Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 97.6%

Time: 23.7s

Alternatives: 10

Speedup: 3.6×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% 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: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{k}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 6e-216)
   (/ 2.0 (/ (* t (pow (* (/ k l) (sin k)) 2.0)) (cos k)))
   (/ 2.0 (* (/ (* t (/ k l)) (/ l k)) (/ (pow (sin k) 2.0) (cos k))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 6e-216) {
		tmp = 2.0 / ((t * pow(((k / l) * sin(k)), 2.0)) / cos(k));
	} else {
		tmp = 2.0 / (((t * (k / l)) / (l / k)) * (pow(sin(k), 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 6d-216) then
        tmp = 2.0d0 / ((t * (((k / l) * sin(k)) ** 2.0d0)) / cos(k))
    else
        tmp = 2.0d0 / (((t * (k / l)) / (l / k)) * ((sin(k) ** 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6e-216) {
		tmp = 2.0 / ((t * Math.pow(((k / l) * Math.sin(k)), 2.0)) / Math.cos(k));
	} else {
		tmp = 2.0 / (((t * (k / l)) / (l / k)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if l <= 6e-216:
		tmp = 2.0 / ((t * math.pow(((k / l) * math.sin(k)), 2.0)) / math.cos(k))
	else:
		tmp = 2.0 / (((t * (k / l)) / (l / k)) * (math.pow(math.sin(k), 2.0) / math.cos(k)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 6e-216)
		tmp = Float64(2.0 / Float64(Float64(t * (Float64(Float64(k / l) * sin(k)) ^ 2.0)) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / l)) / Float64(l / k)) * Float64((sin(k) ^ 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 6e-216)
		tmp = 2.0 / ((t * (((k / l) * sin(k)) ^ 2.0)) / cos(k));
	else
		tmp = 2.0 / (((t * (k / l)) / (l / k)) * ((sin(k) ^ 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 6e-216], N[(2.0 / N[(N[(t * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.00000000000000025e-216

   1. Initial program 29.8%

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

   2. Taylor expanded in t around 0 70.5%

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

      1. times-frac 70.6%

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

   4. Simplified 70.6%

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

      1. add-sqr-sqrt 70.6%

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

      2. sqrt-div 70.6%

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

      3. unpow2 70.6%

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

      4. sqrt-prod 38.2%

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

      5. add-sqr-sqrt 57.8%

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

      6. pow2 57.8%

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

      7. sqrt-prod 6.8%

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

      8. add-sqr-sqrt 46.8%

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

      9. sqrt-div 46.7%

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

      10. unpow2 46.7%

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

      11. sqrt-prod 14.3%

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

      12. add-sqr-sqrt 34.0%

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

      13. pow2 34.0%

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

      14. sqrt-prod 8.8%

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

      15. add-sqr-sqrt 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in k around inf 70.5%

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

      1. *-commutative 70.5%

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

      2. *-commutative 70.5%

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

      3. times-frac 70.6%

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

      4. associate-*l/ 70.5%

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

      5. unpow2 70.5%

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

      6. unpow2 70.5%

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

      7. times-frac 90.9%

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

      8. unpow2 90.9%

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

      9. associate-*l* 91.4%

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

   9. Simplified 91.4%

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

       1. associate-*l/ 91.4%

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

       2. *-commutative 91.4%

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

       3. pow-prod-down 95.3%

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

   11. Applied egg-rr 95.3%

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

   if 6.00000000000000025e-216 < l

   1. Initial program 36.8%

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

   2. Taylor expanded in t around 0 78.3%

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

      1. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. add-sqr-sqrt 82.4%

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

      2. sqrt-div 82.4%

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

      3. unpow2 82.4%

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

      4. sqrt-prod 41.5%

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

      5. add-sqr-sqrt 58.2%

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

      6. pow2 58.2%

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

      7. sqrt-prod 58.2%

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

      8. add-sqr-sqrt 58.2%

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

      9. sqrt-div 58.2%

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

      10. unpow2 58.2%

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

      11. sqrt-prod 41.6%

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

      12. add-sqr-sqrt 82.6%

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

      13. pow2 82.6%

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

      14. sqrt-prod 92.5%

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

      15. add-sqr-sqrt 92.6%

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

   6. Applied egg-rr 92.6%

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

      1. clear-num 92.6%

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

      2. clear-num 92.5%

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

      3. frac-times 92.5%

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

      4. metadata-eval 92.5%

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

   8. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. clear-num 92.5%

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

      3. associate-/r* 92.6%

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

      4. clear-num 92.6%

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

      5. frac-times 97.8%

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

      6. *-un-lft-identity 97.8%

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

   10. Applied egg-rr 97.8%

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

       1. *-commutative 97.8%

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

       2. associate-*r/ 97.8%

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

       3. associate-/l* 97.9%

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

       4. associate-*r* 98.8%

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

       5. times-frac 98.8%

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

       6. *-commutative 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{k}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 2: 92.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

