Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 98.6%

Time: 31.8s

Alternatives: 9

Speedup: 28.1×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.9% 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: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{\frac{\ell}{k}}}{k}\\ \mathbf{if}\;k \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot \left(\ell \cdot \frac{\frac{1}{{\sin k}^{2}}}{k}\right)}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sqrt (/ l k)) k)))
   (if (<= k 1.5e-47)
     (* 2.0 (* t_1 (* t_1 (/ l (* k t)))))
     (*
      2.0
      (/ (* (/ (cos k) t) (* l (/ (/ 1.0 (pow (sin k) 2.0)) k))) (/ k l))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double t_1 = sqrt((l / k)) / k;
	double tmp;
	if (k <= 1.5e-47) {
		tmp = 2.0 * (t_1 * (t_1 * (l / (k * t))));
	} else {
		tmp = 2.0 * (((cos(k) / t) * (l * ((1.0 / pow(sin(k), 2.0)) / k))) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((l / k)) / k
    if (k <= 1.5d-47) then
        tmp = 2.0d0 * (t_1 * (t_1 * (l / (k * t))))
    else
        tmp = 2.0d0 * (((cos(k) / t) * (l * ((1.0d0 / (sin(k) ** 2.0d0)) / k))) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt((l / k)) / k;
	double tmp;
	if (k <= 1.5e-47) {
		tmp = 2.0 * (t_1 * (t_1 * (l / (k * t))));
	} else {
		tmp = 2.0 * (((Math.cos(k) / t) * (l * ((1.0 / Math.pow(Math.sin(k), 2.0)) / k))) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	t_1 = math.sqrt((l / k)) / k
	tmp = 0
	if k <= 1.5e-47:
		tmp = 2.0 * (t_1 * (t_1 * (l / (k * t))))
	else:
		tmp = 2.0 * (((math.cos(k) / t) * (l * ((1.0 / math.pow(math.sin(k), 2.0)) / k))) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	t_1 = Float64(sqrt(Float64(l / k)) / k)
	tmp = 0.0
	if (k <= 1.5e-47)
		tmp = Float64(2.0 * Float64(t_1 * Float64(t_1 * Float64(l / Float64(k * t)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / t) * Float64(l * Float64(Float64(1.0 / (sin(k) ^ 2.0)) / k))) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sqrt((l / k)) / k;
	tmp = 0.0;
	if (k <= 1.5e-47)
		tmp = 2.0 * (t_1 * (t_1 * (l / (k * t))));
	else
		tmp = 2.0 * (((cos(k) / t) * (l * ((1.0 / (sin(k) ^ 2.0)) / k))) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sqrt[N[(l / k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 1.5e-47], N[(2.0 * N[(t$95$1 * N[(t$95$1 * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(l * N[(N[(1.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000008e-47

   1. Initial program 43.3%

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

      1. associate-*l* 43.3%

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

      2. associate-*l* 43.3%

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

      3. +-commutative 43.3%

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

      4. associate--l+ 48.9%

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

      5. metadata-eval 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in k around 0 71.4%

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

      1. unpow2 71.4%

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

      2. times-frac 76.2%

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

   6. Applied egg-rr 76.2%

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

      1. clear-num 76.2%

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

      2. frac-times 76.2%

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

      3. *-commutative 76.2%

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

      4. *-un-lft-identity 76.2%

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

   8. Applied egg-rr 76.2%

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

      1. associate-/r* 76.2%

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

      2. add-exp-log 46.1%

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

      3. associate-/l* 46.9%

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

      4. add-exp-log 76.2%

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

      5. associate-*l/ 71.5%

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

      6. metadata-eval 71.5%

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

      7. pow-sqr 71.5%

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

      8. unpow2 71.5%

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

      9. associate-*l* 71.5%

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

      10. *-commutative 71.5%

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

      11. frac-times 77.2%

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

      12. associate-/l/ 80.1%

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

      13. *-commutative 80.1%

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

      14. associate-*r/ 80.3%

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

      15. add-sqr-sqrt 52.5%

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

      16. associate-*l* 52.5%

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

   10. Applied egg-rr 48.5%

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

   if 1.50000000000000008e-47 < k

   1. Initial program 28.1%

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

      1. associate-*l* 28.1%

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

      2. associate-*l* 28.1%

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

      3. +-commutative 28.1%

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

      4. associate--l+ 38.5%

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

      5. metadata-eval 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in t around 0 72.1%

