Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 87.5%

Time: 16.8s

Alternatives: 12

Speedup: 28.1×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\\ \mathbf{if}\;k \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{{t_1}^{2}}{k} \cdot \frac{t_1}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (pow (cbrt l) 2.0) t)))
   (if (<= k 3.3e-20)
     (* (/ (pow t_1 2.0) k) (/ t_1 k))
     (/ 2.0 (* (* (/ k l) (* t (/ k l))) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0) / t;
	double tmp;
	if (k <= 3.3e-20) {
		tmp = (pow(t_1, 2.0) / k) * (t_1 / k);
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0) / t;
	double tmp;
	if (k <= 3.3e-20) {
		tmp = (Math.pow(t_1, 2.0) / k) * (t_1 / k);
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64((cbrt(l) ^ 2.0) / t)
	tmp = 0.0
	if (k <= 3.3e-20)
		tmp = Float64(Float64((t_1 ^ 2.0) / k) * Float64(t_1 / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[k, 3.3e-20], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / k), $MachinePrecision] * N[(t$95$1 / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3e-20

   1. Initial program 57.3%

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

      1. associate-/l/ 57.5%

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

      2. associate-*l/ 59.1%

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

      3. associate-*l/ 58.9%

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

      4. associate-/r/ 58.1%

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

      5. *-commutative 58.1%

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

      6. associate-/l/ 57.9%

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

      7. associate-*r* 57.9%

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

      8. *-commutative 57.9%

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

      9. associate-*r* 57.9%

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

      10. *-commutative 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in k around 0 56.6%

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

      1. unpow2 56.6%

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

      2. *-commutative 56.6%

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

      3. times-frac 60.0%

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

      4. unpow2 60.0%

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

   6. Simplified 60.0%

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

      1. associate-*r/ 59.0%

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

   8. Applied egg-rr 59.0%

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

      1. add-cube-cbrt 59.0%

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

      2. times-frac 65.0%

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

      3. pow2 65.0%

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

      4. associate-*l/ 62.0%

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

      5. cbrt-div 62.0%

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

      6. cbrt-prod 64.9%

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

      7. pow2 64.9%

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

      8. rem-cbrt-cube 64.9%

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

      9. associate-*l/ 62.0%

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

      10. cbrt-div 63.6%

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

      11. cbrt-prod 68.3%

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

      12. pow2 68.3%

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

      13. rem-cbrt-cube 74.8%

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

   10. Applied egg-rr 74.8%

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

   if 3.3e-20 < k

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-*l* 43.2%

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

      3. associate-*r* 43.2%

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

      4. +-commutative 43.2%

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

      5. associate-+r+ 43.2%

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

      6. metadata-eval 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in k around inf 70.3%

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

      1. unpow2 70.3%

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

      2. associate-*l* 76.7%

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

      3. unpow2 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in k around 0 70.3%

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

      1. unpow2 70.3%

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

      2. associate-*r* 76.7%

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

      3. unpow2 76.7%

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

      4. times-frac 89.5%

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

      5. associate-/l* 88.5%

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

   9. Simplified 88.5%

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

       1. associate-/r/ 93.3%

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

   11. Applied egg-rr 93.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2}}{k} \cdot \frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 2: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot t_1} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l)))) t_1))
        5e+277)
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* (tan k) t_1)))
     (/ 2.0 (* (* (/ k l) (* t (/ k l))) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+277) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))) * t_1)) <= 5d+277) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (tan(k) * t_1))
    else
        tmp = 2.0d0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+277) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (Math.tan(k) * t_1));
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * t_1)) <= 5e+277:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (math.tan(k) * t_1))
	else:
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * t_1)) <= 5e+277)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(tan(k) * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * t_1)) <= 5e+277)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	else
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 5e+277], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 4.99999999999999982e277

   1. Initial program 81.7%

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

      1. associate-*l* 81.7%

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

      2. +-commutative 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in t around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. unpow2 83.8%

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

      3. times-frac 86.4%

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

   6. Simplified 86.4%

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

   if 4.99999999999999982e277 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.4%

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

      1. *-commutative 20.4%

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

      2. associate-*l* 20.4%

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

      3. associate-*r* 20.4%

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

      4. +-commutative 20.4%

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

      5. associate-+r+ 20.4%

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

      6. metadata-eval 20.4%

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

   3. Simplified 20.4%

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

   4. Taylor expanded in k around inf 61.0%

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

      1. unpow2 61.0%

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

      2. associate-*l* 66.0%

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

      3. unpow2 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in k around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 83.6%