2. Taylor expanded in t around 0 73.7%

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

   1. times-frac 75.5%

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

4. Simplified 75.5%

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

   1. add-sqr-sqrt 75.4%

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

   2. sqrt-div 75.5%

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

   3. unpow2 75.5%

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

   4. sqrt-prod 39.6%

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

   5. add-sqr-sqrt 57.9%

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

   6. pow2 57.9%

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

   7. sqrt-prod 28.1%

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

   8. add-sqr-sqrt 51.5%

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

   9. sqrt-div 51.5%

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

   10. unpow2 51.5%

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

   11. sqrt-prod 25.6%

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

   12. add-sqr-sqrt 54.1%

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

   13. pow2 54.1%

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

   14. sqrt-prod 43.5%

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

   15. add-sqr-sqrt 91.6%

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

6. Applied egg-rr 91.6%

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

7. Taylor expanded in k around inf 73.7%

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

   1. *-commutative 73.7%

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

   2. *-commutative 73.7%

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

   3. times-frac 75.5%

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

   4. associate-*l/ 75.5%

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

   5. unpow2 75.5%

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

   6. unpow2 75.5%

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

   7. times-frac 91.6%

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

   8. unpow2 91.6%

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

   9. associate-*l* 91.9%

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

9. Simplified 91.9%

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

    1. associate-*l/ 91.9%

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

    2. *-commutative 91.9%

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

    3. pow-prod-down 94.2%

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

11. Applied egg-rr 94.2%

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

12. Final simplification 94.2%

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

Alternative 3: 93.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.7%

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

2. Taylor expanded in t around 0 73.7%

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

   1. times-frac 75.5%

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

4. Simplified 75.5%

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

   1. add-sqr-sqrt 75.4%

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

   2. sqrt-div 75.5%

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

   3. unpow2 75.5%

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

   4. sqrt-prod 39.6%

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

   5. add-sqr-sqrt 57.9%

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

   6. pow2 57.9%

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

   7. sqrt-prod 28.1%

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

   8. add-sqr-sqrt 51.5%

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

   9. sqrt-div 51.5%

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

   10. unpow2 51.5%

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

   11. sqrt-prod 25.6%

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

   12. add-sqr-sqrt 54.1%

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

   13. pow2 54.1%

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

   14. sqrt-prod 43.5%

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

   15. add-sqr-sqrt 91.6%

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

6. Applied egg-rr 91.6%

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

7. Taylor expanded in k around inf 73.7%

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

   1. *-commutative 73.7%

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

   2. *-commutative 73.7%

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

   3. times-frac 75.5%

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

   4. associate-*l/ 75.5%

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

   5. unpow2 75.5%

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

   6. unpow2 75.5%

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

   7. times-frac 91.6%

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

   8. unpow2 91.6%

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

   9. associate-*l* 91.9%

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

9. Simplified 91.9%

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

    1. *-un-lft-identity 91.9%

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

    2. associate-/r* 92.3%

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

    3. *-commutative 92.3%

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

    4. pow-prod-down 94.5%

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

11. Applied egg-rr 94.5%

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

12. Final simplification 94.5%

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

Alternative 4: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\\ t_2 := t \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;k \leq 0.013:\\ \;\;\;\;\frac{2}{\frac{t_2}{\left(\ell \cdot -0.5 + \left({k}^{2} \cdot \left(\ell \cdot 0.041666666666666664 + \left(-0.3333333333333333 \cdot \left(\ell \cdot -0.3333333333333333 - \ell \cdot -0.5\right) - \ell \cdot 0.044444444444444446\right)\right) + \frac{\ell}{{k}^{2}}\right)\right) - \ell \cdot -0.3333333333333333}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\ell \cdot \frac{\cos k}{t_1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot t_1}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (t_2 (* t (* k (/ k l)))))
   (if (<= k 0.013)
     (/
      2.0
      (/
       t_2
       (-
        (+
         (* l -0.5)
         (+
          (*
           (pow k 2.0)
           (+
            (* l 0.041666666666666664)
            (-
             (* -0.3333333333333333 (- (* l -0.3333333333333333) (* l -0.5)))
             (* l 0.044444444444444446))))
          (/ l (pow k 2.0))))
        (* l -0.3333333333333333))))
     (if (<= k 7.2e+109)
       (/ 2.0 (/ t_2 (* l (/ (cos k) t_1))))
       (/ 2.0 (* (/ 1.0 (* (/ l k) (/ l k))) (/ (* t t_1) (cos k))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 0.5 - (cos((2.0 * k)) / 2.0);
	double t_2 = t * (k * (k / l));
	double tmp;
	if (k <= 0.013) {
		tmp = 2.0 / (t_2 / (((l * -0.5) + ((pow(k, 2.0) * ((l * 0.041666666666666664) + ((-0.3333333333333333 * ((l * -0.3333333333333333) - (l * -0.5))) - (l * 0.044444444444444446)))) + (l / pow(k, 2.0)))) - (l * -0.3333333333333333)));
	} else if (k <= 7.2e+109) {
		tmp = 2.0 / (t_2 / (l * (cos(k) / t_1)));
	} else {
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_1) / cos(k)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.5d0 - (cos((2.0d0 * k)) / 2.0d0)
    t_2 = t * (k * (k / l))
    if (k <= 0.013d0) then
        tmp = 2.0d0 / (t_2 / (((l * (-0.5d0)) + (((k ** 2.0d0) * ((l * 0.041666666666666664d0) + (((-0.3333333333333333d0) * ((l * (-0.3333333333333333d0)) - (l * (-0.5d0)))) - (l * 0.044444444444444446d0)))) + (l / (k ** 2.0d0)))) - (l * (-0.3333333333333333d0))))
    else if (k <= 7.2d+109) then
        tmp = 2.0d0 / (t_2 / (l * (cos(k) / t_1)))
    else
        tmp = 2.0d0 / ((1.0d0 / ((l / k) * (l / k))) * ((t * t_1) / cos(k)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 0.5 - (Math.cos((2.0 * k)) / 2.0);
	double t_2 = t * (k * (k / l));
	double tmp;
	if (k <= 0.013) {
		tmp = 2.0 / (t_2 / (((l * -0.5) + ((Math.pow(k, 2.0) * ((l * 0.041666666666666664) + ((-0.3333333333333333 * ((l * -0.3333333333333333) - (l * -0.5))) - (l * 0.044444444444444446)))) + (l / Math.pow(k, 2.0)))) - (l * -0.3333333333333333)));
	} else if (k <= 7.2e+109) {
		tmp = 2.0 / (t_2 / (l * (Math.cos(k) / t_1)));
	} else {
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_1) / Math.cos(k)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = 0.5 - (math.cos((2.0 * k)) / 2.0)
	t_2 = t * (k * (k / l))
	tmp = 0
	if k <= 0.013:
		tmp = 2.0 / (t_2 / (((l * -0.5) + ((math.pow(k, 2.0) * ((l * 0.041666666666666664) + ((-0.3333333333333333 * ((l * -0.3333333333333333) - (l * -0.5))) - (l * 0.044444444444444446)))) + (l / math.pow(k, 2.0)))) - (l * -0.3333333333333333)))
	elif k <= 7.2e+109:
		tmp = 2.0 / (t_2 / (l * (math.cos(k) / t_1)))
	else:
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_1) / math.cos(k)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))
	t_2 = Float64(t * Float64(k * Float64(k / l)))
	tmp = 0.0
	if (k <= 0.013)
		tmp = Float64(2.0 / Float64(t_2 / Float64(Float64(Float64(l * -0.5) + Float64(Float64((k ^ 2.0) * Float64(Float64(l * 0.041666666666666664) + Float64(Float64(-0.3333333333333333 * Float64(Float64(l * -0.3333333333333333) - Float64(l * -0.5))) - Float64(l * 0.044444444444444446)))) + Float64(l / (k ^ 2.0)))) - Float64(l * -0.3333333333333333))));
	elseif (k <= 7.2e+109)
		tmp = Float64(2.0 / Float64(t_2 / Float64(l * Float64(cos(k) / t_1))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 / Float64(Float64(l / k) * Float64(l / k))) * Float64(Float64(t * t_1) / cos(k))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = 0.5 - (cos((2.0 * k)) / 2.0);
	t_2 = t * (k * (k / l));
	tmp = 0.0;
	if (k <= 0.013)
		tmp = 2.0 / (t_2 / (((l * -0.5) + (((k ^ 2.0) * ((l * 0.041666666666666664) + ((-0.3333333333333333 * ((l * -0.3333333333333333) - (l * -0.5))) - (l * 0.044444444444444446)))) + (l / (k ^ 2.0)))) - (l * -0.3333333333333333)));
	elseif (k <= 7.2e+109)
		tmp = 2.0 / (t_2 / (l * (cos(k) / t_1)));
	else
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_1) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.013], N[(2.0 / N[(t$95$2 / N[(N[(N[(l * -0.5), $MachinePrecision] + N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(l * 0.041666666666666664), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(N[(l * -0.3333333333333333), $MachinePrecision] - N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * 0.044444444444444446), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.2e+109], N[(2.0 / N[(t$95$2 / N[(l * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 / N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t$95$1), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0129999999999999994

