9.8000000000000005e-307

   1. Initial program 17.5%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in t around 0 73.2%

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

      1. times-frac 73.0%

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

   6. Simplified 73.0%

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

      1. expm1-log1p-u 56.5%

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

      2. expm1-udef 10.5%

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

      3. div-inv 10.5%

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

      4. pow-flip 11.8%

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

      5. metadata-eval 11.8%

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

      6. associate-/l* 11.8%

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

   8. Applied egg-rr 11.8%

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

      1. expm1-def 57.0%

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

      2. expm1-log1p 73.6%

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

      3. associate-*l* 74.8%

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

      4. associate-/r/ 75.7%

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

      5. *-commutative 75.7%

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

   10. Simplified 75.7%

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

   11. Taylor expanded in k around 0 65.2%

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

   if -5.20000000000000013e102 < t < -7.2000000000000002e-112

   1. Initial program 66.1%

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

   3. Simplified 71.9%

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

      1. associate-*l* 71.9%

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

      2. +-rgt-identity 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r* 71.9%

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

      5. *-commutative 71.9%

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

      6. *-commutative 71.9%

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

      7. associate-/r* 77.2%

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

      8. associate-*r/ 85.7%

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

   5. Applied egg-rr 85.7%

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

   if 9.8000000000000005e-307 < t < 2.75000000000000003e-73

   1. Initial program 26.0%

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

   3. Simplified 28.5%

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

      1. associate-*l* 26.0%

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

      2. +-rgt-identity 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r* 26.0%

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

      5. *-commutative 26.0%

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

      6. add-sqr-sqrt 26.1%

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

      7. pow2 26.1%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.6%

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

      2. associate-/r/ 76.8%

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

   8. Simplified 76.8%

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

   if 2.75000000000000003e-73 < t < 9.49999999999999908e99

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. associate-*l/ 68.4%

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

      3. associate--l+ 68.4%

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

   3. Simplified 68.4%

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

      1. unpow2 68.4%

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

      2. clear-num 68.4%

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

      3. un-div-inv 68.4%

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

   5. Applied egg-rr 68.5%

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

      1. associate-+r- 68.5%

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

      2. add-exp-log 67.9%

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

      3. log1p-udef 67.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 70.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.3%

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

      6. associate-/r/ 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. associate-*l/ 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-*l/ 71.2%

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

      4. associate-/r* 76.8%

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

      5. associate-*r/ 81.5%

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

   9. Applied egg-rr 81.5%

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

   if 9.49999999999999908e99 < t < 3e193

   1. Initial program 15.0%

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

   3. Simplified 30.5%

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

      1. associate-*l* 30.5%

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

      2. +-rgt-identity 30.5%

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

      3. *-commutative 30.5%

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

      4. associate-*r* 30.5%

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

      5. *-commutative 30.5%

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

      6. add-sqr-sqrt 29.2%

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

      7. pow2 29.2%

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

   5. Applied egg-rr 51.2%

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

   6. Taylor expanded in k around inf 70.8%

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

   if 3e193 < t

   1. Initial program 8.1%

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

   3. Simplified 40.7%

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

      1. associate-*l* 40.7%

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

      2. +-rgt-identity 40.7%

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

      3. *-commutative 40.7%

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

      4. associate-*r* 40.7%

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

      5. *-commutative 40.7%

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

      6. add-sqr-sqrt 10.8%

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

      7. pow2 10.8%

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

   5. Applied egg-rr 19.1%

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

   6. Taylor expanded in k around 0 71.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({k}^{2} \cdot \frac{t}{\cos k}\right)\right)}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{{\left(\left(\sin k \cdot \frac{k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 15: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{{k}^{2}}{{\ell}^{2}}\\ \mathbf{if}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t}{\frac{\cos k}{0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (pow k 2.0) (pow l 2.0))))
   (if (<= k 9.5e-5)
     (/ 2.0 (* (pow k 2.0) (* t t_1)))
     (/ 2.0 (* t_1 (/ t (/ (cos k) (+ 0.5 (* -0.5 (cos (* 2.0 k)))))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow(k, 2.0) / pow(l, 2.0);
	double tmp;
	if (k <= 9.5e-5) {
		tmp = 2.0 / (pow(k, 2.0) * (t * t_1));
	} else {
		tmp = 2.0 / (t_1 * (t / (cos(k) / (0.5 + (-0.5 * cos((2.0 * k)))))));
	}
	return tmp;
}

