Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 78.6%

Time: 32.9s

Alternatives: 19

Speedup: 2.0×

• 35.5% 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: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+156}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left({k}^{-2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+272}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 6.6e-156)
   (* 2.0 (/ (pow (/ l (pow k 2.0)) 2.0) t))
   (if (<= l 1.45e+156)
     (*
      2.0
      (/ (* (pow l 2.0) (* (pow k -2.0) (cos k))) (* t (pow (sin k) 2.0))))
     (if (<= l 9e+272)
       (* 2.0 (pow (* (/ k l) (* (sin k) (sqrt (/ t (cos k))))) -2.0))
       (/
        2.0
        (*
         (* (pow (/ k t) 2.0) (* (sin k) (tan k)))
         (pow (/ t (pow (cbrt l) 2.0)) 3.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.6e-156) {
		tmp = 2.0 * (pow((l / pow(k, 2.0)), 2.0) / t);
	} else if (l <= 1.45e+156) {
		tmp = 2.0 * ((pow(l, 2.0) * (pow(k, -2.0) * cos(k))) / (t * pow(sin(k), 2.0)));
	} else if (l <= 9e+272) {
		tmp = 2.0 * pow(((k / l) * (sin(k) * sqrt((t / cos(k))))), -2.0);
	} else {
		tmp = 2.0 / ((pow((k / t), 2.0) * (sin(k) * tan(k))) * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.6e-156) {
		tmp = 2.0 * (Math.pow((l / Math.pow(k, 2.0)), 2.0) / t);
	} else if (l <= 1.45e+156) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * (Math.pow(k, -2.0) * Math.cos(k))) / (t * Math.pow(Math.sin(k), 2.0)));
	} else if (l <= 9e+272) {
		tmp = 2.0 * Math.pow(((k / l) * (Math.sin(k) * Math.sqrt((t / Math.cos(k))))), -2.0);
	} else {
		tmp = 2.0 / ((Math.pow((k / t), 2.0) * (Math.sin(k) * Math.tan(k))) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 6.6e-156)
		tmp = Float64(2.0 * Float64((Float64(l / (k ^ 2.0)) ^ 2.0) / t));
	elseif (l <= 1.45e+156)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64((k ^ -2.0) * cos(k))) / Float64(t * (sin(k) ^ 2.0))));
	elseif (l <= 9e+272)
		tmp = Float64(2.0 * (Float64(Float64(k / l) * Float64(sin(k) * sqrt(Float64(t / cos(k))))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k / t) ^ 2.0) * Float64(sin(k) * tan(k))) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 6.6e-156], N[(2.0 * N[(N[Power[N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+156], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k, -2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e+272], N[(2.0 * N[Power[N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 6.5999999999999997e-156

   1. Initial program 23.7%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in k around 0 50.6%

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

      1. associate-/r* 50.7%

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

   6. Simplified 50.7%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 50.2%

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

      3. div-inv 49.6%

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

      4. pow-flip 49.6%

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

      5. metadata-eval 49.6%

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

   8. Applied egg-rr 49.6%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 50.1%

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

   10. Simplified 50.1%

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

       1. add-sqr-sqrt 50.1%

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

       2. pow2 50.1%

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

       3. sqrt-prod 50.1%

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

       4. unpow2 50.1%

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

       5. sqrt-prod 14.5%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 65.1%

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

       10. add-sqr-sqrt 65.1%

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

   12. Applied egg-rr 65.1%

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

   13. Taylor expanded in l around 0 65.1%

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

   if 6.5999999999999997e-156 < l < 1.45000000000000005e156

   1. Initial program 45.2%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in t around 0 84.4%

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

      1. associate-/r* 93.6%

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

   6. Simplified 93.6%

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

      1. expm1-log1p-u 85.2%

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

      2. expm1-udef 70.3%

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

      3. div-inv 70.3%

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

      4. pow-flip 70.3%

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

      5. metadata-eval 70.3%

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

   8. Applied egg-rr 70.3%

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

      1. expm1-def 85.1%

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

      2. expm1-log1p 93.6%

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

      3. associate-*l* 93.6%

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

   10. Simplified 93.6%

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

   if 1.45000000000000005e156 < l < 9.00000000000000059e272

   1. Initial program 26.4%

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

   3. Simplified 26.4%

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

      1. associate-*l* 26.4%

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

      2. +-rgt-identity 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-*r* 26.4%

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

      5. *-commutative 26.4%

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

      6. add-sqr-sqrt 13.0%

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

      7. pow2 13.0%

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

   5. Applied egg-rr 26.4%

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

   6. Taylor expanded in k around inf 47.5%

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

      1. associate-*l/ 39.5%

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

      2. associate-*l* 39.5%

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

   8. Simplified 39.5%

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

      1. expm1-log1p-u 38.1%

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

      2. expm1-udef 38.1%

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

      3. div-inv 38.1%

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

      4. pow-flip 38.1%

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

      5. associate-/l* 34.5%

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

      6. metadata-eval 34.5%

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

   10. Applied egg-rr 34.5%

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

       1. expm1-def 42.5%

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

       2. expm1-log1p 43.4%

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

       3. associate-/r/ 47.6%

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

   12. Simplified 47.6%

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

   if 9.00000000000000059e272 < l

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

      1. unpow3 45.5%

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

      2. times-frac 54.7%

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

      3. pow2 54.7%

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

   5. Applied egg-rr 54.7%

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

      1. add-cube-cbrt 54.6%

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

      2. pow3 54.7%

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

   7. Applied egg-rr 72.7%

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

      1. cube-prod 72.3%

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

      2. rem-cube-cbrt 72.3%

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

      3. associate-*r* 72.3%

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

      4. *-commutative 72.3%

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

      5. *-commutative 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+156}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left({k}^{-2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+272}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 2: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-258}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e-258)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= (* l l) 5e+271)
     (*
      2.0
      (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t (pow (sin k) 2.0))))
     (/
      2.0
      (pow
       (*
        (cbrt (* (tan k) (* (sin k) (pow (/ k t) 2.0))))
        (/ t (pow (cbrt l) 2.0)))
       3.0)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-258) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if ((l * l) <= 5e+271) {
		tmp = 2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow((cbrt((tan(k) * (sin(k) * pow((k / t), 2.0)))) * (t / pow(cbrt(l), 2.0))), 3.0);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-258) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if ((l * l) <= 5e+271) {
		tmp = 2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow((Math.cbrt((Math.tan(k) * (Math.sin(k) * Math.pow((k / t), 2.0)))) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-258)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (Float64(l * l) <= 5e+271)
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(sin(k) * (Float64(k / t) ^ 2.0)))) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-258], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+271], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999991e-258

   1. Initial program 30.3%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in k around 0 53.4%

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

      1. associate-/r* 54.9%

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

   6. Simplified 54.9%

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

      1. expm1-log1p-u 54.9%

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

      2. expm1-udef 54.0%

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

      3. div-inv 52.5%

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

      4. pow-flip 52.5%

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

      5. metadata-eval 52.5%

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

   8. Applied egg-rr 52.5%

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

      1. expm1-def 53.4%

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

      2. expm1-log1p 53.4%

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

   10. Simplified 53.4%

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

       1. add-sqr-sqrt 53.4%

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

       2. pow2 53.4%

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

       3. sqrt-prod 53.4%

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

       4. unpow2 53.4%

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

       5. sqrt-prod 37.2%

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

       6. add-sqr-sqrt 72.7%

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

       7. metadata-eval 72.7%

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

       8. pow-prod-up 72.7%

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

       9. sqrt-prod 84.5%

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

       10. add-sqr-sqrt 84.5%

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

   12. Applied egg-rr 84.5%

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

   if 1.99999999999999991e-258 < (*.f64 l l) < 5.0000000000000003e271

   1. Initial program 36.1%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in t around 0 84.8%

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

      1. associate-/r* 93.4%

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

   6. Simplified 93.4%

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

   if 5.0000000000000003e271 < (*.f64 l l)

   1. Initial program 20.9%

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

   3. Simplified 22.4%

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

      1. unpow3 22.4%

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

      2. times-frac 40.2%

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

      3. pow2 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. add-cube-cbrt 40.2%