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

      5. associate-/l* 82.9%

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

   9. Simplified 82.9%

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

       1. associate-/r/ 89.2%

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 3: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
        (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
      5e+277)
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (/ 2.0 (* (* (/ k l) (* t (/ k l))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k / t), 2.0))))) <= 5e+277) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))))) <= 5d+277) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else
        tmp = 2.0d0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k / t), 2.0))))) <= 5e+277) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k / t), 2.0))))) <= 5e+277:
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	else:
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))) <= 5e+277)
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k / t) ^ 2.0))))) <= 5e+277)
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	else
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+277], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 4.99999999999999982e277

   1. Initial program 81.7%

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

      1. associate-/l/ 81.9%

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

      2. associate-*l/ 83.9%

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

      3. associate-*l/ 83.7%

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

      4. associate-/r/ 82.0%

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

      5. *-commutative 82.0%

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

      6. associate-/l/ 81.7%

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

      7. associate-*r* 81.8%

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

      8. *-commutative 81.8%

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

      9. associate-*r* 81.8%

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

      10. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in k around 0 72.8%

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

      1. unpow2 72.8%

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

      2. *-commutative 72.8%

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

      3. times-frac 73.7%

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

      4. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. associate-*r/ 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. associate-/r* 78.5%

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

      2. div-inv 78.5%

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

      3. *-commutative 78.5%

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

      4. div-inv 78.5%

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

      5. pow-flip 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. un-div-inv 79.1%

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

       2. associate-/l* 81.9%

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

   12. Applied egg-rr 81.9%

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

       1. associate-/l/ 82.3%

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

       2. associate-/r* 85.8%

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

   14. Simplified 85.8%

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

   if 4.99999999999999982e277 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.4%

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

      1. *-commutative 20.4%

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

      2. associate-*l* 20.4%

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

      3. associate-*r* 20.4%

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

      4. +-commutative 20.4%

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

      5. associate-+r+ 20.4%

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

      6. metadata-eval 20.4%

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

   3. Simplified 20.4%

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

   4. Taylor expanded in k around inf 61.0%

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

      1. unpow2 61.0%

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

      2. associate-*l* 66.0%

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

      3. unpow2 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in k around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-*r* 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 83.6%

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

      5. associate-/l* 82.9%

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

   9. Simplified 82.9%

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

       1. associate-/r/ 89.2%

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

   11. Applied egg-rr 89.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 4: 72.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e-22)
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-22) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-22) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-22) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e-22:
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e-22)
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e-22)
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	else
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e-22], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e-22

   1. Initial program 57.1%

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

      1. associate-/l/ 57.3%

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

      2. associate-*l/ 58.8%

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

      3. associate-*l/ 58.7%

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

      4. associate-/r/ 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-/l/ 57.7%

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

      7. associate-*r* 57.7%

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

      8. *-commutative 57.7%

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

      9. associate-*r* 57.7%

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

      10. *-commutative 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. unpow2 56.9%

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

      2. *-commutative 56.9%

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

      3. times-frac 60.3%

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

      4. unpow2 60.3%

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

   6. Simplified 60.3%

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

      1. associate-*r/ 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r* 65.3%

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

      2. div-inv 65.3%

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

      3. *-commutative 65.3%

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

      4. div-inv 65.3%

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

      5. pow-flip 65.7%

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

      6. metadata-eval 65.7%

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

   10. Applied egg-rr 65.7%

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

       1. un-div-inv 65.7%

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

       2. associate-/l* 67.9%

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

   12. Applied egg-rr 67.9%

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

       1. associate-/l/ 68.2%

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

       2. associate-/r* 70.8%

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

   14. Simplified 70.8%

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

   if 1.5e-22 < k

   1. Initial program 43.9%

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

      1. associate-/l/ 43.9%

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

      2. associate-*l/ 43.9%

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

      3. associate-*l/ 43.9%

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

      4. associate-/r/ 44.0%

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

      5. *-commutative 44.0%

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

      6. associate-/l/ 44.0%

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

      7. associate-*r* 44.0%

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

      8. *-commutative 44.0%

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

      9. associate-*r* 44.0%

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

      10. *-commutative 44.0%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in k around inf 70.7%