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

      1. times-frac 70.4%

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

   6. Simplified 70.4%

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

      1. unpow2 70.4%

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

      2. unpow2 70.4%

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

      3. times-frac 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. associate-*l* 97.3%

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

      2. clear-num 97.3%

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

      3. associate-*l/ 97.5%

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

      4. *-un-lft-identity 97.5%

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

      5. clear-num 97.5%

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

      6. associate-*l/ 97.5%

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

      7. *-un-lft-identity 97.5%

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

      8. associate-/r* 97.4%

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

   10. Applied egg-rr 97.4%

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

       1. associate-/l/ 99.4%

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

       2. div-inv 99.4%

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

       3. *-commutative 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. associate-/r* 99.4%

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

       2. associate-/r/ 99.4%

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

   14. Simplified 99.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \left(\frac{\sqrt{\frac{\ell}{k}}}{k} \cdot \left(\frac{\sqrt{\frac{\ell}{k}}}{k} \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot \left(\ell \cdot \frac{\frac{1}{{\sin k}^{2}}}{k}\right)}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{-1}{\sin k \cdot \tan k}}{t} \cdot \left(\ell \cdot \frac{1}{-k}\right)}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.25e-8)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (*
    2.0
    (/ (* (/ (/ -1.0 (* (sin k) (tan k))) t) (* l (/ 1.0 (- k)))) (/ k l)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-8) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((-1.0 / (sin(k) * tan(k))) / t) * (l * (1.0 / -k))) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-8) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * (((((-1.0d0) / (sin(k) * tan(k))) / t) * (l * (1.0d0 / -k))) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-8) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((-1.0 / (Math.sin(k) * Math.tan(k))) / t) * (l * (1.0 / -k))) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.25e-8:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * ((((-1.0 / (math.sin(k) * math.tan(k))) / t) * (l * (1.0 / -k))) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.25e-8)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(-1.0 / Float64(sin(k) * tan(k))) / t) * Float64(l * Float64(1.0 / Float64(-k)))) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.25e-8)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * ((((-1.0 / (sin(k) * tan(k))) / t) * (l * (1.0 / -k))) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.25e-8], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(-1.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l * N[(1.0 / (-k)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.2499999999999999e-8

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. associate-*l* 41.1%

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

      3. +-commutative 41.1%

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

      4. associate--l+ 47.5%

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

      5. metadata-eval 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in k around 0 71.5%

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

      1. unpow2 71.5%

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

      2. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. clear-num 77.7%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. *-un-lft-identity 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. associate-/r* 77.7%

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

      2. add-exp-log 45.5%

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

      3. associate-/l* 45.6%

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

      4. add-exp-log 76.1%

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

      5. associate-*l/ 71.7%

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

      6. metadata-eval 71.7%

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

      7. pow-sqr 71.6%

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

      8. unpow2 71.6%

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

      9. associate-*l* 71.6%

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

      10. *-commutative 71.6%

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

      11. frac-times 77.0%

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

      12. associate-/l/ 79.7%

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

      13. associate-*l/ 81.5%

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

      14. associate-*r/ 81.1%

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

      15. unpow2 81.1%

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

      16. times-frac 84.3%

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

      17. associate-/l/ 83.8%

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

      18. *-commutative 83.8%

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

   10. Applied egg-rr 83.8%

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

   if 1.2499999999999999e-8 < k

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. associate-*l* 30.4%

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

      3. +-commutative 30.4%

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

      4. associate--l+ 39.9%

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

      5. metadata-eval 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around 0 72.0%