Fortran

NOTE: l 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 = (k ** 2.0d0) / (l ** 2.0d0)
    if (k <= 9.5d-5) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * t_1))
    else
        tmp = 2.0d0 / (t_1 * (t / (cos(k) / (0.5d0 + ((-0.5d0) * cos((2.0d0 * k)))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(k, 2.0) / Math.pow(l, 2.0);
	double tmp;
	if (k <= 9.5e-5) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * t_1));
	} else {
		tmp = 2.0 / (t_1 * (t / (Math.cos(k) / (0.5 + (-0.5 * Math.cos((2.0 * k)))))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.pow(k, 2.0) / math.pow(l, 2.0)
	tmp = 0
	if k <= 9.5e-5:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * t_1))
	else:
		tmp = 2.0 / (t_1 * (t / (math.cos(k) / (0.5 + (-0.5 * math.cos((2.0 * k)))))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64((k ^ 2.0) / (l ^ 2.0))
	tmp = 0.0
	if (k <= 9.5e-5)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * t_1)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t / Float64(cos(k) / Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * k))))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = (k ^ 2.0) / (l ^ 2.0);
	tmp = 0.0;
	if (k <= 9.5e-5)
		tmp = 2.0 / ((k ^ 2.0) * (t * t_1));
	else
		tmp = 2.0 / (t_1 * (t / (cos(k) / (0.5 + (-0.5 * cos((2.0 * k)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9.5e-5], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t / N[(N[Cos[k], $MachinePrecision] / N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e-5

   1. Initial program 37.8%

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

   3. Simplified 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. Taylor expanded in t around 0 71.3%

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

      1. times-frac 70.8%

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

   6. Simplified 70.8%

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

      1. expm1-log1p-u 55.2%

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

      2. expm1-udef 20.6%

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

      3. div-inv 20.6%

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

      4. pow-flip 20.6%

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

      5. metadata-eval 20.6%

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

      6. associate-/l* 20.6%

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

   8. Applied egg-rr 20.6%

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

      1. expm1-def 54.9%

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

      2. expm1-log1p 70.4%

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

      3. associate-*l* 71.4%

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

      4. associate-/r/ 71.8%

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

      5. *-commutative 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in k around 0 63.6%

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

       1. associate-/l* 62.9%

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

       2. associate-/r/ 64.6%

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

   13. Simplified 64.6%

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

   if 9.5000000000000005e-5 < k

   1. Initial program 26.8%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in t around 0 69.2%

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

      1. times-frac 75.6%

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

   6. Simplified 75.6%

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

      1. unpow2 75.6%

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

      2. sin-mult 75.4%

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

   8. Applied egg-rr 75.4%

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

      1. div-sub 75.4%

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

      2. +-inverses 75.4%

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

      3. cos-0 75.4%

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

      4. metadata-eval 75.4%

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

      5. count-2 75.4%

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

      6. *-commutative 75.4%

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

   10. Simplified 75.4%

       \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\cos k}} \]
   11. Step-by-step derivation

       1. associate-*l/ 69.1%

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

       2. associate-/l* 69.0%

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

       3. div-inv 69.0%

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \frac{t}{\frac{\cos k}{0.5 - \color{blue}{\cos \left(k \cdot 2\right) \cdot \frac{1}{2}}}}}{{\ell}^{2}}} \]

       4. metadata-eval 69.0%

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

   12. Applied egg-rr 69.0%

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

   13. Taylor expanded in k around inf 69.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \cos k}}} \]
   14. Step-by-step derivation

       1. times-frac 75.4%

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

       2. associate-/l* 75.4%

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

       3. cancel-sign-sub-inv 75.4%

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

       4. metadata-eval 75.4%

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

   15. Simplified 75.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t}{\frac{\cos k}{0.5 + -0.5 \cdot \cos \left(2 \cdot k\right)}}}\\ \end{array} \]

Alternative 16: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\frac{t}{\cos k} \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-5)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
   (/
    2.0
    (*
     (pow k 2.0)
     (* (pow l -2.0) (* (/ t (cos k)) (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-5) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * ((t / cos(k)) * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 9.5d-5) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * ((t / cos(k)) * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-5) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * ((t / Math.cos(k)) * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 9.5e-5:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * ((t / math.cos(k)) * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-5)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(Float64(t / cos(k)) * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-5)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	else
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * ((t / cos(k)) * (0.5 - (cos((2.0 * k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 9.5e-5], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e-5