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

      1. unpow2 70.7%

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

   6. Simplified 70.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \]

Alternative 5: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell}} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 7.5e-23)
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (* (/ (/ 2.0 (* (sin k) (tan k))) (/ k l)) (/ l (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.5e-23) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else {
		tmp = ((2.0 / (sin(k) * tan(k))) / (k / l)) * (l / (k * t));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.5d-23) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else
        tmp = ((2.0d0 / (sin(k) * tan(k))) / (k / l)) * (l / (k * t))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.5e-23) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else {
		tmp = ((2.0 / (Math.sin(k) * Math.tan(k))) / (k / l)) * (l / (k * t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 7.5e-23:
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	else:
		tmp = ((2.0 / (math.sin(k) * math.tan(k))) / (k / l)) * (l / (k * t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 7.5e-23)
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / Float64(k / l)) * Float64(l / Float64(k * t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.5e-23)
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	else
		tmp = ((2.0 / (sin(k) * tan(k))) / (k / l)) * (l / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 7.5e-23], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.4999999999999998e-23

   1. Initial program 57.1%

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

      1. associate-/l/ 57.3%

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

      2. associate-*l/ 58.8%

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

      3. associate-*l/ 58.7%

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

      4. associate-/r/ 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-/l/ 57.7%

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

      7. associate-*r* 57.7%

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

      8. *-commutative 57.7%

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

      9. associate-*r* 57.7%

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

      10. *-commutative 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. unpow2 56.9%

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

      2. *-commutative 56.9%

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

      3. times-frac 60.3%

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

      4. unpow2 60.3%

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

   6. Simplified 60.3%

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

      1. associate-*r/ 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r* 65.3%

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

      2. div-inv 65.3%

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

      3. *-commutative 65.3%

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

      4. div-inv 65.3%

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

      5. pow-flip 65.7%

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

      6. metadata-eval 65.7%

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

   10. Applied egg-rr 65.7%

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

       1. un-div-inv 65.7%

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

       2. associate-/l* 67.9%

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

   12. Applied egg-rr 67.9%

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

       1. associate-/l/ 68.2%

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

       2. associate-/r* 70.8%

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

   14. Simplified 70.8%

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

   if 7.4999999999999998e-23 < k

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 43.9%

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

      3. associate-*r* 43.9%

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

      4. +-commutative 43.9%

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

      5. associate-+r+ 43.9%

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

      6. metadata-eval 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in k around inf 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*l* 77.0%

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

      3. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in k around 0 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*r* 77.0%

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

      3. unpow2 77.0%

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

      4. times-frac 89.7%

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

      5. associate-/l* 88.6%

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

   9. Simplified 88.6%

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

       1. associate-/r/ 93.4%

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

   11. Applied egg-rr 93.4%

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

       1. div-inv 93.4%

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

       2. *-commutative 93.4%

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

       3. associate-*l/ 89.7%

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

   13. Applied egg-rr 89.7%

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

       1. associate-*r/ 89.7%

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

       2. metadata-eval 89.7%

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

       3. associate-/r* 89.6%

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

       4. associate-/l* 88.6%

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

       5. associate-*r/ 81.1%

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

       6. associate-/l* 88.5%

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

       7. associate-/r/ 88.6%

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

       8. associate-/l/ 89.7%

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

   15. Simplified 89.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell}} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 6: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -4.2e-101)
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (if (<= t 6.5e-28)
     (/ 2.0 (* (* k k) (* (/ k l) (/ k (/ l t)))))
     (pow (/ (/ l k) (pow t 1.5)) 2.0))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.2e-101) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else if (t <= 6.5e-28) {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	} else {
		tmp = pow(((l / k) / pow(t, 1.5)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-4.2d-101)) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else if (t <= 6.5d-28) then
        tmp = 2.0d0 / ((k * k) * ((k / l) * (k / (l / t))))
    else
        tmp = ((l / k) / (t ** 1.5d0)) ** 2.0d0
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.2e-101) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else if (t <= 6.5e-28) {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	} else {
		tmp = Math.pow(((l / k) / Math.pow(t, 1.5)), 2.0);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -4.2e-101:
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	elif t <= 6.5e-28:
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))))
	else:
		tmp = math.pow(((l / k) / math.pow(t, 1.5)), 2.0)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -4.2e-101)
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	elseif (t <= 6.5e-28)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k / l) * Float64(k / Float64(l / t)))));
	else
		tmp = Float64(Float64(l / k) / (t ^ 1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -4.2e-101)
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	elseif (t <= 6.5e-28)
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	else
		tmp = ((l / k) / (t ^ 1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -4.2e-101], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-28], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.20000000000000031e-101