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

      1. times-frac 70.1%

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

   6. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. frac-2neg 99.4%

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

       2. div-inv 99.4%

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

       3. clear-num 99.3%

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

       4. metadata-eval 99.3%

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

       5. distribute-neg-frac 99.3%

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

       6. metadata-eval 99.3%

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

       7. metadata-eval 99.3%

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

       8. associate-/r/ 99.3%

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

       9. unpow2 99.3%

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

       10. associate-*r/ 99.4%

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

       11. tan-quot 99.4%

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

       12. distribute-neg-frac 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. associate-/r* 99.5%

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

       2. associate-/r/ 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{-1}{\sin k \cdot \tan k}}{t} \cdot \left(\ell \cdot \frac{1}{-k}\right)}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{-1}{\sin k \cdot \tan k}}{t}}{\frac{-k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-9)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (* 2.0 (/ (/ (/ (/ -1.0 (* (sin k) (tan k))) t) (/ (- k) l)) (/ k l)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-9) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((-1.0 / (sin(k) * tan(k))) / t) / (-k / l)) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-9) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * (((((-1.0d0) / (sin(k) * tan(k))) / t) / (-k / l)) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-9) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((-1.0 / (Math.sin(k) * Math.tan(k))) / t) / (-k / l)) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 7e-9:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * ((((-1.0 / (math.sin(k) * math.tan(k))) / t) / (-k / l)) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-9)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(-1.0 / Float64(sin(k) * tan(k))) / t) / Float64(Float64(-k) / l)) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-9)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * ((((-1.0 / (sin(k) * tan(k))) / t) / (-k / l)) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 7e-9], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(-1.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[((-k) / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999998e-9

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. associate-*l* 41.1%

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

      3. +-commutative 41.1%

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

      4. associate--l+ 47.5%

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

      5. metadata-eval 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in k around 0 71.5%

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

      1. unpow2 71.5%

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

      2. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. clear-num 77.7%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. *-un-lft-identity 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. associate-/r* 77.7%

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

      2. add-exp-log 45.5%

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

      3. associate-/l* 45.6%

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

      4. add-exp-log 76.1%

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

      5. associate-*l/ 71.7%

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

      6. metadata-eval 71.7%

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

      7. pow-sqr 71.6%

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

      8. unpow2 71.6%

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

      9. associate-*l* 71.6%

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

      10. *-commutative 71.6%

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

      11. frac-times 77.0%

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

      12. associate-/l/ 79.7%

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

      13. associate-*l/ 81.5%

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

      14. associate-*r/ 81.1%

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

      15. unpow2 81.1%

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

      16. times-frac 84.3%

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

      17. associate-/l/ 83.8%

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

      18. *-commutative 83.8%

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

   10. Applied egg-rr 83.8%

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

   if 6.9999999999999998e-9 < k

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. associate-*l* 30.4%

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

      3. +-commutative 30.4%

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

      4. associate--l+ 39.9%

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

      5. metadata-eval 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around 0 72.0%

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

      1. times-frac 70.1%

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

   6. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. frac-2neg 99.4%

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

       2. div-inv 99.4%

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

       3. clear-num 99.3%

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

       4. metadata-eval 99.3%

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

       5. distribute-neg-frac 99.3%

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

       6. metadata-eval 99.3%

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

       7. metadata-eval 99.3%

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

       8. associate-/r/ 99.3%

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

       9. unpow2 99.3%

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

       10. associate-*r/ 99.4%

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

       11. tan-quot 99.4%

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

       12. distribute-neg-frac 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. *-rgt-identity 99.4%

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

       3. associate-/r* 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{-1}{\sin k \cdot \tan k}}{t}}{\frac{-k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 4: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.16e-33)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (* 2.0 (* (/ l k) (/ l (* k (* t (* (sin k) (tan k)))))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.16e-33) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((l / k) * (l / (k * (t * (sin(k) * tan(k))))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.16d-33) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * ((l / k) * (l / (k * (t * (sin(k) * tan(k))))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.16e-33) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((l / k) * (l / (k * (t * (Math.sin(k) * Math.tan(k))))));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.16e-33:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * ((l / k) * (l / (k * (t * (math.sin(k) * math.tan(k))))))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.16e-33)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(l / Float64(k * Float64(t * Float64(sin(k) * tan(k)))))));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.16e-33)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * ((l / k) * (l / (k * (t * (sin(k) * tan(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.16e-33], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1600000000000001e-33