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. sqrt-div 75.6%

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

      3. unpow2 75.6%

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

      4. sqrt-prod 27.0%

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

      5. add-sqr-sqrt 51.8%

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

      6. pow2 51.8%

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

      7. sqrt-prod 24.7%

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

      8. add-sqr-sqrt 48.9%

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

      9. sqrt-div 48.9%

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

      10. unpow2 48.9%

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

      11. sqrt-prod 13.8%

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

      12. add-sqr-sqrt 52.4%

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

      13. pow2 52.4%

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

      14. sqrt-prod 44.6%

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

      15. add-sqr-sqrt 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate-*r/ 89.8%

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

      2. associate-/l* 89.8%

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

      3. frac-times 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in k around 0 62.5%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(-0.5 \cdot \ell + \left({k}^{2} \cdot \left(0.041666666666666664 \cdot \ell - \left(-0.3333333333333333 \cdot \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right) + 0.044444444444444446 \cdot \ell\right)\right) + \frac{\ell}{{k}^{2}}\right)\right) - -0.3333333333333333 \cdot \ell}}} \]

   if 0.0129999999999999994 < k < 7.2e109

   1. Initial program 17.3%

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

   2. Taylor expanded in t around 0 82.9%

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

      1. times-frac 74.9%

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

   4. Simplified 74.9%

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

      1. add-sqr-sqrt 74.9%

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

      2. sqrt-div 74.9%

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

      3. unpow2 74.9%

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

      4. sqrt-prod 74.9%

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

      5. add-sqr-sqrt 74.9%

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

      6. pow2 74.9%

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

      7. sqrt-prod 35.3%

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

      8. add-sqr-sqrt 48.8%

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

      9. sqrt-div 48.8%

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

      10. unpow2 48.8%

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

      11. sqrt-prod 48.8%

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

      12. add-sqr-sqrt 48.8%

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

      13. pow2 48.8%

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

      14. sqrt-prod 39.4%

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

      15. add-sqr-sqrt 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-/l* 83.2%

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

      3. frac-times 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. unpow2 83.2%

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

      2. sin-mult 83.3%

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

   10. Applied egg-rr 95.7%

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

       1. div-sub 83.3%

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

       2. +-inverses 83.3%

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

       3. cos-0 83.3%

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

       4. metadata-eval 83.3%

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

       5. count-2 83.3%

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

   12. Simplified 95.7%

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

   if 7.2e109 < k

   1. Initial program 29.8%

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

   2. Taylor expanded in t around 0 73.0%

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

      1. times-frac 75.2%

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

   4. Simplified 75.2%

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

      1. add-sqr-sqrt 75.2%

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

      2. sqrt-div 75.2%

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

      3. unpow2 75.2%

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

      4. sqrt-prod 75.2%

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

      5. add-sqr-sqrt 75.2%

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

      6. pow2 75.2%

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

      7. sqrt-prod 38.7%

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

      8. add-sqr-sqrt 64.0%

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

      9. sqrt-div 64.1%

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

      10. unpow2 64.1%

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

      11. sqrt-prod 64.1%

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

      12. add-sqr-sqrt 64.2%

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

      13. pow2 64.2%

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

      14. sqrt-prod 40.8%

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

      15. add-sqr-sqrt 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. clear-num 97.5%