   1. Initial program 66.3%

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

      1. associate-/l/ 66.4%

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

      2. associate-*l/ 67.4%

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

      3. associate-*l/ 67.4%

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

      4. associate-/r/ 65.5%

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

      5. *-commutative 65.5%

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

      6. associate-/l/ 65.2%

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

      7. associate-*r* 65.2%

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

      8. *-commutative 65.2%

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

      9. associate-*r* 65.2%

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

      10. *-commutative 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in k around 0 57.9%

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

      1. unpow2 57.9%

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

      2. *-commutative 57.9%

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

      3. times-frac 61.0%

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

      4. unpow2 61.0%

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

   6. Simplified 61.0%

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

      1. associate-*r/ 59.8%

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

   8. Applied egg-rr 59.8%

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

      1. associate-/r* 65.0%

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

      2. div-inv 65.0%

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

      3. *-commutative 65.0%

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

      4. div-inv 65.0%

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

      5. pow-flip 65.0%

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

      6. metadata-eval 65.0%

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

   10. Applied egg-rr 65.0%

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

       1. un-div-inv 65.0%

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

       2. associate-/l* 69.4%

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

   12. Applied egg-rr 69.4%

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

       1. associate-/l/ 68.3%

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

       2. associate-/r* 70.2%

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

   14. Simplified 70.2%

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

   if -4.20000000000000031e-101 < t < 6.50000000000000043e-28

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*r* 33.7%

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

      4. +-commutative 33.7%

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

      5. associate-+r+ 33.7%

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

      6. metadata-eval 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in k around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*l* 83.2%

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

      3. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 83.2%

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

      3. unpow2 83.2%

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

      4. times-frac 97.5%

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

      5. associate-/l* 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in k around 0 69.0%

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

       1. unpow2 69.0%

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

   12. Simplified 69.0%

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

   if 6.50000000000000043e-28 < t

   1. Initial program 66.9%

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

      1. associate-/l/ 67.1%

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

      2. associate-*l/ 70.1%

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

      3. associate-*l/ 69.7%

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

      4. associate-/r/ 68.5%

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

      5. *-commutative 68.5%

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

      6. associate-/l/ 68.5%

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

      7. associate-*r* 68.5%

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

      8. *-commutative 68.5%

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

      9. associate-*r* 68.5%

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

      10. *-commutative 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in k around 0 59.0%

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

      1. unpow2 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 62.3%

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

      4. unpow2 62.3%

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

   6. Simplified 62.3%

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

      1. associate-*r/ 61.1%

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

   8. Applied egg-rr 61.1%

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

      1. expm1-log1p-u 61.1%

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

      2. expm1-udef 59.5%

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

      3. add-sqr-sqrt 59.5%

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

      4. pow2 59.5%

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

      5. sqrt-div 59.5%

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

      6. associate-*l/ 57.5%

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

      7. sqrt-div 57.6%

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

      8. sqrt-prod 34.5%

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

      9. add-sqr-sqrt 59.7%

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

      10. sqrt-pow1 64.0%

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

      11. metadata-eval 64.0%

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

      12. sqrt-prod 47.9%

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

      13. add-sqr-sqrt 74.4%

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

   10. Applied egg-rr 74.4%

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

       1. expm1-def 81.2%

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

       2. expm1-log1p 81.6%

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

       3. associate-/l/ 83.6%

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

       4. associate-/r* 83.5%

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

   12. Simplified 83.5%

       \[\leadsto \color{blue}{{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 7: 64.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left|k \cdot \frac{t}{\ell}\right|\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.6e-22)
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (/ 2.0 (* (* k k) (* (/ k l) (fabs (* k (/ t l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-22) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * fabs((k * (t / l)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.6d-22) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else
        tmp = 2.0d0 / ((k * k) * ((k / l) * abs((k * (t / l)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-22) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * Math.abs((k * (t / l)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.6e-22:
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	else:
		tmp = 2.0 / ((k * k) * ((k / l) * math.fabs((k * (t / l)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.6e-22)
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k / l) * abs(Float64(k * Float64(t / l))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.6e-22)
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	else
		tmp = 2.0 / ((k * k) * ((k / l) * abs((k * (t / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.6e-22], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[Abs[N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999994e-22