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. associate-*l* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate--l+ 48.3%

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

      5. metadata-eval 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 70.5%

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

      1. unpow2 70.5%

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

      2. times-frac 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. clear-num 76.5%

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

      2. frac-times 76.5%

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

      3. *-commutative 76.5%

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

      4. *-un-lft-identity 76.5%

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

   8. Applied egg-rr 76.5%

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

      1. associate-/r* 76.5%

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

      2. add-exp-log 46.1%

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

      3. associate-/l* 46.3%

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

      4. add-exp-log 75.3%

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

      5. associate-*l/ 70.7%

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

      6. metadata-eval 70.7%

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

      7. pow-sqr 70.6%

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

      8. unpow2 70.6%

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

      9. associate-*l* 70.6%

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

      10. *-commutative 70.6%

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

      11. frac-times 76.3%

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

      12. associate-/l/ 79.2%

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

      13. associate-*l/ 80.5%

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

      14. associate-*r/ 80.0%

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

      15. unpow2 80.0%

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

      16. times-frac 83.5%

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

      17. associate-/l/ 82.9%

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

      18. *-commutative 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 1.1600000000000001e-33 < k

   1. Initial program 28.7%

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

      1. associate-*l* 28.7%

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

      2. associate-*l* 28.7%

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

      3. +-commutative 28.7%

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

      4. associate--l+ 39.3%

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

      5. metadata-eval 39.3%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. times-frac 71.9%

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

   6. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. div-inv 99.4%

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

       2. div-inv 99.3%

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

       3. clear-num 99.2%

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

       4. metadata-eval 99.2%

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

       5. clear-num 99.2%

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

       6. frac-times 92.4%

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

       7. metadata-eval 92.4%

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

       8. *-un-lft-identity 92.4%

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

       9. associate-/r/ 92.5%

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

       10. unpow2 92.5%

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

       11. associate-*r/ 92.5%

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

       12. tan-quot 92.5%

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

       13. clear-num 92.5%

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

   12. Applied egg-rr 92.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \end{array} \]

Alternative 5: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e-33)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (* 2.0 (/ (/ l (* k (* t (* (sin k) (tan k))))) (/ k l)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-33) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((l / (k * (t * (sin(k) * tan(k))))) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d-33) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * ((l / (k * (t * (sin(k) * tan(k))))) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e-33) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((l / (k * (t * (Math.sin(k) * Math.tan(k))))) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.3e-33:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * ((l / (k * (t * (math.sin(k) * math.tan(k))))) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e-33)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(t * Float64(sin(k) * tan(k))))) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e-33)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * ((l / (k * (t * (sin(k) * tan(k))))) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.3e-33], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.29999999999999997e-33

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. associate-*l* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate--l+ 48.3%

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

      5. metadata-eval 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 70.5%

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

      1. unpow2 70.5%

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

      2. times-frac 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. clear-num 76.5%

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

      2. frac-times 76.5%

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

      3. *-commutative 76.5%

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

      4. *-un-lft-identity 76.5%

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

   8. Applied egg-rr 76.5%

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

      1. associate-/r* 76.5%

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

      2. add-exp-log 46.1%

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

      3. associate-/l* 46.3%

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

      4. add-exp-log 75.3%

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

      5. associate-*l/ 70.7%

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

      6. metadata-eval 70.7%

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

      7. pow-sqr 70.6%

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

      8. unpow2 70.6%

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

      9. associate-*l* 70.6%

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

      10. *-commutative 70.6%

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

      11. frac-times 76.3%

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

      12. associate-/l/ 79.2%

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

      13. associate-*l/ 80.5%

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

      14. associate-*r/ 80.0%

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

      15. unpow2 80.0%

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

      16. times-frac 83.5%

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

      17. associate-/l/ 82.9%

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

      18. *-commutative 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 1.29999999999999997e-33 < k