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

      2. clear-num 97.3%

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

      3. frac-times 97.6%

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

      4. metadata-eval 97.6%

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

   8. Applied egg-rr 97.6%

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

      1. unpow2 97.5%

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

      2. sin-mult 96.9%

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

   10. Applied egg-rr 97.0%

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

       1. div-sub 96.9%

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

       2. +-inverses 96.9%

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

       3. cos-0 96.9%

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

       4. metadata-eval 96.9%

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

       5. count-2 96.9%

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

   12. Simplified 97.0%

       \[\leadsto \frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}}{\cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.013:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\left(\ell \cdot -0.5 + \left({k}^{2} \cdot \left(\ell \cdot 0.041666666666666664 + \left(-0.3333333333333333 \cdot \left(\ell \cdot -0.3333333333333333 - \ell \cdot -0.5\right) - \ell \cdot 0.044444444444444446\right)\right) + \frac{\ell}{{k}^{2}}\right)\right) - \ell \cdot -0.3333333333333333}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\ell \cdot \frac{\cos k}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 5: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \left(k \cdot \frac{k}{\ell}\right)\\ t_2 := 0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\\ \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \frac{\cos k}{t_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot t_2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (* k (/ k l)))) (t_2 (- 0.5 (/ (cos (* 2.0 k)) 2.0))))
   (if (<= k 0.0025)
     (/ 2.0 (/ t_1 (+ (/ l (pow k 2.0)) (* l -0.16666666666666666))))
     (if (<= k 5e+109)
       (/ 2.0 (/ t_1 (* l (/ (cos k) t_2))))
       (/ 2.0 (* (/ 1.0 (* (/ l k) (/ l k))) (/ (* t t_2) (cos k))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t * (k * (k / l));
	double t_2 = 0.5 - (cos((2.0 * k)) / 2.0);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 / ((l / pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else if (k <= 5e+109) {
		tmp = 2.0 / (t_1 / (l * (cos(k) / t_2)));
	} else {
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_2) / cos(k)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (k * (k / l))
    t_2 = 0.5d0 - (cos((2.0d0 * k)) / 2.0d0)
    if (k <= 0.0025d0) then
        tmp = 2.0d0 / (t_1 / ((l / (k ** 2.0d0)) + (l * (-0.16666666666666666d0))))
    else if (k <= 5d+109) then
        tmp = 2.0d0 / (t_1 / (l * (cos(k) / t_2)))
    else
        tmp = 2.0d0 / ((1.0d0 / ((l / k) * (l / k))) * ((t * t_2) / cos(k)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t * (k * (k / l));
	double t_2 = 0.5 - (Math.cos((2.0 * k)) / 2.0);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 / ((l / Math.pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else if (k <= 5e+109) {
		tmp = 2.0 / (t_1 / (l * (Math.cos(k) / t_2)));
	} else {
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_2) / Math.cos(k)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t * (k * (k / l))
	t_2 = 0.5 - (math.cos((2.0 * k)) / 2.0)
	tmp = 0
	if k <= 0.0025:
		tmp = 2.0 / (t_1 / ((l / math.pow(k, 2.0)) + (l * -0.16666666666666666)))
	elif k <= 5e+109:
		tmp = 2.0 / (t_1 / (l * (math.cos(k) / t_2)))
	else:
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_2) / math.cos(k)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t * Float64(k * Float64(k / l)))
	t_2 = Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))
	tmp = 0.0
	if (k <= 0.0025)
		tmp = Float64(2.0 / Float64(t_1 / Float64(Float64(l / (k ^ 2.0)) + Float64(l * -0.16666666666666666))));
	elseif (k <= 5e+109)
		tmp = Float64(2.0 / Float64(t_1 / Float64(l * Float64(cos(k) / t_2))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 / Float64(Float64(l / k) * Float64(l / k))) * Float64(Float64(t * t_2) / cos(k))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t * (k * (k / l));
	t_2 = 0.5 - (cos((2.0 * k)) / 2.0);
	tmp = 0.0;
	if (k <= 0.0025)
		tmp = 2.0 / (t_1 / ((l / (k ^ 2.0)) + (l * -0.16666666666666666)));
	elseif (k <= 5e+109)
		tmp = 2.0 / (t_1 / (l * (cos(k) / t_2)));
	else
		tmp = 2.0 / ((1.0 / ((l / k) * (l / k))) * ((t * t_2) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.0025], N[(2.0 / N[(t$95$1 / N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+109], N[(2.0 / N[(t$95$1 / N[(l * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 / N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00250000000000000005

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. sqrt-div 75.6%

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

      3. unpow2 75.6%

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

      4. sqrt-prod 27.0%

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

      5. add-sqr-sqrt 51.8%

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

      6. pow2 51.8%

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

      7. sqrt-prod 24.7%

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

      8. add-sqr-sqrt 48.9%

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

      9. sqrt-div 48.9%

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

      10. unpow2 48.9%

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

      11. sqrt-prod 13.8%

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

      12. add-sqr-sqrt 52.4%

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

      13. pow2 52.4%

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

      14. sqrt-prod 44.6%

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

      15. add-sqr-sqrt 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate-*r/ 89.8%

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

      2. associate-/l* 89.8%

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

      3. frac-times 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in k around 0 81.4%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}} \]
   10. Step-by-step derivation

       1. +-commutative 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right)} - -0.3333333333333333 \cdot \ell}} \]

       2. associate--l+ 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}} \]

       3. distribute-rgt-out-- 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}}} \]

       4. metadata-eval 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \ell \cdot \color{blue}{-0.16666666666666666}}} \]