   1. Initial program 57.1%

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

      1. associate-/l/ 57.3%

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

      2. associate-*l/ 58.8%

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

      3. associate-*l/ 58.7%

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

      4. associate-/r/ 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-/l/ 57.7%

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

      7. associate-*r* 57.7%

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

      8. *-commutative 57.7%

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

      9. associate-*r* 57.7%

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

      10. *-commutative 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. unpow2 56.9%

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

      2. *-commutative 56.9%

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

      3. times-frac 60.3%

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

      4. unpow2 60.3%

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

   6. Simplified 60.3%

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

      1. associate-*r/ 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-/r* 65.3%

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

      2. div-inv 65.3%

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

      3. *-commutative 65.3%

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

      4. div-inv 65.3%

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

      5. pow-flip 65.7%

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

      6. metadata-eval 65.7%

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

   10. Applied egg-rr 65.7%

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

       1. un-div-inv 65.7%

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

       2. associate-/l* 67.9%

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

   12. Applied egg-rr 67.9%

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

       1. associate-/l/ 68.2%

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

       2. associate-/r* 70.8%

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

   14. Simplified 70.8%

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

   if 1.59999999999999994e-22 < k

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. associate-*l* 43.9%

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

      3. associate-*r* 43.9%

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

      4. +-commutative 43.9%

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

      5. associate-+r+ 43.9%

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

      6. metadata-eval 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in k around inf 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*l* 77.0%

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

      3. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in k around 0 70.7%

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

      1. unpow2 70.7%

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

      2. associate-*r* 77.0%

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

      3. unpow2 77.0%

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

      4. times-frac 89.7%

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

      5. associate-/l* 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in k around 0 54.7%

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

       1. unpow2 54.7%

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

   12. Simplified 54.7%

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

       1. associate-/l* 55.9%

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

       2. add-sqr-sqrt 30.3%

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

       3. sqrt-unprod 57.2%

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

       4. pow2 57.2%

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

       5. associate-/l* 57.1%

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

   14. Applied egg-rr 57.1%

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

       1. unpow2 57.1%

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

       2. rem-sqrt-square 55.9%

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

       3. associate-/l* 57.1%

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

       4. associate-*r/ 55.8%

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

   16. Simplified 55.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \left|k \cdot \frac{t}{\ell}\right|\right)}\\ \end{array} \]

Alternative 8: 68.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101} \lor \neg \left(t \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.2e-101) (not (<= t 6.5e-28)))
   (* (/ l k) (/ l (* k (pow t 3.0))))
   (/ 2.0 (* (* k k) (* (/ k l) (/ k (/ l t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-101) || !(t <= 6.5e-28)) {
		tmp = (l / k) * (l / (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-4.2d-101)) .or. (.not. (t <= 6.5d-28))) then
        tmp = (l / k) * (l / (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 / ((k * k) * ((k / l) * (k / (l / t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-101) || !(t <= 6.5e-28)) {
		tmp = (l / k) * (l / (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -4.2e-101) or not (t <= 6.5e-28):
		tmp = (l / k) * (l / (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.2e-101) || !(t <= 6.5e-28))
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * (t ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k / l) * Float64(k / Float64(l / t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -4.2e-101) || ~((t <= 6.5e-28)))
		tmp = (l / k) * (l / (k * (t ^ 3.0)));
	else
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -4.2e-101], N[Not[LessEqual[t, 6.5e-28]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.20000000000000031e-101 or 6.50000000000000043e-28 < t

   1. Initial program 66.5%

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

      1. associate-/l/ 66.7%

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

      2. associate-*l/ 68.6%

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

      3. associate-*l/ 68.4%

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

      4. associate-/r/ 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-/l/ 66.6%

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

      7. associate-*r* 66.6%

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

      8. *-commutative 66.6%

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

      9. associate-*r* 66.6%

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

      10. *-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 58.4%

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

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. times-frac 61.5%

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

      4. unpow2 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in l around 0 58.4%

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

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. unpow2 58.4%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. times-frac 72.7%

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

   9. Simplified 72.7%

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

   if -4.20000000000000031e-101 < t < 6.50000000000000043e-28

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*r* 33.7%

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

      4. +-commutative 33.7%

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

      5. associate-+r+ 33.7%

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

      6. metadata-eval 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in k around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*l* 83.2%