   1. Initial program 28.7%

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

      1. associate-*l* 28.7%

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

      2. associate-*l* 28.7%

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

      3. +-commutative 28.7%

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

      4. associate--l+ 39.3%

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

      5. metadata-eval 39.3%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. times-frac 71.9%

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

   6. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. *-un-lft-identity 99.4%

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

       2. metadata-eval 99.4%

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

       3. *-commutative 99.4%

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

       4. div-inv 99.4%

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

       5. clear-num 99.3%

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

       6. metadata-eval 99.3%

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

       7. clear-num 99.3%

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

       8. frac-times 92.4%

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

       9. metadata-eval 92.4%

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

       10. *-un-lft-identity 92.4%

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

       11. associate-/r/ 92.5%

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

       12. unpow2 92.5%

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

       13. associate-*r/ 92.4%

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

       14. tan-quot 92.5%

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

       15. metadata-eval 92.5%

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

   12. Applied egg-rr 92.5%

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

       1. *-rgt-identity 92.5%

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

       2. *-commutative 92.5%

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

       3. *-commutative 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 6: 75.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{1}{t \cdot {k}^{2}} + 0.16666666666666666 \cdot \frac{-1}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-34)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (*
    2.0
    (/
     (/
      (+ (/ 1.0 (* t (pow k 2.0))) (* 0.16666666666666666 (/ -1.0 t)))
      (/ k l))
     (/ k l)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-34) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((1.0 / (t * pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))) / (k / l)) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-34) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * ((((1.0d0 / (t * (k ** 2.0d0))) + (0.16666666666666666d0 * ((-1.0d0) / t))) / (k / l)) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-34) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * ((((1.0 / (t * Math.pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))) / (k / l)) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-34:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * ((((1.0 / (t * math.pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))) / (k / l)) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-34)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(1.0 / Float64(t * (k ^ 2.0))) + Float64(0.16666666666666666 * Float64(-1.0 / t))) / Float64(k / l)) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-34)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * ((((1.0 / (t * (k ^ 2.0))) + (0.16666666666666666 * (-1.0 / t))) / (k / l)) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-34], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000003e-34

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. associate-*l* 42.7%

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

      3. +-commutative 42.7%

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

      4. associate--l+ 48.3%

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

      5. metadata-eval 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in k around 0 70.5%

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

      1. unpow2 70.5%

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

      2. times-frac 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. clear-num 76.5%

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

      2. frac-times 76.5%

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

      3. *-commutative 76.5%

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

      4. *-un-lft-identity 76.5%

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

   8. Applied egg-rr 76.5%

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

      1. associate-/r* 76.5%

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

      2. add-exp-log 46.1%

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

      3. associate-/l* 46.3%

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

      4. add-exp-log 75.3%

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

      5. associate-*l/ 70.7%

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

      6. metadata-eval 70.7%

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

      7. pow-sqr 70.6%

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

      8. unpow2 70.6%

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

      9. associate-*l* 70.6%

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

      10. *-commutative 70.6%

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

      11. frac-times 76.3%

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

      12. associate-/l/ 79.2%

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

      13. associate-*l/ 80.5%

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

      14. associate-*r/ 80.0%

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

      15. unpow2 80.0%

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

      16. times-frac 83.5%

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

      17. associate-/l/ 82.9%

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

      18. *-commutative 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 5.0000000000000003e-34 < k