   1. Initial program 37.8%

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

   3. Simplified 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. Taylor expanded in t around 0 71.3%

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

      1. times-frac 70.8%

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

   6. Simplified 70.8%

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

      1. expm1-log1p-u 55.2%

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

      2. expm1-udef 20.6%

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

      3. div-inv 20.6%

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

      4. pow-flip 20.6%

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

      5. metadata-eval 20.6%

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

      6. associate-/l* 20.6%

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

   8. Applied egg-rr 20.6%

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

      1. expm1-def 54.9%

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

      2. expm1-log1p 70.4%

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

      3. associate-*l* 71.4%

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

      4. associate-/r/ 71.8%

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

      5. *-commutative 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in k around 0 63.6%

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

       1. associate-/l* 62.9%

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

       2. associate-/r/ 64.6%

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

   13. Simplified 64.6%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   if 9.5000000000000005e-5 < k

   1. Initial program 26.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 43.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 69.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 75.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 75.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 37.3%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 27.9%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 27.9%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 29.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 29.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 29.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 29.2%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 38.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 76.9%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 76.8%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 76.8%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 76.8%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 76.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]
   11. Step-by-step derivation

       1. unpow2 75.6%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}}{\cos k}} \]

       2. sin-mult 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}}{\cos k}} \]

   12. Applied egg-rr 76.6%

       \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}} \cdot \frac{t}{\cos k}\right)\right)} \]
   13. Step-by-step derivation

       1. div-sub 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}}{\cos k}} \]

       2. +-inverses 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

       3. cos-0 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

       4. metadata-eval 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

       5. count-2 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\cos k}} \]

       6. *-commutative 75.4%

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}{\cos k}} \]

   14. Simplified 76.6%

       \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)} \cdot \frac{t}{\cos k}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(\frac{t}{\cos k} \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 17: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -4.6e+101)
   (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* t (pow k 2.0)))))
   (if (<= t -1.35e-103)
     (/
      2.0
      (* (/ (pow t 3.0) (* l l)) (* (tan k) (* (sin k) (/ (/ k t) (/ t k))))))
     (if (<= t 2.5e-309)
       (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
       (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.6e+101) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (t * pow(k, 2.0))));
	} else if (t <= -1.35e-103) {
		tmp = 2.0 / ((pow(t, 3.0) / (l * l)) * (tan(k) * (sin(k) * ((k / t) / (t / k)))));
	} else if (t <= 2.5e-309) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 <= (-4.6d+101)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * (t * (k ** 2.0d0))))
    else if (t <= (-1.35d-103)) then
        tmp = 2.0d0 / (((t ** 3.0d0) / (l * l)) * (tan(k) * (sin(k) * ((k / t) / (t / k)))))
    else if (t <= 2.5d-309) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.6e+101) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (t * Math.pow(k, 2.0))));
	} else if (t <= -1.35e-103) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / (l * l)) * (Math.tan(k) * (Math.sin(k) * ((k / t) / (t / k)))));
	} else if (t <= 2.5e-309) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= -4.6e+101:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (t * math.pow(k, 2.0))))
	elif t <= -1.35e-103:
		tmp = 2.0 / ((math.pow(t, 3.0) / (l * l)) * (math.tan(k) * (math.sin(k) * ((k / t) / (t / k)))))
	elif t <= 2.5e-309:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= -4.6e+101)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(t * (k ^ 2.0)))));
	elseif (t <= -1.35e-103)
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / Float64(l * l)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64(k / t) / Float64(t / k))))));
	elseif (t <= 2.5e-309)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4.6e+101)
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * (t * (k ^ 2.0))));
	elseif (t <= -1.35e-103)
		tmp = 2.0 / (((t ^ 3.0) / (l * l)) * (tan(k) * (sin(k) * ((k / t) / (t / k)))));
	elseif (t <= 2.5e-309)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -4.6e+101], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.35e-103], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-309], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.6000000000000003e101

   1. Initial program 8.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 72.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 77.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 77.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 52.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 5.7%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 5.7%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 8.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 53.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 78.7%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 76.3%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 78.4%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 78.4%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 78.4%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]

   11. Taylor expanded in k around 0 75.2%

       \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   if -4.6000000000000003e101 < t < -1.35000000000000005e-103

   1. Initial program 66.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. unpow2 72.2%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)\right)} \]

      2. clear-num 72.2%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}\right)\right)\right)} \]