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

      3. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 83.2%

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

      3. unpow2 83.2%

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

      4. times-frac 97.5%

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

      5. associate-/l* 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in k around 0 69.0%

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

       1. unpow2 69.0%

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

   12. Simplified 69.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101} \lor \neg \left(t \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 9: 68.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101} \lor \neg \left(t \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.2e-101) (not (<= t 6.5e-28)))
   (/ l (* k (/ (/ k l) (pow t -3.0))))
   (/ 2.0 (* (* k k) (* (/ k l) (/ k (/ l t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-101) || !(t <= 6.5e-28)) {
		tmp = l / (k * ((k / l) / pow(t, -3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-4.2d-101)) .or. (.not. (t <= 6.5d-28))) then
        tmp = l / (k * ((k / l) / (t ** (-3.0d0))))
    else
        tmp = 2.0d0 / ((k * k) * ((k / l) * (k / (l / t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.2e-101) || !(t <= 6.5e-28)) {
		tmp = l / (k * ((k / l) / Math.pow(t, -3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -4.2e-101) or not (t <= 6.5e-28):
		tmp = l / (k * ((k / l) / math.pow(t, -3.0)))
	else:
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.2e-101) || !(t <= 6.5e-28))
		tmp = Float64(l / Float64(k * Float64(Float64(k / l) / (t ^ -3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k / l) * Float64(k / Float64(l / t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -4.2e-101) || ~((t <= 6.5e-28)))
		tmp = l / (k * ((k / l) / (t ^ -3.0)));
	else
		tmp = 2.0 / ((k * k) * ((k / l) * (k / (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -4.2e-101], N[Not[LessEqual[t, 6.5e-28]], $MachinePrecision]], N[(l / N[(k * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.20000000000000031e-101 or 6.50000000000000043e-28 < t

   1. Initial program 66.5%

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

      1. associate-/l/ 66.7%

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

      2. associate-*l/ 68.6%

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

      3. associate-*l/ 68.4%

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

      4. associate-/r/ 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-/l/ 66.6%

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

      7. associate-*r* 66.6%

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

      8. *-commutative 66.6%

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

      9. associate-*r* 66.6%

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

      10. *-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 58.4%

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

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. times-frac 61.5%

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

      4. unpow2 61.5%

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

   6. Simplified 61.5%

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

      1. associate-*r/ 60.3%

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

   8. Applied egg-rr 60.3%

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

      1. associate-/r* 66.7%

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

      2. div-inv 66.7%

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

      3. *-commutative 66.7%

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

      4. div-inv 66.7%

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

      5. pow-flip 67.2%

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

      6. metadata-eval 67.2%

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

   10. Applied egg-rr 67.2%

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

       1. un-div-inv 67.2%

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

       2. associate-/l* 69.8%

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

   12. Applied egg-rr 69.8%

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

       1. associate-/l/ 70.2%

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

       2. associate-/r* 73.2%

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

   14. Simplified 73.2%

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

   if -4.20000000000000031e-101 < t < 6.50000000000000043e-28

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-*r* 33.7%

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

      4. +-commutative 33.7%

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

      5. associate-+r+ 33.7%

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

      6. metadata-eval 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in k around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*l* 83.2%

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

      3. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 83.2%

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

      3. unpow2 83.2%

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

      4. times-frac 97.5%

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

      5. associate-/l* 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in k around 0 69.0%

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

       1. unpow2 69.0%

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

   12. Simplified 69.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-101} \lor \neg \left(t \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 10: 60.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 1.95e-67)
   (/ 2.0 (* (* k k) (/ (* k k) (* l (/ l t)))))
   (* (* l (pow t -3.0)) (/ l (* k k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.95e-67) {
		tmp = 2.0 / ((k * k) * ((k * k) / (l * (l / t))));
	} else {
		tmp = (l * pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.95d-67) then
        tmp = 2.0d0 / ((k * k) * ((k * k) / (l * (l / t))))
    else
        tmp = (l * (t ** (-3.0d0))) * (l / (k * k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.95e-67) {
		tmp = 2.0 / ((k * k) * ((k * k) / (l * (l / t))));
	} else {
		tmp = (l * Math.pow(t, -3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= 1.95e-67:
		tmp = 2.0 / ((k * k) * ((k * k) / (l * (l / t))))
	else:
		tmp = (l * math.pow(t, -3.0)) * (l / (k * k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= 1.95e-67)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * k) / Float64(l * Float64(l / t)))));
	else
		tmp = Float64(Float64(l * (t ^ -3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.95e-67)
		tmp = 2.0 / ((k * k) * ((k * k) / (l * (l / t))));
	else
		tmp = (l * (t ^ -3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, 1.95e-67], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9499999999999999e-67