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

      2. pow3 40.1%

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

   7. Applied egg-rr 61.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-258}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+271}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 3: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-154}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left({k}^{-2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+273}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{\sin k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.45e-154)
   (* 2.0 (/ (pow (/ l (pow k 2.0)) 2.0) t))
   (if (<= l 8.6e+155)
     (*
      2.0
      (/ (* (pow l 2.0) (* (pow k -2.0) (cos k))) (* t (pow (sin k) 2.0))))
     (if (<= l 1.65e+273)
       (* 2.0 (pow (* (/ k l) (* (sin k) (sqrt (/ t (cos k))))) -2.0))
       (/
        (/ 2.0 (* (tan k) (* (pow (/ t (cbrt l)) 3.0) (/ (sin k) l))))
        (* (/ k t) (/ k t)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.45e-154) {
		tmp = 2.0 * (pow((l / pow(k, 2.0)), 2.0) / t);
	} else if (l <= 8.6e+155) {
		tmp = 2.0 * ((pow(l, 2.0) * (pow(k, -2.0) * cos(k))) / (t * pow(sin(k), 2.0)));
	} else if (l <= 1.65e+273) {
		tmp = 2.0 * pow(((k / l) * (sin(k) * sqrt((t / cos(k))))), -2.0);
	} else {
		tmp = (2.0 / (tan(k) * (pow((t / cbrt(l)), 3.0) * (sin(k) / l)))) / ((k / t) * (k / t));
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.45e-154) {
		tmp = 2.0 * (Math.pow((l / Math.pow(k, 2.0)), 2.0) / t);
	} else if (l <= 8.6e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * (Math.pow(k, -2.0) * Math.cos(k))) / (t * Math.pow(Math.sin(k), 2.0)));
	} else if (l <= 1.65e+273) {
		tmp = 2.0 * Math.pow(((k / l) * (Math.sin(k) * Math.sqrt((t / Math.cos(k))))), -2.0);
	} else {
		tmp = (2.0 / (Math.tan(k) * (Math.pow((t / Math.cbrt(l)), 3.0) * (Math.sin(k) / l)))) / ((k / t) * (k / t));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.45e-154)
		tmp = Float64(2.0 * Float64((Float64(l / (k ^ 2.0)) ^ 2.0) / t));
	elseif (l <= 8.6e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64((k ^ -2.0) * cos(k))) / Float64(t * (sin(k) ^ 2.0))));
	elseif (l <= 1.65e+273)
		tmp = Float64(2.0 * (Float64(Float64(k / l) * Float64(sin(k) * sqrt(Float64(t / cos(k))))) ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) * Float64(sin(k) / l)))) / Float64(Float64(k / t) * Float64(k / t)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.45e-154], N[(2.0 * N[(N[Power[N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.6e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k, -2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+273], N[(2.0 * N[Power[N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.45e-154

   1. Initial program 23.7%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in k around 0 50.6%

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

      1. associate-/r* 50.7%

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

   6. Simplified 50.7%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 50.2%

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

      3. div-inv 49.6%

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

      4. pow-flip 49.6%

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

      5. metadata-eval 49.6%

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

   8. Applied egg-rr 49.6%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 50.1%

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

   10. Simplified 50.1%

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

       1. add-sqr-sqrt 50.1%

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

       2. pow2 50.1%

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

       3. sqrt-prod 50.1%

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

       4. unpow2 50.1%

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

       5. sqrt-prod 14.5%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 65.1%

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

       10. add-sqr-sqrt 65.1%

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

   12. Applied egg-rr 65.1%

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

   13. Taylor expanded in l around 0 65.1%

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

   if 1.45e-154 < l < 8.6000000000000005e155

   1. Initial program 45.2%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in t around 0 84.4%

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

      1. associate-/r* 93.6%

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

   6. Simplified 93.6%

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

      1. expm1-log1p-u 85.2%

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

      2. expm1-udef 70.3%

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

      3. div-inv 70.3%

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

      4. pow-flip 70.3%

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

      5. metadata-eval 70.3%

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

   8. Applied egg-rr 70.3%

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

      1. expm1-def 85.1%

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

      2. expm1-log1p 93.6%

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

      3. associate-*l* 93.6%

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

   10. Simplified 93.6%

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

   if 8.6000000000000005e155 < l < 1.64999999999999993e273

   1. Initial program 26.4%

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

   3. Simplified 26.4%

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

      1. associate-*l* 26.4%

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

      2. +-rgt-identity 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-*r* 26.4%

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

      5. *-commutative 26.4%

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

      6. add-sqr-sqrt 13.0%

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

      7. pow2 13.0%

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

   5. Applied egg-rr 26.4%

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

   6. Taylor expanded in k around inf 47.5%

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

      1. associate-*l/ 39.5%

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

      2. associate-*l* 39.5%

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

   8. Simplified 39.5%

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

      1. expm1-log1p-u 38.1%

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

      2. expm1-udef 38.1%

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

      3. div-inv 38.1%

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

      4. pow-flip 38.1%

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

      5. associate-/l* 34.5%

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

      6. metadata-eval 34.5%

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

   10. Applied egg-rr 34.5%

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

       1. expm1-def 42.5%

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

       2. expm1-log1p 43.4%

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

       3. associate-/r/ 47.6%

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

   12. Simplified 47.6%

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

   if 1.64999999999999993e273 < l

   1. Initial program 45.5%

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

      1. associate-/r* 45.5%

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

      2. associate-*l/ 45.5%

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

      3. associate--l+ 45.5%

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

   3. Simplified 45.5%

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

      1. times-frac 45.8%

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

   5. Applied egg-rr 45.8%

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

      1. +-commutative 45.8%

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

      2. associate-+l- 46.0%

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

      3. metadata-eval 46.0%

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

      4. --rgt-identity 46.0%

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

      5. unpow2 46.0%

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

   7. Applied egg-rr 46.0%

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

      1. add-cube-cbrt 46.0%

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

      2. pow3 46.0%

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

      3. cbrt-div 46.0%

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

      4. unpow3 46.0%

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

      5. add-cbrt-cube 54.5%

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

   9. Applied egg-rr 54.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-154}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{elif}\;\ell \leq 8.6 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left({k}^{-2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+273}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{\sin k}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]

Alternative 4: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.5e-5)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (*
    2.0
    (/
     (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
     (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.5e-5) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * (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 <= 6.5d-5) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 * ((((l ** 2.0d0) * cos(k)) / (k ** 2.0d0)) / (t * (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 <= 6.5e-5) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 6.5e-5:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 * (((math.pow(l, 2.0) * math.cos(k)) / math.pow(k, 2.0)) / (t * (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 <= 6.5e-5)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * 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 <= 6.5e-5)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 * ((((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / (t * (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, 6.5e-5], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.49999999999999943e-5

   1. Initial program 29.6%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in k around 0 57.8%

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

      1. associate-/r* 57.9%

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

   6. Simplified 57.9%

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

      1. expm1-log1p-u 57.6%

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

      2. expm1-udef 55.7%

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

      3. div-inv 55.1%

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

      4. pow-flip 55.2%

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

      5. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. expm1-def 57.1%

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

      2. expm1-log1p 57.4%

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

   10. Simplified 57.4%

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

       1. add-sqr-sqrt 57.4%

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

       2. pow2 57.4%

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

       3. sqrt-prod 57.4%

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

       4. unpow2 57.4%

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

       5. sqrt-prod 35.9%

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

       6. add-sqr-sqrt 65.3%

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

       7. metadata-eval 65.3%

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

       8. pow-prod-up 65.3%

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

       9. sqrt-prod 71.5%

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

       10. add-sqr-sqrt 71.6%

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

   12. Applied egg-rr 71.6%

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

   if 6.49999999999999943e-5 < k

   1. Initial program 32.6%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 66.9%

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

      1. associate-/r* 70.3%

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

   6. Simplified 70.3%

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

      1. unpow2 70.3%

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

      2. sin-mult 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. div-sub 69.8%

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

      2. +-inverses 69.8%

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

      3. cos-0 69.8%

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

      4. metadata-eval 69.8%

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

      5. count-2 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 1.9e-209)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= t 4.1e+216)
     (/ 2.0 (* (sin k) (* (tan k) (pow (* (/ k t) (/ (pow t 1.5) l)) 2.0))))
     (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.9e-209) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 4.1e+216) {
		tmp = 2.0 / (sin(k) * (tan(k) * pow(((k / t) * (pow(t, 1.5) / l)), 2.0)));
	} else {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 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 <= 1.9d-209) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 4.1d+216) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (((k / t) * ((t ** 1.5d0) / l)) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 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 <= 1.9e-209) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 4.1e+216) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(((k / t) * (Math.pow(t, 1.5) / l)), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 1.9e-209:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 4.1e+216:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * math.pow(((k / t) * (math.pow(t, 1.5) / l)), 2.0)))
	else:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 1.9e-209)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 4.1e+216)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * (Float64(Float64(k / t) * Float64((t ^ 1.5) / l)) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.9e-209)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 4.1e+216)
		tmp = 2.0 / (sin(k) * (tan(k) * (((k / t) * ((t ^ 1.5) / l)) ^ 2.0)));
	else
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, 1.9e-209], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+216], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(k / t), $MachinePrecision] * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-209