   11. Simplified 81.4%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}} \]

   if 0.00250000000000000005 < k < 5.0000000000000001e109

   1. Initial program 17.3%

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

   2. Taylor expanded in t around 0 82.9%

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

      1. times-frac 74.9%

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

   4. Simplified 74.9%

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

      1. add-sqr-sqrt 74.9%

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

      2. sqrt-div 74.9%

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

      3. unpow2 74.9%

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

      4. sqrt-prod 74.9%

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

      5. add-sqr-sqrt 74.9%

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

      6. pow2 74.9%

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

      7. sqrt-prod 35.3%

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

      8. add-sqr-sqrt 48.8%

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

      9. sqrt-div 48.8%

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

      10. unpow2 48.8%

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

      11. sqrt-prod 48.8%

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

      12. add-sqr-sqrt 48.8%

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

      13. pow2 48.8%

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

      14. sqrt-prod 39.4%

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

      15. add-sqr-sqrt 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-*r/ 83.2%

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

      2. associate-/l* 83.2%

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

      3. frac-times 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. unpow2 83.2%

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

      2. sin-mult 83.3%

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

   10. Applied egg-rr 95.7%

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

       1. div-sub 83.3%

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

       2. +-inverses 83.3%

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

       3. cos-0 83.3%

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

       4. metadata-eval 83.3%

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

       5. count-2 83.3%

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

   12. Simplified 95.7%

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

   if 5.0000000000000001e109 < k

   1. Initial program 29.8%

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

   2. Taylor expanded in t around 0 73.0%

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

      1. times-frac 75.2%

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

   4. Simplified 75.2%

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

      1. add-sqr-sqrt 75.2%

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

      2. sqrt-div 75.2%

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

      3. unpow2 75.2%

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

      4. sqrt-prod 75.2%

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

      5. add-sqr-sqrt 75.2%

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

      6. pow2 75.2%

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

      7. sqrt-prod 38.7%

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

      8. add-sqr-sqrt 64.0%

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

      9. sqrt-div 64.1%

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

      10. unpow2 64.1%

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

      11. sqrt-prod 64.1%

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

      12. add-sqr-sqrt 64.2%

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

      13. pow2 64.2%

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

      14. sqrt-prod 40.8%

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

      15. add-sqr-sqrt 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. clear-num 97.5%

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

      2. clear-num 97.3%

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

      3. frac-times 97.6%

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

      4. metadata-eval 97.6%

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

   8. Applied egg-rr 97.6%

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

      1. unpow2 97.5%

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

      2. sin-mult 96.9%

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

   10. Applied egg-rr 97.0%

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

       1. div-sub 96.9%

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

       2. +-inverses 96.9%

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

       3. cos-0 96.9%

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

       4. metadata-eval 96.9%

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

       5. count-2 96.9%

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

   12. Simplified 97.0%

       \[\leadsto \frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}}{\cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\ell \cdot \frac{\cos k}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 6: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \left(k \cdot \frac{k}{\ell}\right)\\ t_2 := 0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\\ \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\ell \cdot \frac{\cos k}{t_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot t_2}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (* k (/ k l)))) (t_2 (- 0.5 (/ (cos (* 2.0 k)) 2.0))))
   (if (<= k 0.0025)
     (/ 2.0 (/ t_1 (+ (/ l (pow k 2.0)) (* l -0.16666666666666666))))
     (if (<= k 5.8e+79)
       (/ 2.0 (/ t_1 (* l (/ (cos k) t_2))))
       (/ 2.0 (* (/ (* t t_2) (cos k)) (* (/ k l) (/ k l))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = t * (k * (k / l));
	double t_2 = 0.5 - (cos((2.0 * k)) / 2.0);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 / ((l / pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else if (k <= 5.8e+79) {
		tmp = 2.0 / (t_1 / (l * (cos(k) / t_2)));
	} else {
		tmp = 2.0 / (((t * t_2) / cos(k)) * ((k / l) * (k / l)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * (k * (k / l))
    t_2 = 0.5d0 - (cos((2.0d0 * k)) / 2.0d0)
    if (k <= 0.0025d0) then
        tmp = 2.0d0 / (t_1 / ((l / (k ** 2.0d0)) + (l * (-0.16666666666666666d0))))
    else if (k <= 5.8d+79) then
        tmp = 2.0d0 / (t_1 / (l * (cos(k) / t_2)))
    else
        tmp = 2.0d0 / (((t * t_2) / cos(k)) * ((k / l) * (k / l)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = t * (k * (k / l));
	double t_2 = 0.5 - (Math.cos((2.0 * k)) / 2.0);
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / (t_1 / ((l / Math.pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else if (k <= 5.8e+79) {
		tmp = 2.0 / (t_1 / (l * (Math.cos(k) / t_2)));
	} else {
		tmp = 2.0 / (((t * t_2) / Math.cos(k)) * ((k / l) * (k / l)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = t * (k * (k / l))
	t_2 = 0.5 - (math.cos((2.0 * k)) / 2.0)
	tmp = 0
	if k <= 0.0025:
		tmp = 2.0 / (t_1 / ((l / math.pow(k, 2.0)) + (l * -0.16666666666666666)))
	elif k <= 5.8e+79:
		tmp = 2.0 / (t_1 / (l * (math.cos(k) / t_2)))
	else:
		tmp = 2.0 / (((t * t_2) / math.cos(k)) * ((k / l) * (k / l)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(t * Float64(k * Float64(k / l)))
	t_2 = Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))
	tmp = 0.0
	if (k <= 0.0025)
		tmp = Float64(2.0 / Float64(t_1 / Float64(Float64(l / (k ^ 2.0)) + Float64(l * -0.16666666666666666))));
	elseif (k <= 5.8e+79)
		tmp = Float64(2.0 / Float64(t_1 / Float64(l * Float64(cos(k) / t_2))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * t_2) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = t * (k * (k / l));
	t_2 = 0.5 - (cos((2.0 * k)) / 2.0);
	tmp = 0.0;
	if (k <= 0.0025)
		tmp = 2.0 / (t_1 / ((l / (k ^ 2.0)) + (l * -0.16666666666666666)));
	elseif (k <= 5.8e+79)
		tmp = 2.0 / (t_1 / (l * (cos(k) / t_2)));
	else
		tmp = 2.0 / (((t * t_2) / cos(k)) * ((k / l) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.0025], N[(2.0 / N[(t$95$1 / N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e+79], N[(2.0 / N[(t$95$1 / N[(l * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00250000000000000005