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-*l* 45.4%

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

      3. associate-*r* 45.4%

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

      4. +-commutative 45.4%

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

      5. associate-+r+ 45.4%

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

      6. metadata-eval 45.4%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in k around inf 67.6%

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

      1. unpow2 67.6%

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

      2. associate-*l* 70.8%

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

      3. unpow2 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in k around 0 67.6%

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

      1. unpow2 67.6%

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

      2. associate-*r* 70.8%

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

      3. unpow2 70.8%

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

      4. times-frac 79.9%

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

      5. associate-/l* 81.1%

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

   9. Simplified 81.1%

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

   10. Taylor expanded in k around 0 63.5%

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

       1. unpow2 63.5%

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

   12. Simplified 63.5%

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

       1. frac-times 63.8%

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

   14. Applied egg-rr 63.8%

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

   if 1.9499999999999999e-67 < t

   1. Initial program 66.0%

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

      1. associate-/l/ 66.3%

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

      2. associate-*l/ 68.9%

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

      3. associate-*l/ 68.5%

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

      4. associate-/r/ 67.5%

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

      5. *-commutative 67.5%

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

      6. associate-/l/ 67.5%

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

      7. associate-*r* 67.5%

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

      8. *-commutative 67.5%

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

      9. associate-*r* 67.5%

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

      10. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in k around 0 59.3%

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

      1. unpow2 59.3%

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

      2. *-commutative 59.3%

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

      3. times-frac 62.2%

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

      4. unpow2 62.2%

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

   6. Simplified 62.2%

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

      1. expm1-log1p-u 59.4%

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

      2. expm1-udef 56.9%

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

      3. div-inv 56.9%

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

      4. pow-flip 56.9%

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

      5. metadata-eval 56.9%

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

   8. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k} \]
   9. Step-by-step derivation

      1. expm1-def 60.5%

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

      2. expm1-log1p 63.2%

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

   10. Simplified 63.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot {t}^{-3}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 11: 59.2% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((k * k) * ((k / l) * (k / (l / t))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. *-commutative 53.2%

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

   2. associate-*l* 49.5%

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

   3. associate-*r* 49.5%

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

   4. +-commutative 49.5%

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

   5. associate-+r+ 49.5%

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

   6. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in k around inf 62.9%

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

   1. unpow2 62.9%

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

   2. associate-*l* 65.2%

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

   3. unpow2 65.2%

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

6. Simplified 65.2%

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

7. Taylor expanded in k around 0 62.9%

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

   1. unpow2 62.9%

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

   2. associate-*r* 65.2%

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

   3. unpow2 65.2%

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

   4. times-frac 75.2%

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

   5. associate-/l* 76.8%

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

9. Simplified 76.8%

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

10. Taylor expanded in k around 0 61.3%

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

    1. unpow2 61.3%

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

12. Simplified 61.3%

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

13. Final simplification 61.3%

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

Alternative 12: 58.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((k * k) * ((k * k) / (l * (l / t))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. *-commutative 53.2%

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

   2. associate-*l* 49.5%

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

   3. associate-*r* 49.5%

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

   4. +-commutative 49.5%

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

   5. associate-+r+ 49.5%

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

   6. metadata-eval 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in k around inf 62.9%

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

   1. unpow2 62.9%

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

   2. associate-*l* 65.2%

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

   3. unpow2 65.2%

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

6. Simplified 65.2%

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

7. Taylor expanded in k around 0 62.9%

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

   1. unpow2 62.9%

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

   2. associate-*r* 65.2%

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

   3. unpow2 65.2%

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

   4. times-frac 75.2%

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

   5. associate-/l* 76.8%

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

9. Simplified 76.8%

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

10. Taylor expanded in k around 0 61.3%

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

    1. unpow2 61.3%

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

12. Simplified 61.3%

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

    1. frac-times 61.4%

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

14. Applied egg-rr 61.4%

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

15. Final simplification 61.4%

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

Reproduce

herbie shell --seed 2023213 
(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))))