   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 36.7%

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

   4. Taylor expanded in k around 0 52.8%

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

      1. associate-/r* 53.5%

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

   6. Simplified 53.5%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 50.9%

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

      3. div-inv 50.3%

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

      4. pow-flip 50.3%

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

      5. metadata-eval 50.3%

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

   8. Applied egg-rr 50.3%

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

      1. expm1-def 52.7%

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

      2. expm1-log1p 52.9%

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

   10. Simplified 52.9%

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

       1. add-sqr-sqrt 52.9%

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

       2. pow2 52.9%

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

       3. sqrt-prod 52.9%

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

       4. unpow2 52.9%

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

       5. sqrt-prod 35.1%

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

       6. add-sqr-sqrt 62.2%

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

       7. metadata-eval 62.2%

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

       8. pow-prod-up 62.2%

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

       9. sqrt-prod 67.9%

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

       10. add-sqr-sqrt 67.9%

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

   12. Applied egg-rr 67.9%

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

   if 1.8999999999999999e-209 < t < 4.0999999999999998e216

   1. Initial program 42.1%

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

   3. Simplified 46.5%

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

      1. associate-*l* 46.5%

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

      2. +-rgt-identity 46.5%

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

      3. *-commutative 46.5%

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

      4. associate-*r* 46.5%

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

      5. *-commutative 46.5%

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

      6. add-sqr-sqrt 33.7%

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

      7. pow2 33.7%

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

   5. Applied egg-rr 59.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 58.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]

      3. associate-*l* 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{{\color{blue}{\left(\sqrt{\tan k \cdot \sin k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}}\right)} - 1 \]

      4. unpow-prod-down 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{\left(\sqrt{\tan k \cdot \sin k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}}\right)} - 1 \]

      5. pow2 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sqrt{\tan k \cdot \sin k} \cdot \sqrt{\tan k \cdot \sin k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      6. add-sqr-sqrt 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      7. *-commutative 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 79.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 87.1%

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

      3. associate-*l* 87.1%

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

   9. Simplified 87.1%

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

   if 4.0999999999999998e216 < t

   1. Initial program 26.3%

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

   3. Simplified 42.7%

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

      1. associate-*l* 42.7%

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

      2. +-rgt-identity 42.7%

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

      3. *-commutative 42.7%

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

      4. associate-*r* 42.7%

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

      5. *-commutative 42.7%

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

      6. add-sqr-sqrt 16.2%

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

      7. pow2 16.2%

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

   5. Applied egg-rr 17.1%

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

   6. Taylor expanded in k around inf 47.2%

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

      1. associate-*l/ 47.4%

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

      2. associate-*l* 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in k around 0 66.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 3.7e-209)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= t 4.1e+216)
     (/ 2.0 (* (* (sin k) (tan k)) (pow (* (/ k t) (/ (pow t 1.5) l)) 2.0)))
     (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.7e-209) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 4.1e+216) {
		tmp = 2.0 / ((sin(k) * tan(k)) * pow(((k / t) * (pow(t, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 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 <= 3.7d-209) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 4.1d+216) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (((k / t) * ((t ** 1.5d0) / l)) ** 2.0d0))
    else
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 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 <= 3.7e-209) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 4.1e+216) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow(((k / t) * (Math.pow(t, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 3.7e-209:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 4.1e+216:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * math.pow(((k / t) * (math.pow(t, 1.5) / l)), 2.0))
	else:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 3.7e-209)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 4.1e+216)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(Float64(k / t) * Float64((t ^ 1.5) / l)) ^ 2.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.7e-209)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 4.1e+216)
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k / t) * ((t ^ 1.5) / l)) ^ 2.0));
	else
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, 3.7e-209], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+216], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k / t), $MachinePrecision] * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.6999999999999998e-209

   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 36.7%

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

   4. Taylor expanded in k around 0 52.8%

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

      1. associate-/r* 53.5%

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

   6. Simplified 53.5%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 50.9%

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

      3. div-inv 50.3%

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

      4. pow-flip 50.3%

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

      5. metadata-eval 50.3%

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

   8. Applied egg-rr 50.3%

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

      1. expm1-def 52.7%

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

      2. expm1-log1p 52.9%

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

   10. Simplified 52.9%

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

       1. add-sqr-sqrt 52.9%

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

       2. pow2 52.9%

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

       3. sqrt-prod 52.9%

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

       4. unpow2 52.9%

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

       5. sqrt-prod 35.1%

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

       6. add-sqr-sqrt 62.2%

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

       7. metadata-eval 62.2%

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

       8. pow-prod-up 62.2%

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

       9. sqrt-prod 67.9%

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

       10. add-sqr-sqrt 67.9%

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

   12. Applied egg-rr 67.9%

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

   if 3.6999999999999998e-209 < t < 4.0999999999999998e216

   1. Initial program 42.1%

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

   3. Simplified 46.5%

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

      1. associate-*l* 46.5%

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

      2. +-rgt-identity 46.5%

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

      3. *-commutative 46.5%

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

      4. associate-*r* 46.5%

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

      5. *-commutative 46.5%

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

      6. add-sqr-sqrt 33.7%

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

      7. pow2 33.7%

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

   5. Applied egg-rr 59.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 58.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 51.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]

      3. associate-*l* 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{{\color{blue}{\left(\sqrt{\tan k \cdot \sin k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}}^{2}}\right)} - 1 \]

      4. unpow-prod-down 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{\left(\sqrt{\tan k \cdot \sin k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}}\right)} - 1 \]

      5. pow2 52.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sqrt{\tan k \cdot \sin k} \cdot \sqrt{\tan k \cdot \sin k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      6. add-sqr-sqrt 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      7. *-commutative 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 79.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)\right)} \]

      2. expm1-log1p 87.1%

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

   9. Simplified 87.1%

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

   if 4.0999999999999998e216 < t

   1. Initial program 26.3%

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

   3. Simplified 42.7%

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

      1. associate-*l* 42.7%

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

      2. +-rgt-identity 42.7%

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

      3. *-commutative 42.7%

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

      4. associate-*r* 42.7%

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

      5. *-commutative 42.7%

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

      6. add-sqr-sqrt 16.2%

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

      7. pow2 16.2%

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

   5. Applied egg-rr 17.1%

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

   6. Taylor expanded in k around inf 47.2%

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

      1. associate-*l/ 47.4%

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

      2. associate-*l* 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in k around 0 66.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-209}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 7: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t}{\cos k}} \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{t_1}^{3}}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (pow k -2.0))))
   (if (<= k 2.2e+17)
     (* 2.0 (/ (pow t_1 2.0) t))
     (if (<= k 5e+145)
       (* 2.0 (pow (* (sqrt (/ t (cos k))) (* (sin k) (/ k l))) -2.0))
       (* 2.0 (/ (pow (cbrt (pow t_1 3.0)) 2.0) t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = l * pow(k, -2.0);
	double tmp;
	if (k <= 2.2e+17) {
		tmp = 2.0 * (pow(t_1, 2.0) / t);
	} else if (k <= 5e+145) {
		tmp = 2.0 * pow((sqrt((t / cos(k))) * (sin(k) * (k / l))), -2.0);
	} else {
		tmp = 2.0 * (pow(cbrt(pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = l * Math.pow(k, -2.0);
	double tmp;
	if (k <= 2.2e+17) {
		tmp = 2.0 * (Math.pow(t_1, 2.0) / t);
	} else if (k <= 5e+145) {
		tmp = 2.0 * Math.pow((Math.sqrt((t / Math.cos(k))) * (Math.sin(k) * (k / l))), -2.0);
	} else {
		tmp = 2.0 * (Math.pow(Math.cbrt(Math.pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l * (k ^ -2.0))
	tmp = 0.0
	if (k <= 2.2e+17)
		tmp = Float64(2.0 * Float64((t_1 ^ 2.0) / t));
	elseif (k <= 5e+145)
		tmp = Float64(2.0 * (Float64(sqrt(Float64(t / cos(k))) * Float64(sin(k) * Float64(k / l))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64((cbrt((t_1 ^ 3.0)) ^ 2.0) / t));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.2e+17], N[(2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+145], N[(2.0 * N[Power[N[(N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2e17

   1. Initial program 29.6%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. associate-/r* 57.4%

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

   6. Simplified 57.4%

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

      1. expm1-log1p-u 57.1%

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

      2. expm1-udef 54.9%

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

      3. div-inv 54.4%

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

      4. pow-flip 54.4%

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

      5. metadata-eval 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 56.9%

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

   10. Simplified 56.9%

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

       1. add-sqr-sqrt 56.9%

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

       2. pow2 56.9%

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

       3. sqrt-prod 56.9%

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

       4. unpow2 56.9%

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

       5. sqrt-prod 35.6%

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

       6. add-sqr-sqrt 64.7%

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

       7. metadata-eval 64.7%

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

       8. pow-prod-up 64.7%

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

       9. sqrt-prod 70.8%

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

       10. add-sqr-sqrt 70.8%

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

   12. Applied egg-rr 70.8%

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

   if 2.2e17 < k < 4.99999999999999967e145

   1. Initial program 26.1%

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

   3. Simplified 48.8%

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

      1. associate-*l* 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.7%

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

      5. *-commutative 48.7%

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

      6. add-sqr-sqrt 19.7%

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

      7. pow2 19.7%

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

   5. Applied egg-rr 22.9%

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

   6. Taylor expanded in k around inf 48.0%

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

      1. associate-*l/ 48.2%

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

      2. associate-*l* 48.3%

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

   8. Simplified 48.3%

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

      1. div-inv 48.3%

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

      2. pow-flip 48.2%

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

      3. associate-/l* 48.0%

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

      4. metadata-eval 48.0%

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

   10. Applied egg-rr 48.0%

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

   11. Taylor expanded in k around inf 48.0%

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

       1. *-commutative 48.0%

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

       2. associate-*l/ 48.0%

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

       3. *-commutative 48.0%

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

   13. Simplified 48.0%

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

   if 4.99999999999999967e145 < k

   1. Initial program 38.0%

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

   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. Taylor expanded in k around 0 54.8%