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. sqrt-div 75.6%

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

      3. unpow2 75.6%

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

      4. sqrt-prod 27.0%

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

      5. add-sqr-sqrt 51.8%

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

      6. pow2 51.8%

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

      7. sqrt-prod 24.7%

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

      8. add-sqr-sqrt 48.9%

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

      9. sqrt-div 48.9%

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

      10. unpow2 48.9%

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

      11. sqrt-prod 13.8%

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

      12. add-sqr-sqrt 52.4%

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

      13. pow2 52.4%

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

      14. sqrt-prod 44.6%

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

      15. add-sqr-sqrt 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate-*r/ 89.8%

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

      2. associate-/l* 89.8%

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

      3. frac-times 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in k around 0 81.4%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}} \]
   10. Step-by-step derivation

       1. +-commutative 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right)} - -0.3333333333333333 \cdot \ell}} \]

       2. associate--l+ 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}} \]

       3. distribute-rgt-out-- 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}}} \]

       4. metadata-eval 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \ell \cdot \color{blue}{-0.16666666666666666}}} \]

   11. Simplified 81.4%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}} \]

   if 0.00250000000000000005 < k < 5.79999999999999984e79

   1. Initial program 17.6%

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

   2. Taylor expanded in t around 0 82.7%

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

      1. times-frac 71.8%

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

   4. Simplified 71.8%

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

      1. add-sqr-sqrt 71.8%

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

      2. sqrt-div 71.8%

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

      3. unpow2 71.8%

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

      4. sqrt-prod 71.8%

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

      5. add-sqr-sqrt 71.8%

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

      6. pow2 71.8%

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

      7. sqrt-prod 35.8%

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

      8. add-sqr-sqrt 48.2%

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

      9. sqrt-div 48.2%

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

      10. unpow2 48.2%

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

      11. sqrt-prod 48.2%

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

      12. add-sqr-sqrt 48.2%

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

      13. pow2 48.2%

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

      14. sqrt-prod 35.8%

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

      15. add-sqr-sqrt 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. associate-*r/ 77.4%

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

      2. associate-/l* 77.4%

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

      3. frac-times 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. unpow2 77.3%

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

      2. sin-mult 77.5%

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

   10. Applied egg-rr 94.2%

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

       1. div-sub 77.5%

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

       2. +-inverses 77.5%

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

       3. cos-0 77.5%

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

       4. metadata-eval 77.5%

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

       5. count-2 77.5%

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

   12. Simplified 94.2%

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

   if 5.79999999999999984e79 < k

   1. Initial program 28.2%

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

   2. Taylor expanded in t around 0 74.3%

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

      1. times-frac 76.2%

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

   4. Simplified 76.2%

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

      1. add-sqr-sqrt 76.2%

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

      2. sqrt-div 76.2%

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

      3. unpow2 76.2%

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

      4. sqrt-prod 76.2%

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

      5. add-sqr-sqrt 76.2%

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

      6. pow2 76.2%

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

      7. sqrt-prod 38.1%

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

      8. add-sqr-sqrt 62.4%

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

      9. sqrt-div 62.4%

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

      10. unpow2 62.4%

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

      11. sqrt-prod 62.5%

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

      12. add-sqr-sqrt 62.6%

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

      13. pow2 62.6%

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

      14. sqrt-prod 41.9%

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

      15. add-sqr-sqrt 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. unpow2 97.7%

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

      2. sin-mult 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. div-sub 97.3%

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

      2. +-inverses 97.3%

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

      3. cos-0 97.3%

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

      4. metadata-eval 97.3%

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

      5. count-2 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\ell \cdot \frac{\cos k}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 7: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.017:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.017)
   (/
    2.0
    (/ (* t (* k (/ k l))) (+ (/ l (pow k 2.0)) (* l -0.16666666666666666))))
   (/
    2.0
    (*
     (/ (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k))
     (* (/ k l) (/ k l))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.017) {
		tmp = 2.0 / ((t * (k * (k / l))) / ((l / pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else {
		tmp = 2.0 / (((t * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)) * ((k / l) * (k / l)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.017d0) then
        tmp = 2.0d0 / ((t * (k * (k / l))) / ((l / (k ** 2.0d0)) + (l * (-0.16666666666666666d0))))
    else
        tmp = 2.0d0 / (((t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / cos(k)) * ((k / l) * (k / l)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.017) {
		tmp = 2.0 / ((t * (k * (k / l))) / ((l / Math.pow(k, 2.0)) + (l * -0.16666666666666666)));
	} else {
		tmp = 2.0 / (((t * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k)) * ((k / l) * (k / l)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 0.017:
		tmp = 2.0 / ((t * (k * (k / l))) / ((l / math.pow(k, 2.0)) + (l * -0.16666666666666666)))
	else:
		tmp = 2.0 / (((t * (0.5 - (math.cos((2.0 * k)) / 2.0))) / math.cos(k)) * ((k / l) * (k / l)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.017)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) / Float64(Float64(l / (k ^ 2.0)) + Float64(l * -0.16666666666666666))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.017)
		tmp = 2.0 / ((t * (k * (k / l))) / ((l / (k ^ 2.0)) + (l * -0.16666666666666666)));
	else
		tmp = 2.0 / (((t * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)) * ((k / l) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.017], N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.017000000000000001