      3. un-div-inv 72.2%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}\right)\right)} \]

   5. Applied egg-rr 72.2%

      \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}\right)\right)} \]

   if -1.35000000000000005e-103 < t < 2.5000000000000022e-309

   1. Initial program 27.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 27.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 73.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 69.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 69.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.3%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 13.5%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 13.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 13.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 13.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 13.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 13.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 60.3%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 69.1%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 73.1%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 73.1%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 73.1%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 73.1%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]

   11. Taylor expanded in k around 0 50.8%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 54.6%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]

       2. associate-/r/ 54.7%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   13. Simplified 54.7%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   if 2.5000000000000022e-309 < t

   1. Initial program 31.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 43.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 25.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 25.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 41.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 18: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -4.3e+102)
   (/ 2.0 (* (pow k 2.0) (* (pow l -2.0) (* t (pow k 2.0)))))
   (if (<= t -7.2e-112)
     (/
      (/ 2.0 (/ (* (tan k) (/ (* (sin k) (pow t 3.0)) l)) l))
      (* k (/ (/ k t) t)))
     (if (<= t 6e-309)
       (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
       (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.3e+102) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(l, -2.0) * (t * pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = (2.0 / ((tan(k) * ((sin(k) * pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	} else if (t <= 6e-309) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 <= (-4.3d+102)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((l ** (-2.0d0)) * (t * (k ** 2.0d0))))
    else if (t <= (-7.2d-112)) then
        tmp = (2.0d0 / ((tan(k) * ((sin(k) * (t ** 3.0d0)) / l)) / l)) / (k * ((k / t) / t))
    else if (t <= 6d-309) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.3e+102) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(l, -2.0) * (t * Math.pow(k, 2.0))));
	} else if (t <= -7.2e-112) {
		tmp = (2.0 / ((Math.tan(k) * ((Math.sin(k) * Math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t));
	} else if (t <= 6e-309) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= -4.3e+102:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(l, -2.0) * (t * math.pow(k, 2.0))))
	elif t <= -7.2e-112:
		tmp = (2.0 / ((math.tan(k) * ((math.sin(k) * math.pow(t, 3.0)) / l)) / l)) / (k * ((k / t) / t))
	elif t <= 6e-309:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= -4.3e+102)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((l ^ -2.0) * Float64(t * (k ^ 2.0)))));
	elseif (t <= -7.2e-112)
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(sin(k) * (t ^ 3.0)) / l)) / l)) / Float64(k * Float64(Float64(k / t) / t)));
	elseif (t <= 6e-309)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4.3e+102)
		tmp = 2.0 / ((k ^ 2.0) * ((l ^ -2.0) * (t * (k ^ 2.0))));
	elseif (t <= -7.2e-112)
		tmp = (2.0 / ((tan(k) * ((sin(k) * (t ^ 3.0)) / l)) / l)) / (k * ((k / t) / t));
	elseif (t <= 6e-309)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -4.3e+102], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.2e-112], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(k / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-309], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.3000000000000001e102

   1. Initial program 8.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 14.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 72.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 77.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 77.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 52.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 5.7%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 5.7%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 8.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 8.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 53.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 78.7%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 76.3%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 78.4%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 78.4%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 78.4%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]

   11. Taylor expanded in k around 0 75.2%

       \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   if -4.3000000000000001e102 < t < -7.2000000000000002e-112

   1. Initial program 66.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l/ 66.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate--l+ 66.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]
   4. Step-by-step derivation

      1. unpow2 71.9%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)\right)} \]

      2. clear-num 71.9%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}\right)\right)\right)} \]

      3. un-div-inv 71.9%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}\right)\right)} \]

   5. Applied egg-rr 66.1%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{1 + \left(\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}} - 1\right)} \]
   6. Step-by-step derivation

      1. associate-+r- 66.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) - 1}} \]

      2. add-exp-log 65.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{e^{\log \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      3. log1p-udef 65.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{e^{\color{blue}{\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)}} - 1} \]

      4. expm1-udef 71.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}} \]

      5. expm1-log1p-u 71.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\frac{\frac{k}{t}}{\frac{t}{k}}}} \]

      6. associate-/r/ 71.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\frac{\frac{k}{t}}{t} \cdot k}} \]

   7. Applied egg-rr 71.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{\frac{\frac{k}{t}}{t} \cdot k}} \]
   8. Step-by-step derivation

      1. associate-*l/ 71.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k}}{\frac{\frac{k}{t}}{t} \cdot k} \]