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

      1. associate-/r* 54.8%

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

   6. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

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

      2. expm1-udef 54.8%

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

      3. div-inv 54.8%

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

      4. pow-flip 54.8%

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

      5. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. expm1-def 54.8%

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

      2. expm1-log1p 54.8%

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

   10. Simplified 54.8%

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

       1. add-sqr-sqrt 54.8%

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

       2. pow2 54.8%

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

       3. sqrt-prod 54.8%

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

       4. unpow2 54.8%

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

       5. sqrt-prod 30.3%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 58.2%

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

       10. add-sqr-sqrt 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. add-cbrt-cube 58.3%

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

       2. pow3 58.3%

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

   14. Applied egg-rr 58.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\sqrt{\frac{t}{\cos k}} \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{\left(\ell \cdot {k}^{-2}\right)}^{3}}\right)}^{2}}{t}\\ \end{array} \]

Alternative 8: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;k \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{t_1}^{3}}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (pow k -2.0))))
   (if (<= k 7.5e+16)
     (* 2.0 (/ (pow t_1 2.0) t))
     (if (<= k 3.1e+145)
       (* 2.0 (pow (* (/ k l) (* (sin k) (sqrt (/ t (cos k))))) -2.0))
       (* 2.0 (/ (pow (cbrt (pow t_1 3.0)) 2.0) t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = l * pow(k, -2.0);
	double tmp;
	if (k <= 7.5e+16) {
		tmp = 2.0 * (pow(t_1, 2.0) / t);
	} else if (k <= 3.1e+145) {
		tmp = 2.0 * pow(((k / l) * (sin(k) * sqrt((t / cos(k))))), -2.0);
	} else {
		tmp = 2.0 * (pow(cbrt(pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = l * Math.pow(k, -2.0);
	double tmp;
	if (k <= 7.5e+16) {
		tmp = 2.0 * (Math.pow(t_1, 2.0) / t);
	} else if (k <= 3.1e+145) {
		tmp = 2.0 * Math.pow(((k / l) * (Math.sin(k) * Math.sqrt((t / Math.cos(k))))), -2.0);
	} else {
		tmp = 2.0 * (Math.pow(Math.cbrt(Math.pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l * (k ^ -2.0))
	tmp = 0.0
	if (k <= 7.5e+16)
		tmp = Float64(2.0 * Float64((t_1 ^ 2.0) / t));
	elseif (k <= 3.1e+145)
		tmp = Float64(2.0 * (Float64(Float64(k / l) * Float64(sin(k) * sqrt(Float64(t / cos(k))))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64((cbrt((t_1 ^ 3.0)) ^ 2.0) / t));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 7.5e+16], N[(2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+145], N[(2.0 * N[Power[N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.5e16

   1. Initial program 29.6%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. associate-/r* 57.4%

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

   6. Simplified 57.4%

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

      1. expm1-log1p-u 57.1%

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

      2. expm1-udef 54.9%

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

      3. div-inv 54.4%

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

      4. pow-flip 54.4%

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

      5. metadata-eval 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 56.9%

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

   10. Simplified 56.9%

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

       1. add-sqr-sqrt 56.9%

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

       2. pow2 56.9%

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

       3. sqrt-prod 56.9%

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

       4. unpow2 56.9%

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

       5. sqrt-prod 35.6%

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

       6. add-sqr-sqrt 64.7%

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

       7. metadata-eval 64.7%

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

       8. pow-prod-up 64.7%

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

       9. sqrt-prod 70.8%

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

       10. add-sqr-sqrt 70.8%

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

   12. Applied egg-rr 70.8%

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

   if 7.5e16 < k < 3.09999999999999988e145

   1. Initial program 26.1%

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

   3. Simplified 48.8%

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

      1. associate-*l* 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.7%

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

      5. *-commutative 48.7%

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

      6. add-sqr-sqrt 19.7%

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

      7. pow2 19.7%

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

   5. Applied egg-rr 22.9%

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

   6. Taylor expanded in k around inf 48.0%

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

      1. associate-*l/ 48.2%

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

      2. associate-*l* 48.3%

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

   8. Simplified 48.3%

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

      1. expm1-log1p-u 47.6%

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

      2. expm1-udef 32.6%

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

      3. div-inv 32.6%

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

      4. pow-flip 32.6%

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

      5. associate-/l* 32.6%

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

      6. metadata-eval 32.6%

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

   10. Applied egg-rr 32.6%

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

       1. expm1-def 47.5%

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

       2. expm1-log1p 48.0%

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

       3. associate-/r/ 48.1%

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

   12. Simplified 48.1%

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

   if 3.09999999999999988e145 < k

   1. Initial program 38.0%

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

   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. Taylor expanded in k around 0 54.8%

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

      1. associate-/r* 54.8%

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

   6. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

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

      2. expm1-udef 54.8%

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

      3. div-inv 54.8%

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

      4. pow-flip 54.8%

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

      5. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. expm1-def 54.8%

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

      2. expm1-log1p 54.8%

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

   10. Simplified 54.8%

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

       1. add-sqr-sqrt 54.8%

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

       2. pow2 54.8%

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

       3. sqrt-prod 54.8%

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

       4. unpow2 54.8%

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

       5. sqrt-prod 30.3%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 58.2%

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

       10. add-sqr-sqrt 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. add-cbrt-cube 58.3%

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

       2. pow3 58.3%

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

   14. Applied egg-rr 58.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{\left(\ell \cdot {k}^{-2}\right)}^{3}}\right)}^{2}}{t}\\ \end{array} \]

Alternative 9: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k} \cdot \sqrt{\frac{\cos k}{t}}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{t_1}^{3}}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (pow k -2.0))))
   (if (<= k 2e+17)
     (* 2.0 (/ (pow t_1 2.0) t))
     (if (<= k 8.5e+145)
       (* 2.0 (pow (/ k (* (/ l (sin k)) (sqrt (/ (cos k) t)))) -2.0))
       (* 2.0 (/ (pow (cbrt (pow t_1 3.0)) 2.0) t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = l * pow(k, -2.0);
	double tmp;
	if (k <= 2e+17) {
		tmp = 2.0 * (pow(t_1, 2.0) / t);
	} else if (k <= 8.5e+145) {
		tmp = 2.0 * pow((k / ((l / sin(k)) * sqrt((cos(k) / t)))), -2.0);
	} else {
		tmp = 2.0 * (pow(cbrt(pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = l * Math.pow(k, -2.0);
	double tmp;
	if (k <= 2e+17) {
		tmp = 2.0 * (Math.pow(t_1, 2.0) / t);
	} else if (k <= 8.5e+145) {
		tmp = 2.0 * Math.pow((k / ((l / Math.sin(k)) * Math.sqrt((Math.cos(k) / t)))), -2.0);
	} else {
		tmp = 2.0 * (Math.pow(Math.cbrt(Math.pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l * (k ^ -2.0))
	tmp = 0.0
	if (k <= 2e+17)
		tmp = Float64(2.0 * Float64((t_1 ^ 2.0) / t));
	elseif (k <= 8.5e+145)
		tmp = Float64(2.0 * (Float64(k / Float64(Float64(l / sin(k)) * sqrt(Float64(cos(k) / t)))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64((cbrt((t_1 ^ 3.0)) ^ 2.0) / t));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e+17], N[(2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.5e+145], N[(2.0 * N[Power[N[(k / N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e17

   1. Initial program 29.6%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. associate-/r* 57.4%

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

   6. Simplified 57.4%

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

      1. expm1-log1p-u 57.1%

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

      2. expm1-udef 54.9%

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

      3. div-inv 54.4%

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

      4. pow-flip 54.4%

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

      5. metadata-eval 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 56.9%

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

   10. Simplified 56.9%

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

       1. add-sqr-sqrt 56.9%

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

       2. pow2 56.9%

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

       3. sqrt-prod 56.9%

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

       4. unpow2 56.9%

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

       5. sqrt-prod 35.6%

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

       6. add-sqr-sqrt 64.7%

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

       7. metadata-eval 64.7%

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

       8. pow-prod-up 64.7%

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

       9. sqrt-prod 70.8%

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

       10. add-sqr-sqrt 70.8%

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

   12. Applied egg-rr 70.8%

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

   if 2e17 < k < 8.49999999999999977e145

   1. Initial program 26.1%

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

   3. Simplified 48.8%

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

      1. associate-*l* 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.7%

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

      5. *-commutative 48.7%

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

      6. add-sqr-sqrt 19.7%

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

      7. pow2 19.7%

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

   5. Applied egg-rr 22.9%

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

   6. Taylor expanded in k around inf 48.0%

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

      1. associate-*l/ 48.2%

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

      2. associate-*l* 48.3%

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

   8. Simplified 48.3%

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

      1. div-inv 48.3%

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

      2. pow-flip 48.2%

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

      3. associate-/l* 48.0%

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

      4. metadata-eval 48.0%

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

   10. Applied egg-rr 48.0%

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

   11. Taylor expanded in l around 0 48.0%

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

   if 8.49999999999999977e145 < k

   1. Initial program 38.0%

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

   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. Taylor expanded in k around 0 54.8%