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 72.7%

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

      1. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. add-sqr-sqrt 75.6%

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

      2. sqrt-div 75.6%

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

      3. unpow2 75.6%

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

      4. sqrt-prod 27.0%

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

      5. add-sqr-sqrt 51.8%

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

      6. pow2 51.8%

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

      7. sqrt-prod 24.7%

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

      8. add-sqr-sqrt 48.9%

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

      9. sqrt-div 48.9%

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

      10. unpow2 48.9%

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

      11. sqrt-prod 13.8%

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

      12. add-sqr-sqrt 52.4%

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

      13. pow2 52.4%

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

      14. sqrt-prod 44.6%

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

      15. add-sqr-sqrt 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. associate-*r/ 89.8%

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

      2. associate-/l* 89.8%

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

      3. frac-times 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in k around 0 81.4%

      \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}} \]
   10. Step-by-step derivation

       1. +-commutative 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right)} - -0.3333333333333333 \cdot \ell}} \]

       2. associate--l+ 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}} \]

       3. distribute-rgt-out-- 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}}} \]

       4. metadata-eval 81.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \ell \cdot \color{blue}{-0.16666666666666666}}} \]

   11. Simplified 81.4%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}} \]

   if 0.017000000000000001 < k

   1. Initial program 25.5%

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

   2. Taylor expanded in t around 0 76.4%

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

      1. times-frac 75.1%

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

   4. Simplified 75.1%

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

      1. add-sqr-sqrt 75.1%

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

      2. sqrt-div 75.1%

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

      3. unpow2 75.1%

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

      4. sqrt-prod 75.1%

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

      5. add-sqr-sqrt 75.1%

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

      6. pow2 75.1%

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

      7. sqrt-prod 37.5%

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

      8. add-sqr-sqrt 58.8%

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

      9. sqrt-div 58.8%

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

      10. unpow2 58.8%

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

      11. sqrt-prod 58.9%

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

      12. add-sqr-sqrt 58.9%

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

      13. pow2 58.9%

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

      14. sqrt-prod 40.3%

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

      15. add-sqr-sqrt 92.6%

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

   6. Applied egg-rr 92.6%

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

      1. unpow2 92.6%

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

      2. sin-mult 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. div-sub 92.3%

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

      2. +-inverses 92.3%

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

      3. cos-0 92.3%

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

      4. metadata-eval 92.3%

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

      5. count-2 92.3%

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

   10. Simplified 92.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.017:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 8: 74.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (/ (* t (* k (/ k l))) (+ (/ l (pow k 2.0)) (* l -0.16666666666666666)))))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 / ((t * (k * (k / l))) / ((l / pow(k, 2.0)) + (l * -0.16666666666666666)));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((t * (k * (k / l))) / ((l / (k ** 2.0d0)) + (l * (-0.16666666666666666d0))))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return 2.0 / ((t * (k * (k / l))) / ((l / Math.pow(k, 2.0)) + (l * -0.16666666666666666)));
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 / ((t * (k * (k / l))) / ((l / math.pow(k, 2.0)) + (l * -0.16666666666666666)))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) / Float64(Float64(l / (k ^ 2.0)) + Float64(l * -0.16666666666666666))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 / ((t * (k * (k / l))) / ((l / (k ^ 2.0)) + (l * -0.16666666666666666)));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

2. Taylor expanded in t around 0 73.7%

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

   1. times-frac 75.5%

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

4. Simplified 75.5%

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

   1. add-sqr-sqrt 75.4%

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

   2. sqrt-div 75.5%

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

   3. unpow2 75.5%

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

   4. sqrt-prod 39.6%

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

   5. add-sqr-sqrt 57.9%

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

   6. pow2 57.9%

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

   7. sqrt-prod 28.1%

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

   8. add-sqr-sqrt 51.5%

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

   9. sqrt-div 51.5%

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

   10. unpow2 51.5%

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

   11. sqrt-prod 25.6%

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

   12. add-sqr-sqrt 54.1%

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

   13. pow2 54.1%

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

   14. sqrt-prod 43.5%

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

   15. add-sqr-sqrt 91.6%

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

6. Applied egg-rr 91.6%

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

   1. associate-*r/ 89.8%

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

   2. associate-/l* 89.8%

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

   3. frac-times 92.4%

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

8. Applied egg-rr 92.4%

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

9. Taylor expanded in k around 0 75.2%

   \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(-0.5 \cdot \ell + \frac{\ell}{{k}^{2}}\right) - -0.3333333333333333 \cdot \ell}}} \]
10. Step-by-step derivation

    1. +-commutative 75.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right)} - -0.3333333333333333 \cdot \ell}} \]

    2. associate--l+ 75.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}} \]

    3. distribute-rgt-out-- 75.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}}} \]

    4. metadata-eval 75.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\ell}{{k}^{2}} + \ell \cdot \color{blue}{-0.16666666666666666}}} \]

11. Simplified 75.2%

    \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\color{blue}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}}} \]

12. Final simplification 75.2%

    \[\leadsto \frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}} + \ell \cdot -0.16666666666666666}} \]