      2. *-commutative 71.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\frac{\frac{k}{t}}{t} \cdot k} \]

      3. associate-*l/ 71.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\frac{\frac{k}{t}}{t} \cdot k} \]

      4. associate-/r* 80.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}}}{\frac{\frac{k}{t}}{t} \cdot k} \]

      5. associate-*r/ 80.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}}}{\frac{\frac{k}{t}}{t} \cdot k} \]

   9. Applied egg-rr 80.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}}}{\frac{\frac{k}{t}}{t} \cdot k} \]

   if -7.2000000000000002e-112 < t < 6.000000000000001e-309

   1. Initial program 25.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 73.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 69.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 69.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 59.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 14.5%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 14.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 14.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 14.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 14.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 14.5%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 59.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 69.3%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 73.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 73.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 73.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 73.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]

   11. Taylor expanded in k around 0 49.7%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 53.8%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]

       2. associate-/r/ 53.9%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   13. Simplified 53.9%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   if 6.000000000000001e-309 < t

   1. Initial program 31.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 43.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 25.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 25.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 41.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left(t \cdot {k}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k \cdot \frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{k \cdot \frac{\frac{k}{t}}{t}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 19: 66.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -2e-310)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow k 2.0) (pow l 2.0)))))
   (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-310) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(k, 2.0) / pow(l, 2.0))));
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 <= (-2d-310)) then
        tmp = 2.0d0 / ((k ** 2.0d0) * (t * ((k ** 2.0d0) / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-310) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(k, 2.0) / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= -2e-310:
		tmp = 2.0 / (math.pow(k, 2.0) * (t * (math.pow(k, 2.0) / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((k ^ 2.0) / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = 2.0 / ((k ^ 2.0) * (t * ((k ^ 2.0) / (l ^ 2.0))));
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -2e-310], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 74.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 74.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 74.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 54.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)\right)}} \]

      2. expm1-udef 10.5%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1}} \]

      3. div-inv 10.5%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      4. pow-flip 11.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      5. metadata-eval 11.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)} - 1} \]

      6. associate-/l* 11.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \color{blue}{\frac{t}{\frac{\cos k}{{\sin k}^{2}}}}\right)} - 1} \]

   8. Applied egg-rr 11.2%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 54.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)\right)}} \]

      2. expm1-log1p 75.0%

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {\ell}^{-2}\right) \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}}} \]

      3. associate-*l* 75.8%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \frac{t}{\frac{\cos k}{{\sin k}^{2}}}\right)}} \]

      4. associate-/r/ 76.3%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}\right)} \]

      5. *-commutative 76.3%

         \[\leadsto \frac{2}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \color{blue}{\left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)}\right)} \]

   10. Simplified 76.3%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({\ell}^{-2} \cdot \left({\sin k}^{2} \cdot \frac{t}{\cos k}\right)\right)}} \]

   11. Taylor expanded in k around 0 59.8%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 59.7%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]

       2. associate-/r/ 61.1%

          \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   13. Simplified 61.1%

       \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 31.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 43.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 25.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 25.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 41.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 20: 64.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -2e-310)
   (* (/ 2.0 t) (/ (pow l 2.0) (pow k 4.0)))
   (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t)) 2.0))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-310) {
		tmp = (2.0 / t) * (pow(l, 2.0) / pow(k, 4.0));
	} else {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: l 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 <= (-2d-310)) then
        tmp = (2.0d0 / t) * ((l ** 2.0d0) / (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2e-310) {
		tmp = (2.0 / t) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= -2e-310:
		tmp = (2.0 / t) * (math.pow(l, 2.0) / math.pow(k, 4.0))
	else:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t)), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(Float64(2.0 / t) * Float64((l ^ 2.0) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = (2.0 / t) * ((l ^ 2.0) / (k ^ 4.0));
	else
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -2e-310], N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 41.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in t around 0 74.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. times-frac 74.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   6. Simplified 74.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   7. Taylor expanded in k around 0 56.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. associate-/l* 57.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

      3. associate-/r/ 56.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

   9. Simplified 56.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

   10. Taylor expanded in t around 0 56.8%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-*r/ 56.8%

          \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

       2. *-commutative 56.8%

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

       3. times-frac 57.6%

          \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   12. Simplified 57.6%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 31.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 43.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 43.1%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 25.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 25.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 41.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 21: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \end{array} \]