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

      1. associate-/r* 54.8%

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

   6. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

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

      2. expm1-udef 54.8%

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

      3. div-inv 54.8%

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

      4. pow-flip 54.8%

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

      5. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. expm1-def 54.8%

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

      2. expm1-log1p 54.8%

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

   10. Simplified 54.8%

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

       1. add-sqr-sqrt 54.8%

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

       2. pow2 54.8%

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

       3. sqrt-prod 54.8%

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

       4. unpow2 54.8%

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

       5. sqrt-prod 30.3%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 58.2%

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

       10. add-sqr-sqrt 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. add-cbrt-cube 58.3%

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

       2. pow3 58.3%

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

   14. Applied egg-rr 58.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+145}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k} \cdot \sqrt{\frac{\cos k}{t}}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{\left(\ell \cdot {k}^{-2}\right)}^{3}}\right)}^{2}}{t}\\ \end{array} \]

Alternative 10: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{t_1}^{3}}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (pow k -2.0))))
   (if (<= k 2.2e+17)
     (* 2.0 (/ (pow t_1 2.0) t))
     (if (<= k 1.4e+146)
       (* 2.0 (pow (/ k (/ l (* (sin k) (sqrt (/ t (cos k)))))) -2.0))
       (* 2.0 (/ (pow (cbrt (pow t_1 3.0)) 2.0) t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = l * pow(k, -2.0);
	double tmp;
	if (k <= 2.2e+17) {
		tmp = 2.0 * (pow(t_1, 2.0) / t);
	} else if (k <= 1.4e+146) {
		tmp = 2.0 * pow((k / (l / (sin(k) * sqrt((t / cos(k)))))), -2.0);
	} else {
		tmp = 2.0 * (pow(cbrt(pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = l * Math.pow(k, -2.0);
	double tmp;
	if (k <= 2.2e+17) {
		tmp = 2.0 * (Math.pow(t_1, 2.0) / t);
	} else if (k <= 1.4e+146) {
		tmp = 2.0 * Math.pow((k / (l / (Math.sin(k) * Math.sqrt((t / Math.cos(k)))))), -2.0);
	} else {
		tmp = 2.0 * (Math.pow(Math.cbrt(Math.pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l * (k ^ -2.0))
	tmp = 0.0
	if (k <= 2.2e+17)
		tmp = Float64(2.0 * Float64((t_1 ^ 2.0) / t));
	elseif (k <= 1.4e+146)
		tmp = Float64(2.0 * (Float64(k / Float64(l / Float64(sin(k) * sqrt(Float64(t / cos(k)))))) ^ -2.0));
	else
		tmp = Float64(2.0 * Float64((cbrt((t_1 ^ 3.0)) ^ 2.0) / t));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.2e+17], N[(2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+146], N[(2.0 * N[Power[N[(k / N[(l / N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2e17

   1. Initial program 29.6%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. associate-/r* 57.4%

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

   6. Simplified 57.4%

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

      1. expm1-log1p-u 57.1%

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

      2. expm1-udef 54.9%

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

      3. div-inv 54.4%

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

      4. pow-flip 54.4%

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

      5. metadata-eval 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 56.9%

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

   10. Simplified 56.9%

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

       1. add-sqr-sqrt 56.9%

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

       2. pow2 56.9%

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

       3. sqrt-prod 56.9%

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

       4. unpow2 56.9%

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

       5. sqrt-prod 35.6%

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

       6. add-sqr-sqrt 64.7%

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

       7. metadata-eval 64.7%

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

       8. pow-prod-up 64.7%

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

       9. sqrt-prod 70.8%

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

       10. add-sqr-sqrt 70.8%

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

   12. Applied egg-rr 70.8%

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

   if 2.2e17 < k < 1.4e146

   1. Initial program 26.1%

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

   3. Simplified 48.8%

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

      1. associate-*l* 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.7%

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

      5. *-commutative 48.7%

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

      6. add-sqr-sqrt 19.7%

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

      7. pow2 19.7%

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

   5. Applied egg-rr 22.9%

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

   6. Taylor expanded in k around inf 48.0%

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

      1. associate-*l/ 48.2%

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

      2. associate-*l* 48.3%

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

   8. Simplified 48.3%

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

      1. div-inv 48.3%

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

      2. pow-flip 48.2%

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

      3. associate-/l* 48.0%

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

      4. metadata-eval 48.0%

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

   10. Applied egg-rr 48.0%

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

   if 1.4e146 < k

   1. Initial program 38.0%

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

   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. Taylor expanded in k around 0 54.8%

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

      1. associate-/r* 54.8%

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

   6. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

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

      2. expm1-udef 54.8%

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

      3. div-inv 54.8%

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

      4. pow-flip 54.8%

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

      5. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. expm1-def 54.8%

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

      2. expm1-log1p 54.8%

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

   10. Simplified 54.8%

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

       1. add-sqr-sqrt 54.8%

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

       2. pow2 54.8%

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

       3. sqrt-prod 54.8%

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

       4. unpow2 54.8%

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

       5. sqrt-prod 30.3%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 58.2%

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

       10. add-sqr-sqrt 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. add-cbrt-cube 58.3%

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

       2. pow3 58.3%

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

   14. Applied egg-rr 58.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{\left(\ell \cdot {k}^{-2}\right)}^{3}}\right)}^{2}}{t}\\ \end{array} \]

Alternative 11: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{t_1}^{3}}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (pow k -2.0))))
   (if (<= k 2e+17)
     (* 2.0 (/ (pow t_1 2.0) t))
     (if (<= k 2.9e+147)
       (/ 2.0 (pow (/ (* k (* (sin k) (sqrt (/ t (cos k))))) l) 2.0))
       (* 2.0 (/ (pow (cbrt (pow t_1 3.0)) 2.0) t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = l * pow(k, -2.0);
	double tmp;
	if (k <= 2e+17) {
		tmp = 2.0 * (pow(t_1, 2.0) / t);
	} else if (k <= 2.9e+147) {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt((t / cos(k))))) / l), 2.0);
	} else {
		tmp = 2.0 * (pow(cbrt(pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = l * Math.pow(k, -2.0);
	double tmp;
	if (k <= 2e+17) {
		tmp = 2.0 * (Math.pow(t_1, 2.0) / t);
	} else if (k <= 2.9e+147) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt((t / Math.cos(k))))) / l), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(Math.cbrt(Math.pow(t_1, 3.0)), 2.0) / t);
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l * (k ^ -2.0))
	tmp = 0.0
	if (k <= 2e+17)
		tmp = Float64(2.0 * Float64((t_1 ^ 2.0) / t));
	elseif (k <= 2.9e+147)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(Float64(t / cos(k))))) / l) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((cbrt((t_1 ^ 3.0)) ^ 2.0) / t));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e+17], N[(2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.9e+147], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[Power[N[Power[t$95$1, 3.0], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e17

   1. Initial program 29.6%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in k around 0 57.3%

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

      1. associate-/r* 57.4%

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

   6. Simplified 57.4%

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

      1. expm1-log1p-u 57.1%

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

      2. expm1-udef 54.9%

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

      3. div-inv 54.4%

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

      4. pow-flip 54.4%

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

      5. metadata-eval 54.4%

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

   8. Applied egg-rr 54.4%

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

      1. expm1-def 56.5%

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

      2. expm1-log1p 56.9%

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

   10. Simplified 56.9%

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

       1. add-sqr-sqrt 56.9%

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

       2. pow2 56.9%

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

       3. sqrt-prod 56.9%

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

       4. unpow2 56.9%

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

       5. sqrt-prod 35.6%

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

       6. add-sqr-sqrt 64.7%

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

       7. metadata-eval 64.7%

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

       8. pow-prod-up 64.7%

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

       9. sqrt-prod 70.8%

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

       10. add-sqr-sqrt 70.8%

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

   12. Applied egg-rr 70.8%

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

   if 2e17 < k < 2.8999999999999998e147

   1. Initial program 26.1%

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

   3. Simplified 48.8%

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

      1. associate-*l* 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-*r* 48.7%

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

      5. *-commutative 48.7%

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

      6. add-sqr-sqrt 19.7%

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

      7. pow2 19.7%

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

   5. Applied egg-rr 22.9%

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

   6. Taylor expanded in k around inf 48.0%

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

      1. associate-*l/ 48.2%

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

      2. associate-*l* 48.3%

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

   8. Simplified 48.3%

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

   if 2.8999999999999998e147 < k

   1. Initial program 38.0%

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

   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. Taylor expanded in k around 0 54.8%