Alternative 9: 72.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.5e+17)
   (/ 2.0 (/ (* t (* k (/ k l))) (/ l (pow k 2.0))))
   (/ -0.3333333333333333 (* t (* (/ k l) (/ k l))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e+17) {
		tmp = 2.0 / ((t * (k * (k / l))) / (l / pow(k, 2.0)));
	} else {
		tmp = -0.3333333333333333 / (t * ((k / l) * (k / l)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.5d+17) then
        tmp = 2.0d0 / ((t * (k * (k / l))) / (l / (k ** 2.0d0)))
    else
        tmp = (-0.3333333333333333d0) / (t * ((k / l) * (k / l)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e+17) {
		tmp = 2.0 / ((t * (k * (k / l))) / (l / Math.pow(k, 2.0)));
	} else {
		tmp = -0.3333333333333333 / (t * ((k / l) * (k / l)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 4.5e+17:
		tmp = 2.0 / ((t * (k * (k / l))) / (l / math.pow(k, 2.0)))
	else:
		tmp = -0.3333333333333333 / (t * ((k / l) * (k / l)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.5e+17)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) / Float64(l / (k ^ 2.0))));
	else
		tmp = Float64(-0.3333333333333333 / Float64(t * Float64(Float64(k / l) * Float64(k / l))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.5e+17)
		tmp = 2.0 / ((t * (k * (k / l))) / (l / (k ^ 2.0)));
	else
		tmp = -0.3333333333333333 / (t * ((k / l) * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.5e+17], N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5e17

   1. Initial program 34.8%

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

   2. Taylor expanded in t around 0 72.9%

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

      1. times-frac 75.7%

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

   4. Simplified 75.7%

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

      1. add-sqr-sqrt 75.7%

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

      2. sqrt-div 75.7%

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

      3. unpow2 75.7%

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

      4. sqrt-prod 28.3%

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

      5. add-sqr-sqrt 52.6%

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

      6. pow2 52.6%

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

      7. sqrt-prod 25.1%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-div 49.2%

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

      10. unpow2 49.2%

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

      11. sqrt-prod 15.0%

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

      12. add-sqr-sqrt 52.6%

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

      13. pow2 52.6%

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

      14. sqrt-prod 44.5%

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

      15. add-sqr-sqrt 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. associate-*r/ 89.5%

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

      2. associate-/l* 89.6%

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

      3. frac-times 93.4%

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

   8. Applied egg-rr 93.4%

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

   9. Taylor expanded in k around 0 77.7%

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

   if 4.5e17 < k

   1. Initial program 26.0%

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

      1. associate-/r* 26.0%

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

      2. associate-*l/ 26.0%

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

      3. associate--l+ 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in k around 0 51.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   5. Taylor expanded in k around inf 57.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. pow2 57.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

      2. associate-*r/ 57.8%

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

      3. pow2 57.8%

         \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot t} \]

      4. *-commutative 57.8%

         \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

   7. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t \cdot {k}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 57.8%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{t \cdot {k}^{2}}{{\ell}^{2}}}} \]

      2. *-commutative 57.8%

         \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}}} \]

      3. unpow2 57.8%

         \[\leadsto \frac{-0.3333333333333333}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      4. associate-/r* 60.1%

         \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}}} \]

      5. associate-*l/ 60.1%

         \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot t}}{\ell}} \]

      6. unpow2 60.1%

         \[\leadsto \frac{-0.3333333333333333}{\frac{\frac{\color{blue}{k \cdot k}}{\ell} \cdot t}{\ell}} \]

      7. associate-*l/ 60.2%

         \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t}{\ell}} \]

      8. associate-*l/ 60.2%

         \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{\frac{k}{\ell} \cdot k}{\ell} \cdot t}} \]

      9. associate-*r/ 60.4%

         \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

      10. unpow2 60.4%

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

   9. Simplified 60.4%

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

       1. pow2 60.4%

          \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

   11. Applied egg-rr 60.4%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 10: 35.1% accurate, 38.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{-0.3333333333333333}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ -0.3333333333333333 (* t (* (/ k l) (/ k l)))))

C

l = abs(l);
double code(double t, double l, double k) {
	return -0.3333333333333333 / (t * ((k / l) * (k / l)));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) / (t * ((k / l) * (k / l)))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return -0.3333333333333333 / (t * ((k / l) * (k / l)));
}

Python

l = abs(l)
def code(t, l, k):
	return -0.3333333333333333 / (t * ((k / l) * (k / l)))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(-0.3333333333333333 / Float64(t * Float64(Float64(k / l) * Float64(k / l))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = -0.3333333333333333 / (t * ((k / l) * (k / l)));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(-0.3333333333333333 / N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

   1. associate-/r* 32.7%

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

   2. associate-*l/ 32.4%

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

   3. associate--l+ 32.4%

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

3. Simplified 32.4%

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

4. Taylor expanded in k around 0 39.2%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

5. Taylor expanded in k around inf 35.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. pow2 35.0%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

   2. associate-*r/ 35.0%

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

   3. pow2 35.0%

      \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{{\ell}^{2}}}{{k}^{2} \cdot t} \]

   4. *-commutative 35.0%

      \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

7. Applied egg-rr 35.0%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t \cdot {k}^{2}}} \]
8. Step-by-step derivation

   1. associate-/l* 35.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{t \cdot {k}^{2}}{{\ell}^{2}}}} \]

   2. *-commutative 35.0%

      \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}}} \]

   3. unpow2 35.0%

      \[\leadsto \frac{-0.3333333333333333}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   4. associate-/r* 36.5%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}}} \]

   5. associate-*l/ 36.8%

      \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{\frac{{k}^{2}}{\ell} \cdot t}}{\ell}} \]

   6. unpow2 36.8%

      \[\leadsto \frac{-0.3333333333333333}{\frac{\frac{\color{blue}{k \cdot k}}{\ell} \cdot t}{\ell}} \]

   7. associate-*l/ 37.0%

      \[\leadsto \frac{-0.3333333333333333}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t}{\ell}} \]

   8. associate-*l/ 36.9%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{\frac{k}{\ell} \cdot k}{\ell} \cdot t}} \]

   9. associate-*r/ 37.0%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

   10. unpow2 37.0%

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

9. Simplified 37.0%

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

    1. pow2 37.0%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

11. Applied egg-rr 37.0%

    \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

12. Final simplification 37.0%

    \[\leadsto \frac{-0.3333333333333333}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

Reproduce

herbie shell --seed 2023309 
(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))))