FPCore

NOTE: l 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

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * (pow(l, 2.0) / (t * pow(k, 4.0)));
}

Fortran

NOTE: l 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

l = Math.abs(l);
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

l = abs(l)
def code(t, l, k):
	return 2.0 * (math.pow(l, 2.0) / (t * math.pow(k, 4.0)))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k ^ 4.0))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l ^ 2.0) / (t * (k ^ 4.0)));
end

Wolfram

NOTE: l 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 34.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 34.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. associate-*l/ 34.5%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate--l+ 34.6%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

3. Simplified 34.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

4. Taylor expanded in k around 0 55.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

5. Final simplification 55.8%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]

Alternative 22: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{2}{t} \cdot \left({\ell}^{2} \cdot {k}^{-4}\right) \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ 2.0 t) (* (pow l 2.0) (pow k -4.0))))

C

l = abs(l);
double code(double t, double l, double k) {
	return (2.0 / t) * (pow(l, 2.0) * pow(k, -4.0));
}

Fortran

NOTE: l 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 / t) * ((l ** 2.0d0) * (k ** (-4.0d0)))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return (2.0 / t) * (Math.pow(l, 2.0) * Math.pow(k, -4.0));
}

Python

l = abs(l)
def code(t, l, k):
	return (2.0 / t) * (math.pow(l, 2.0) * math.pow(k, -4.0))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(2.0 / t) * Float64((l ^ 2.0) * (k ^ -4.0)))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = (2.0 / t) * ((l ^ 2.0) * (k ^ -4.0));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.9%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in t around 0 70.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. times-frac 72.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

6. Simplified 72.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

7. Taylor expanded in k around 0 55.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
8. Step-by-step derivation

   1. *-commutative 55.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

   2. associate-/l* 56.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   3. associate-/r/ 54.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

9. Simplified 54.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

10. Taylor expanded in t around 0 55.8%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. associate-*r/ 55.8%

       \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

    2. *-commutative 55.8%

       \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 56.2%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

12. Simplified 56.2%

    \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 56.1%

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)} \]

    2. expm1-udef 55.0%

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1\right)} \]

    3. div-inv 55.0%

       \[\leadsto \frac{2}{t} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

    4. pow-flip 55.0%

       \[\leadsto \frac{2}{t} \cdot \left(e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

    5. metadata-eval 55.0%

       \[\leadsto \frac{2}{t} \cdot \left(e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

14. Applied egg-rr 55.0%

    \[\leadsto \frac{2}{t} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1\right)} \]
15. Step-by-step derivation

    1. expm1-def 56.1%

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)} \]

    2. expm1-log1p 56.2%

       \[\leadsto \frac{2}{t} \cdot \color{blue}{\left({\ell}^{2} \cdot {k}^{-4}\right)} \]

16. Simplified 56.2%

    \[\leadsto \frac{2}{t} \cdot \color{blue}{\left({\ell}^{2} \cdot {k}^{-4}\right)} \]

17. Final simplification 56.2%

    \[\leadsto \frac{2}{t} \cdot \left({\ell}^{2} \cdot {k}^{-4}\right) \]

Alternative 23: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ 2.0 t) (/ (pow l 2.0) (pow k 4.0))))

C

l = abs(l);
double code(double t, double l, double k) {
	return (2.0 / t) * (pow(l, 2.0) / pow(k, 4.0));
}

Fortran

NOTE: l 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 / t) * ((l ** 2.0d0) / (k ** 4.0d0))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return (2.0 / t) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
}

Python

l = abs(l)
def code(t, l, k):
	return (2.0 / t) * (math.pow(l, 2.0) / math.pow(k, 4.0))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(2.0 / t) * Float64((l ^ 2.0) / (k ^ 4.0)))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = (2.0 / t) * ((l ^ 2.0) / (k ^ 4.0));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 42.9%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in t around 0 70.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. times-frac 72.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

6. Simplified 72.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

7. Taylor expanded in k around 0 55.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
8. Step-by-step derivation

   1. *-commutative 55.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

   2. associate-/l* 56.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

   3. associate-/r/ 54.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

9. Simplified 54.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

10. Taylor expanded in t around 0 55.8%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. associate-*r/ 55.8%

       \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

    2. *-commutative 55.8%

       \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 56.2%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

12. Simplified 56.2%

    \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

13. Final simplification 56.2%

    \[\leadsto \frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}} \]

Reproduce

herbie shell --seed 2023315 
(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))))