   1. Initial program 28.7%

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

      1. associate-*l* 28.7%

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

      2. associate-*l* 28.7%

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

      3. +-commutative 28.7%

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

      4. associate--l+ 39.3%

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

      5. metadata-eval 39.3%

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

   3. Simplified 39.3%

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

   4. Taylor expanded in t around 0 73.6%

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

      1. times-frac 71.9%

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

   6. Simplified 71.9%

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

      1. unpow2 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in k around 0 61.5%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{1}{t \cdot {k}^{2}} + 0.16666666666666666 \cdot \frac{-1}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 7: 74.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e-12)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (- (/ 1.0 (* t (pow k 2.0))) (/ 0.16666666666666666 t))))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-12) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * pow(k, 2.0))) - (0.16666666666666666 / t)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d-12) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (t * (k ** 2.0d0))) - (0.16666666666666666d0 / t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-12) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * Math.pow(k, 2.0))) - (0.16666666666666666 / t)));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.75e-12:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * math.pow(k, 2.0))) - (0.16666666666666666 / t)))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e-12)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(t * (k ^ 2.0))) - Float64(0.16666666666666666 / t))));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e-12)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k ^ 2.0))) - (0.16666666666666666 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.75e-12], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e-12

   1. Initial program 41.2%

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

      1. associate-*l* 41.2%

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

      2. associate-*l* 41.2%

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

      3. +-commutative 41.2%

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

      4. associate--l+ 47.1%

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

      5. metadata-eval 47.1%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in k around 0 71.0%

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

      1. unpow2 71.0%

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

      2. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. clear-num 77.3%

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

      2. frac-times 77.3%

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

      3. *-commutative 77.3%

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

      4. *-un-lft-identity 77.3%

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

   8. Applied egg-rr 77.3%

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

      1. associate-/r* 77.4%

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

      2. add-exp-log 45.1%

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

      3. associate-/l* 45.3%

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

      4. add-exp-log 75.7%

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

      5. associate-*l/ 71.2%

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

      6. metadata-eval 71.2%

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

      7. pow-sqr 71.1%

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

      8. unpow2 71.1%

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

      9. associate-*l* 71.1%

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

      10. *-commutative 71.1%

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

      11. frac-times 76.6%

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

      12. associate-/l/ 79.4%

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

      13. associate-*l/ 81.2%

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

      14. associate-*r/ 80.7%

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

      15. unpow2 80.7%

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

      16. times-frac 84.1%

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

      17. associate-/l/ 83.5%

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

      18. *-commutative 83.5%

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

   10. Applied egg-rr 83.5%

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

   if 1.75e-12 < k

   1. Initial program 30.5%

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

      1. associate-*l* 30.5%

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

      2. associate-*l* 30.5%

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

      3. +-commutative 30.5%

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

      4. associate--l+ 40.9%

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

      5. metadata-eval 40.9%

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

   3. Simplified 40.9%

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

   4. Taylor expanded in t around 0 72.9%

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

      1. times-frac 71.1%

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

   6. Simplified 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 91.4%

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

   8. Applied egg-rr 91.4%

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

   9. Taylor expanded in k around 0 58.5%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 58.5%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{{k}^{2} \cdot t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

       2. metadata-eval 58.5%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{{k}^{2} \cdot t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   11. Simplified 58.5%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - \frac{0.16666666666666666}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 8: 75.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} l = |l|\\ k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e-8)
   (* 2.0 (* (/ (/ l (* k t)) k) (/ (/ l k) k)))
   (*
    2.0
    (/
     (/ (- (/ (/ 1.0 (pow k 2.0)) t) (/ 0.16666666666666666 t)) (/ k l))
     (/ k l)))))

C

l = abs(l);
k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-8) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((((1.0 / pow(k, 2.0)) / t) - (0.16666666666666666 / t)) / (k / l)) / (k / l));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-8) then
        tmp = 2.0d0 * (((l / (k * t)) / k) * ((l / k) / k))
    else
        tmp = 2.0d0 * (((((1.0d0 / (k ** 2.0d0)) / t) - (0.16666666666666666d0 / t)) / (k / l)) / (k / l))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-8) {
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((((1.0 / Math.pow(k, 2.0)) / t) - (0.16666666666666666 / t)) / (k / l)) / (k / l));
	}
	return tmp;
}

Python

l = abs(l)
k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e-8:
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k))
	else:
		tmp = 2.0 * (((((1.0 / math.pow(k, 2.0)) / t) - (0.16666666666666666 / t)) / (k / l)) / (k / l))
	return tmp