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

      1. associate-/r* 54.8%

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

   6. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

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

      2. expm1-udef 54.8%

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

      3. div-inv 54.8%

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

      4. pow-flip 54.8%

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

      5. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. expm1-def 54.8%

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

      2. expm1-log1p 54.8%

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

   10. Simplified 54.8%

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

       1. add-sqr-sqrt 54.8%

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

       2. pow2 54.8%

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

       3. sqrt-prod 54.8%

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

       4. unpow2 54.8%

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

       5. sqrt-prod 30.3%

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

       6. add-sqr-sqrt 58.3%

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

       7. metadata-eval 58.3%

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

       8. pow-prod-up 58.3%

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

       9. sqrt-prod 58.2%

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

       10. add-sqr-sqrt 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. add-cbrt-cube 58.3%

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

       2. pow3 58.3%

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

   14. Applied egg-rr 58.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{{\left(\ell \cdot {k}^{-2}\right)}^{3}}\right)}^{2}}{t}\\ \end{array} \]

Alternative 12: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 7.2e-297)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= t 9.2e-99)
     (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0))
     (if (<= t 1.45e+81)
       (/
        (/ 2.0 (* (tan k) (* (/ (sin k) l) (/ (pow t 3.0) l))))
        (* (/ k t) (/ k t)))
       (* 2.0 (pow (/ k (* (/ l k) (sqrt (/ 1.0 t)))) -2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.2e-297) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-99) {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 2.0);
	} else if (t <= 1.45e+81) {
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * (pow(t, 3.0) / l)))) / ((k / t) * (k / t));
	} else {
		tmp = 2.0 * pow((k / ((l / k) * sqrt((1.0 / 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 <= 7.2d-297) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 9.2d-99) then
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 2.0d0)
    else if (t <= 1.45d+81) then
        tmp = (2.0d0 / (tan(k) * ((sin(k) / l) * ((t ** 3.0d0) / l)))) / ((k / t) * (k / t))
    else
        tmp = 2.0d0 * ((k / ((l / k) * sqrt((1.0d0 / 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 <= 7.2e-297) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-99) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	} else if (t <= 1.45e+81) {
		tmp = (2.0 / (Math.tan(k) * ((Math.sin(k) / l) * (Math.pow(t, 3.0) / l)))) / ((k / t) * (k / t));
	} else {
		tmp = 2.0 * Math.pow((k / ((l / k) * Math.sqrt((1.0 / t)))), -2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 7.2e-297:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 9.2e-99:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	elif t <= 1.45e+81:
		tmp = (2.0 / (math.tan(k) * ((math.sin(k) / l) * (math.pow(t, 3.0) / l)))) / ((k / t) * (k / t))
	else:
		tmp = 2.0 * math.pow((k / ((l / k) * math.sqrt((1.0 / t)))), -2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 7.2e-297)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 9.2e-99)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	elseif (t <= 1.45e+81)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(sin(k) / l) * Float64((t ^ 3.0) / l)))) / Float64(Float64(k / t) * Float64(k / t)));
	else
		tmp = Float64(2.0 * (Float64(k / Float64(Float64(l / k) * sqrt(Float64(1.0 / t)))) ^ -2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7.2e-297)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 9.2e-99)
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	elseif (t <= 1.45e+81)
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * ((t ^ 3.0) / l)))) / ((k / t) * (k / t));
	else
		tmp = 2.0 * ((k / ((l / k) * sqrt((1.0 / 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, 7.2e-297], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-99], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+81], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(k / N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 7.19999999999999988e-297

   1. Initial program 27.3%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in k around 0 53.1%

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

      1. associate-/r* 53.8%

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

   6. Simplified 53.8%

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

      1. expm1-log1p-u 53.6%

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

      2. expm1-udef 51.6%

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

      3. div-inv 51.0%

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

      4. pow-flip 51.0%

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

      5. metadata-eval 51.0%

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

   8. Applied egg-rr 51.0%

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

      1. expm1-def 53.0%

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

      2. expm1-log1p 53.2%

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

   10. Simplified 53.2%

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

       1. add-sqr-sqrt 53.2%

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

       2. pow2 53.2%

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

       3. sqrt-prod 53.2%

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

       4. unpow2 53.2%

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

       5. sqrt-prod 37.0%

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

       6. add-sqr-sqrt 62.7%

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

       7. metadata-eval 62.7%

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

       8. pow-prod-up 62.7%

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

       9. sqrt-prod 69.0%

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

       10. add-sqr-sqrt 69.0%

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

   12. Applied egg-rr 69.0%

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

   if 7.19999999999999988e-297 < t < 9.1999999999999994e-99

   1. Initial program 21.0%

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

   3. Simplified 21.0%

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

      1. associate-*l* 21.0%

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

      2. +-rgt-identity 21.0%

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

      3. *-commutative 21.0%

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

      4. associate-*r* 21.0%

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

      5. *-commutative 21.0%

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

      6. add-sqr-sqrt 21.0%

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

      7. pow2 21.0%

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

   5. Applied egg-rr 57.9%

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

   6. Taylor expanded in k around inf 71.5%

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

      1. associate-*l/ 71.5%

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

      2. associate-*l* 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in k around 0 79.7%

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

   if 9.1999999999999994e-99 < t < 1.45e81

   1. Initial program 63.5%

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

      1. associate-/r* 61.2%

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

      2. associate-*l/ 61.2%

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

      3. associate--l+ 61.2%

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

   3. Simplified 61.2%

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

      1. times-frac 65.1%

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

   5. Applied egg-rr 65.1%

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

      1. +-commutative 65.1%

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

      2. associate-+l- 72.6%

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

      3. metadata-eval 72.6%

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

      4. --rgt-identity 72.6%

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

      5. unpow2 72.6%

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

   7. Applied egg-rr 72.6%

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

   if 1.45e81 < t

   1. Initial program 29.7%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-*l* 41.4%

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

      2. +-rgt-identity 41.4%

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

      3. *-commutative 41.4%

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

      4. associate-*r* 41.4%

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

      5. *-commutative 41.4%

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

      6. add-sqr-sqrt 23.0%

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

      7. pow2 23.0%

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

   5. Applied egg-rr 37.0%

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

   6. Taylor expanded in k around inf 56.5%

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

      1. associate-*l/ 54.5%

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

      2. associate-*l* 56.8%

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

   8. Simplified 56.8%

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

      1. div-inv 56.8%

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

      2. pow-flip 56.8%

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

      3. associate-/l* 60.9%

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

      4. metadata-eval 60.9%

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

   10. Applied egg-rr 60.9%

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

   11. Taylor expanded in k around 0 65.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \]

Alternative 13: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}}{\frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 1.8e-296)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= t 6.2e-104)
     (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0))
     (if (<= t 7.8e+85)
       (/
        (/ 2.0 (* (tan k) (* (/ (sin k) l) (/ (pow t 3.0) l))))
        (/ k (* t (/ t k))))
       (* 2.0 (pow (/ k (* (/ l k) (sqrt (/ 1.0 t)))) -2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.8e-296) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 6.2e-104) {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 2.0);
	} else if (t <= 7.8e+85) {
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * (pow(t, 3.0) / l)))) / (k / (t * (t / k)));
	} else {
		tmp = 2.0 * pow((k / ((l / k) * sqrt((1.0 / 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 <= 1.8d-296) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 6.2d-104) then
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 2.0d0)
    else if (t <= 7.8d+85) then
        tmp = (2.0d0 / (tan(k) * ((sin(k) / l) * ((t ** 3.0d0) / l)))) / (k / (t * (t / k)))
    else
        tmp = 2.0d0 * ((k / ((l / k) * sqrt((1.0d0 / 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 <= 1.8e-296) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 6.2e-104) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	} else if (t <= 7.8e+85) {
		tmp = (2.0 / (Math.tan(k) * ((Math.sin(k) / l) * (Math.pow(t, 3.0) / l)))) / (k / (t * (t / k)));
	} else {
		tmp = 2.0 * Math.pow((k / ((l / k) * Math.sqrt((1.0 / t)))), -2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 1.8e-296:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 6.2e-104:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	elif t <= 7.8e+85:
		tmp = (2.0 / (math.tan(k) * ((math.sin(k) / l) * (math.pow(t, 3.0) / l)))) / (k / (t * (t / k)))
	else:
		tmp = 2.0 * math.pow((k / ((l / k) * math.sqrt((1.0 / t)))), -2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 1.8e-296)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 6.2e-104)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	elseif (t <= 7.8e+85)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(sin(k) / l) * Float64((t ^ 3.0) / l)))) / Float64(k / Float64(t * Float64(t / k))));
	else
		tmp = Float64(2.0 * (Float64(k / Float64(Float64(l / k) * sqrt(Float64(1.0 / t)))) ^ -2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.8e-296)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 6.2e-104)
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	elseif (t <= 7.8e+85)
		tmp = (2.0 / (tan(k) * ((sin(k) / l) * ((t ^ 3.0) / l)))) / (k / (t * (t / k)));
	else
		tmp = 2.0 * ((k / ((l / k) * sqrt((1.0 / 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, 1.8e-296], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-104], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e+85], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / N[(t * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(k / N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.7999999999999999e-296