Julia

l = abs(l)
k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e-8)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * t)) / k) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) - Float64(0.16666666666666666 / t)) / Float64(k / l)) / Float64(k / l)));
	end
	return tmp
end

MATLAB

l = abs(l)
k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e-8)
		tmp = 2.0 * (((l / (k * t)) / k) * ((l / k) / k));
	else
		tmp = 2.0 * (((((1.0 / (k ^ 2.0)) / t) - (0.16666666666666666 / t)) / (k / l)) / (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e-8], N[(2.0 * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999987e-8

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. associate-*l* 41.1%

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

      3. +-commutative 41.1%

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

      4. associate--l+ 47.5%

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

      5. metadata-eval 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in k around 0 71.5%

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

      1. unpow2 71.5%

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

      2. times-frac 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. clear-num 77.7%

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

      2. frac-times 77.7%

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

      3. *-commutative 77.7%

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

      4. *-un-lft-identity 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. associate-/r* 77.7%

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

      2. add-exp-log 45.5%

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

      3. associate-/l* 45.6%

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

      4. add-exp-log 76.1%

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

      5. associate-*l/ 71.7%

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

      6. metadata-eval 71.7%

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

      7. pow-sqr 71.6%

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

      8. unpow2 71.6%

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

      9. associate-*l* 71.6%

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

      10. *-commutative 71.6%

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

      11. frac-times 77.0%

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

      12. associate-/l/ 79.7%

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

      13. associate-*l/ 81.5%

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

      14. associate-*r/ 81.1%

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

      15. unpow2 81.1%

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

      16. times-frac 84.3%

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

      17. associate-/l/ 83.8%

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

      18. *-commutative 83.8%

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

   10. Applied egg-rr 83.8%

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

   if 1.49999999999999987e-8 < k

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. associate-*l* 30.4%

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

      3. +-commutative 30.4%

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

      4. associate--l+ 39.9%

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

      5. metadata-eval 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around 0 72.0%

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

      1. times-frac 70.1%

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

   6. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*l* 99.3%

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

      2. clear-num 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in k around 0 57.5%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]
   12. Step-by-step derivation

       1. associate-/r* 57.5%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]

       2. associate-*r/ 57.5%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]

       3. metadata-eval 57.5%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]

   13. Simplified 57.5%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}}}{\frac{k}{\ell}}}{\frac{k}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k \cdot t}}{k} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}}{\frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 9: 73.3% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. associate-*l* 37.5%

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

   2. associate-*l* 37.5%

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

   3. +-commutative 37.5%

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

   4. associate--l+ 44.9%

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

   5. metadata-eval 44.9%

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

3. Simplified 44.9%

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

4. Taylor expanded in k around 0 62.9%

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

   1. unpow2 62.9%

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

   2. times-frac 68.0%

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

6. Applied egg-rr 68.0%

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

   1. clear-num 67.9%

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

   2. frac-times 67.9%

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

   3. *-commutative 67.9%

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

   4. *-un-lft-identity 67.9%

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

8. Applied egg-rr 67.9%

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

   1. associate-/r* 68.0%

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

   2. add-exp-log 45.8%

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

   3. associate-/l* 46.0%

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

   4. add-exp-log 66.9%

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

   5. associate-*l/ 63.0%

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

   6. metadata-eval 63.0%

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

   7. pow-sqr 63.0%

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

   8. unpow2 63.0%

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

   9. associate-*l* 63.0%

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

   10. *-commutative 63.0%

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

   11. frac-times 67.5%

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

   12. associate-/l/ 69.3%

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

   13. associate-*l/ 70.5%

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

   14. associate-*r/ 70.2%

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

   15. unpow2 70.2%

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

   16. times-frac 72.3%

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

   17. associate-/l/ 72.0%

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

   18. *-commutative 72.0%

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

10. Applied egg-rr 72.0%

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

11. Final simplification 72.0%

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

Reproduce

herbie shell --seed 2023301 
(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))))