   1. Initial program 27.3%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in k around 0 53.1%

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

      1. associate-/r* 53.8%

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

   6. Simplified 53.8%

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

      1. expm1-log1p-u 53.6%

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

      2. expm1-udef 51.6%

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

      3. div-inv 51.0%

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

      4. pow-flip 51.0%

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

      5. metadata-eval 51.0%

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

   8. Applied egg-rr 51.0%

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

      1. expm1-def 53.0%

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

      2. expm1-log1p 53.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

   10. Simplified 53.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{t} \]

       2. pow2 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\sqrt{{\ell}^{2} \cdot {k}^{-4}}\right)}^{2}}}{t} \]

       3. sqrt-prod 53.2%

          \[\leadsto 2 \cdot \frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}\right)}}^{2}}{t} \]

       4. unpow2 53.2%

          \[\leadsto 2 \cdot \frac{{\left(\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       5. sqrt-prod 37.0%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       6. add-sqr-sqrt 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       7. metadata-eval 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{{k}^{\color{blue}{\left(-2 + -2\right)}}}\right)}^{2}}{t} \]

       8. pow-prod-up 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{\color{blue}{{k}^{-2} \cdot {k}^{-2}}}\right)}^{2}}{t} \]

       9. sqrt-prod 69.0%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{\left(\sqrt{{k}^{-2}} \cdot \sqrt{{k}^{-2}}\right)}\right)}^{2}}{t} \]

       10. add-sqr-sqrt 69.0%

           \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{{k}^{-2}}\right)}^{2}}{t} \]

   12. Applied egg-rr 69.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]

   if 1.7999999999999999e-296 < t < 6.19999999999999951e-104

   1. Initial program 21.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 21.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 21.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 21.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 21.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 21.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 21.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 21.4%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 21.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 56.7%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 70.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l/ 70.6%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

      2. associate-*l* 70.7%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}}{\ell}\right)}^{2}} \]

   8. Simplified 70.7%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}}^{2}} \]

   9. Taylor expanded in k around 0 79.1%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\color{blue}{t}}\right)}{\ell}\right)}^{2}} \]

   if 6.19999999999999951e-104 < t < 7.80000000000000067e85

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l/ 59.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate--l+ 59.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]
   4. Step-by-step derivation

      1. times-frac 66.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)} \]

   5. Applied egg-rr 66.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)} \]
   6. Step-by-step derivation

      1. +-commutative 66.4%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} - 1\right) + 1}} \]

      2. associate-+l- 73.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} - \left(1 - 1\right)}} \]

      3. metadata-eval 73.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}} \]

      4. --rgt-identity 73.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      5. unpow2 73.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      6. clear-num 73.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t}} \]

      7. frac-times 73.7%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{1 \cdot k}{\frac{t}{k} \cdot t}}} \]

      8. *-un-lft-identity 73.7%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\frac{\color{blue}{k}}{\frac{t}{k} \cdot t}} \]

   7. Applied egg-rr 73.7%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{k}{\frac{t}{k} \cdot t}}} \]

   if 7.80000000000000067e85 < t

   1. Initial program 29.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 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. Step-by-step derivation

      1. associate-*l* 41.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 23.0%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 23.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 37.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 56.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l/ 54.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

      2. associate-*l* 56.8%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}}{\ell}\right)}^{2}} \]

   8. Simplified 56.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. div-inv 56.8%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{2}}} \]

      2. pow-flip 56.8%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{\left(-2\right)}} \]

      3. associate-/l* 60.9%

         \[\leadsto 2 \cdot {\color{blue}{\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}}^{\left(-2\right)} \]

      4. metadata-eval 60.9%

         \[\leadsto 2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{\color{blue}{-2}} \]

   10. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{-2}} \]

   11. Taylor expanded in k around 0 65.0%

       \[\leadsto 2 \cdot {\left(\frac{k}{\color{blue}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}}\right)}^{-2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-296}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)}}{\frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \]

Alternative 14: 72.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 5.4e-297)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (if (<= t 3.3e-99)
     (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0))
     (if (<= t 4.2e+84)
       (/
        (/ 2.0 (* (tan k) (/ (/ (* (sin k) (pow t 3.0)) l) l)))
        (* (/ k t) (/ k t)))
       (* 2.0 (pow (/ k (* (/ l k) (sqrt (/ 1.0 t)))) -2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.4e-297) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 3.3e-99) {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 2.0);
	} else if (t <= 4.2e+84) {
		tmp = (2.0 / (tan(k) * (((sin(k) * pow(t, 3.0)) / l) / l))) / ((k / t) * (k / t));
	} else {
		tmp = 2.0 * pow((k / ((l / k) * sqrt((1.0 / 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 <= 5.4d-297) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 3.3d-99) then
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 2.0d0)
    else if (t <= 4.2d+84) then
        tmp = (2.0d0 / (tan(k) * (((sin(k) * (t ** 3.0d0)) / l) / l))) / ((k / t) * (k / t))
    else
        tmp = 2.0d0 * ((k / ((l / k) * sqrt((1.0d0 / 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 <= 5.4e-297) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 3.3e-99) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	} else if (t <= 4.2e+84) {
		tmp = (2.0 / (Math.tan(k) * (((Math.sin(k) * Math.pow(t, 3.0)) / l) / l))) / ((k / t) * (k / t));
	} else {
		tmp = 2.0 * Math.pow((k / ((l / k) * Math.sqrt((1.0 / t)))), -2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 5.4e-297:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 3.3e-99:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	elif t <= 4.2e+84:
		tmp = (2.0 / (math.tan(k) * (((math.sin(k) * math.pow(t, 3.0)) / l) / l))) / ((k / t) * (k / t))
	else:
		tmp = 2.0 * math.pow((k / ((l / k) * math.sqrt((1.0 / t)))), -2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 5.4e-297)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 3.3e-99)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	elseif (t <= 4.2e+84)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(Float64(sin(k) * (t ^ 3.0)) / l) / l))) / Float64(Float64(k / t) * Float64(k / t)));
	else
		tmp = Float64(2.0 * (Float64(k / Float64(Float64(l / k) * sqrt(Float64(1.0 / t)))) ^ -2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.4e-297)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 3.3e-99)
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	elseif (t <= 4.2e+84)
		tmp = (2.0 / (tan(k) * (((sin(k) * (t ^ 3.0)) / l) / l))) / ((k / t) * (k / t));
	else
		tmp = 2.0 * ((k / ((l / k) * sqrt((1.0 / 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, 5.4e-297], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-99], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+84], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(k / N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5.4000000000000002e-297

   1. Initial program 27.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in k around 0 53.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   6. Simplified 53.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 53.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 51.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   8. Applied egg-rr 51.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   9. Step-by-step derivation

      1. expm1-def 53.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

      2. expm1-log1p 53.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

   10. Simplified 53.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{t} \]

       2. pow2 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\sqrt{{\ell}^{2} \cdot {k}^{-4}}\right)}^{2}}}{t} \]

       3. sqrt-prod 53.2%

          \[\leadsto 2 \cdot \frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}\right)}}^{2}}{t} \]

       4. unpow2 53.2%

          \[\leadsto 2 \cdot \frac{{\left(\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       5. sqrt-prod 37.0%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       6. add-sqr-sqrt 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       7. metadata-eval 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{{k}^{\color{blue}{\left(-2 + -2\right)}}}\right)}^{2}}{t} \]

       8. pow-prod-up 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{\color{blue}{{k}^{-2} \cdot {k}^{-2}}}\right)}^{2}}{t} \]

       9. sqrt-prod 69.0%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{\left(\sqrt{{k}^{-2}} \cdot \sqrt{{k}^{-2}}\right)}\right)}^{2}}{t} \]

       10. add-sqr-sqrt 69.0%

           \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{{k}^{-2}}\right)}^{2}}{t} \]

   12. Applied egg-rr 69.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]

   if 5.4000000000000002e-297 < t < 3.29999999999999986e-99

   1. Initial program 21.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 21.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 21.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 21.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 21.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 21.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 21.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 21.0%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 21.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 57.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 71.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l/ 71.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

      2. associate-*l* 71.6%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}}{\ell}\right)}^{2}} \]

   8. Simplified 71.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}}^{2}} \]

   9. Taylor expanded in k around 0 79.7%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\color{blue}{t}}\right)}{\ell}\right)}^{2}} \]

   if 3.29999999999999986e-99 < t < 4.20000000000000037e84

   1. Initial program 63.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-*l/ 61.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate--l+ 61.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]
   4. Step-by-step derivation

      1. times-frac 65.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)} \]

   5. Applied egg-rr 65.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k}}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)} \]
   6. Step-by-step derivation

      1. +-commutative 65.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} - 1\right) + 1}} \]

      2. associate-+l- 72.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} - \left(1 - 1\right)}} \]

      3. metadata-eval 72.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}} \]

      4. --rgt-identity 72.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      5. unpow2 72.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   7. Applied egg-rr 72.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]
   8. Step-by-step derivation

      1. frac-times 65.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}} \]

      2. associate-/r* 72.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}} \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}} \]

      3. *-commutative 72.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\sin k \cdot {t}^{3}}}{\ell}}{\ell} \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   9. Applied egg-rr 72.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}} \]

   if 4.20000000000000037e84 < t

   1. Initial program 29.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 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. Step-by-step derivation

      1. associate-*l* 41.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 41.4%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 23.0%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 23.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 37.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 56.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l/ 54.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

      2. associate-*l* 56.8%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}}{\ell}\right)}^{2}} \]

   8. Simplified 56.8%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}}^{2}} \]
   9. Step-by-step derivation

      1. div-inv 56.8%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{2}}} \]

      2. pow-flip 56.8%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}^{\left(-2\right)}} \]

      3. associate-/l* 60.9%

         \[\leadsto 2 \cdot {\color{blue}{\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}}^{\left(-2\right)} \]

      4. metadata-eval 60.9%

         \[\leadsto 2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{\color{blue}{-2}} \]

   10. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{2 \cdot {\left(\frac{k}{\frac{\ell}{\sin k \cdot \sqrt{\frac{t}{\cos k}}}}\right)}^{-2}} \]

   11. Taylor expanded in k around 0 65.0%

       \[\leadsto 2 \cdot {\left(\frac{k}{\color{blue}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}}\right)}^{-2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\frac{\sin k \cdot {t}^{3}}{\ell}}{\ell}}}{\frac{k}{t} \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{k}{\frac{\ell}{k} \cdot \sqrt{\frac{1}{t}}}\right)}^{-2}\\ \end{array} \]

Alternative 15: 71.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 6.5e-297)
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (/ 2.0 (pow (/ (* k (* (sin k) (sqrt t))) l) 2.0))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 6.5e-297) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / pow(((k * (sin(k) * sqrt(t))) / l), 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 <= 6.5d-297) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 / (((k * (sin(k) * sqrt(t))) / l) ** 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 <= 6.5e-297) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) * Math.sqrt(t))) / l), 2.0);
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if t <= 6.5e-297:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 / math.pow(((k * (math.sin(k) * math.sqrt(t))) / l), 2.0)
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (t <= 6.5e-297)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) * sqrt(t))) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 6.5e-297)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 / (((k * (sin(k) * sqrt(t))) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, 6.5e-297], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.5000000000000002e-297

   1. Initial program 27.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

   4. Taylor expanded in k around 0 53.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

   6. Simplified 53.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 53.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 51.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 51.0%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   8. Applied egg-rr 51.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   9. Step-by-step derivation

      1. expm1-def 53.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

      2. expm1-log1p 53.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

   10. Simplified 53.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{t} \]

       2. pow2 53.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\sqrt{{\ell}^{2} \cdot {k}^{-4}}\right)}^{2}}}{t} \]

       3. sqrt-prod 53.2%

          \[\leadsto 2 \cdot \frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}\right)}}^{2}}{t} \]

       4. unpow2 53.2%

          \[\leadsto 2 \cdot \frac{{\left(\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       5. sqrt-prod 37.0%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       6. add-sqr-sqrt 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

       7. metadata-eval 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{{k}^{\color{blue}{\left(-2 + -2\right)}}}\right)}^{2}}{t} \]

       8. pow-prod-up 62.7%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{\color{blue}{{k}^{-2} \cdot {k}^{-2}}}\right)}^{2}}{t} \]

       9. sqrt-prod 69.0%

          \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{\left(\sqrt{{k}^{-2}} \cdot \sqrt{{k}^{-2}}\right)}\right)}^{2}}{t} \]

       10. add-sqr-sqrt 69.0%

           \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{{k}^{-2}}\right)}^{2}}{t} \]

   12. Applied egg-rr 69.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]

   if 6.5000000000000002e-297 < t

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

      3. *-commutative 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\right)} \]

      4. associate-*r* 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}} \]

      5. *-commutative 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-sqr-sqrt 27.5%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow2 27.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{2}}} \]

   5. Applied egg-rr 48.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 61.2%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}}^{2}} \]
   7. Step-by-step derivation

      1. associate-*l/ 59.5%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sin k\right) \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}}^{2}} \]

      2. associate-*l* 60.5%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}}{\ell}\right)}^{2}} \]

   8. Simplified 60.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)}{\ell}\right)}}^{2}} \]

   9. Taylor expanded in k around 0 66.6%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{\color{blue}{t}}\right)}{\ell}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 16: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t)))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t);
}

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) * (k ** (-4.0d0))) / t)
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) * Math.pow(k, -4.0)) / t);
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t)

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t);
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in k around 0 54.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. associate-/r* 54.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

6. Simplified 54.7%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 54.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 52.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 52.4%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 52.5%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 52.5%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

8. Applied egg-rr 52.5%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
9. Step-by-step derivation

   1. expm1-def 54.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

   2. expm1-log1p 54.3%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

10. Simplified 54.3%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

11. Final simplification 54.3%

    \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \]

Alternative 17: 59.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \end{array} \]

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(Float64((l ^ 2.0) / 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[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in t around 0 67.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
5. Step-by-step derivation

   1. *-commutative 67.6%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   2. times-frac 71.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}}} \]

6. Simplified 71.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}}} \]
7. Step-by-step derivation

   1. associate-*l/ 71.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}}{\cos k}}} \]

   2. div-inv 70.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot \frac{1}{{\ell}^{2}}\right)}}{\cos k}} \]

   3. pow-flip 71.1%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)}{\cos k}} \]

   4. metadata-eval 71.1%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot {\ell}^{\color{blue}{-2}}\right)}{\cos k}} \]

8. Applied egg-rr 71.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot {\ell}^{-2}\right)}{\cos k}}} \]

9. Taylor expanded in k around 0 54.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
10. Step-by-step derivation

    1. *-commutative 54.6%

       \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

    2. associate-/r* 54.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

11. Simplified 54.5%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

12. Final simplification 54.5%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]

Alternative 18: 60.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t)))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t);
}

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) / (k ** 4.0d0)) / t)
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) / Math.pow(k, 4.0)) / t);
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t)

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t);
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in k around 0 54.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. associate-/r* 54.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

6. Simplified 54.7%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

7. Final simplification 54.7%

   \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t} \]

Alternative 19: 69.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t)))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
}

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 * (k ** (-2.0d0))) ** 2.0d0) / t)
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]

4. Taylor expanded in k around 0 54.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. associate-/r* 54.7%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]

6. Simplified 54.7%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 54.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 52.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 52.4%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 52.5%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 52.5%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

8. Applied egg-rr 52.5%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
9. Step-by-step derivation

   1. expm1-def 54.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

   2. expm1-log1p 54.3%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]

10. Simplified 54.3%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{-4}}}{t} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 54.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}} \cdot \sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{t} \]

    2. pow2 54.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\sqrt{{\ell}^{2} \cdot {k}^{-4}}\right)}^{2}}}{t} \]

    3. sqrt-prod 54.3%

       \[\leadsto 2 \cdot \frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}\right)}}^{2}}{t} \]

    4. unpow2 54.3%

       \[\leadsto 2 \cdot \frac{{\left(\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    5. sqrt-prod 33.5%

       \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    6. add-sqr-sqrt 60.2%

       \[\leadsto 2 \cdot \frac{{\left(\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}\right)}^{2}}{t} \]

    7. metadata-eval 60.2%

       \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{{k}^{\color{blue}{\left(-2 + -2\right)}}}\right)}^{2}}{t} \]

    8. pow-prod-up 60.2%

       \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \sqrt{\color{blue}{{k}^{-2} \cdot {k}^{-2}}}\right)}^{2}}{t} \]

    9. sqrt-prod 64.7%

       \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{\left(\sqrt{{k}^{-2}} \cdot \sqrt{{k}^{-2}}\right)}\right)}^{2}}{t} \]

    10. add-sqr-sqrt 64.8%

        \[\leadsto 2 \cdot \frac{{\left(\ell \cdot \color{blue}{{k}^{-2}}\right)}^{2}}{t} \]

12. Applied egg-rr 64.8%

    \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\ell \cdot {k}^{-2}\right)}^{2}}}{t} \]

13. Final simplification 64.8%

    \[\leadsto 2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t} \]

Reproduce

herbie shell --seed 2023299 
(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))))