Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.3% → 95.0%

Time: 23.2s

Alternatives: 9

Speedup: 38.3×

• 34.8% 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: 34.3% 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: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 1.16 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{t_1}{\frac{\ell}{t}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell}}}{-\frac{t \cdot t_1}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 1.16e-97)
     (* -2.0 (/ (pow (/ l k) 2.0) (/ (* k (* k (- t))) (cos k))))
     (if (<= k 9.5e+150)
       (/ 2.0 (/ (* (* k k) (/ t_1 (/ l t))) (* l (cos k))))
       (* -2.0 (/ (/ (/ l k) (/ k l)) (- (/ (* t t_1) (cos k)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 1.16e-97) {
		tmp = -2.0 * (pow((l / k), 2.0) / ((k * (k * -t)) / cos(k)));
	} else if (k <= 9.5e+150) {
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)));
	} else {
		tmp = -2.0 * (((l / k) / (k / l)) / -((t * t_1) / cos(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 1.16d-97) then
        tmp = (-2.0d0) * (((l / k) ** 2.0d0) / ((k * (k * -t)) / cos(k)))
    else if (k <= 9.5d+150) then
        tmp = 2.0d0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)))
    else
        tmp = (-2.0d0) * (((l / k) / (k / l)) / -((t * t_1) / cos(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 1.16e-97) {
		tmp = -2.0 * (Math.pow((l / k), 2.0) / ((k * (k * -t)) / Math.cos(k)));
	} else if (k <= 9.5e+150) {
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * Math.cos(k)));
	} else {
		tmp = -2.0 * (((l / k) / (k / l)) / -((t * t_1) / Math.cos(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 1.16e-97:
		tmp = -2.0 * (math.pow((l / k), 2.0) / ((k * (k * -t)) / math.cos(k)))
	elif k <= 9.5e+150:
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * math.cos(k)))
	else:
		tmp = -2.0 * (((l / k) / (k / l)) / -((t * t_1) / math.cos(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 1.16e-97)
		tmp = Float64(-2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(Float64(k * Float64(k * Float64(-t))) / cos(k))));
	elseif (k <= 9.5e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_1 / Float64(l / t))) / Float64(l * cos(k))));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64(l / k) / Float64(k / l)) / Float64(-Float64(Float64(t * t_1) / cos(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.16e-97)
		tmp = -2.0 * (((l / k) ^ 2.0) / ((k * (k * -t)) / cos(k)));
	elseif (k <= 9.5e+150)
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)));
	else
		tmp = -2.0 * (((l / k) / (k / l)) / -((t * t_1) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.16e-97], N[(-2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.5e+150], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$1 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / (-N[(N[(t * t$95$1), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.16e-97

   1. Initial program 28.3%

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

   3. Applied egg-rr 16.6%

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

      1. expm1-def 17.2%

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

      2. expm1-log1p 17.4%

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

      3. *-commutative 17.4%

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

      4. associate-*l* 17.3%

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

   5. Simplified 17.3%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 88.7%

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

      9. *-commutative 88.7%

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

      10. associate-*l* 88.7%

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

   8. Simplified 88.7%

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

      1. associate-*r/ 89.2%

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

      2. pow2 89.2%

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

   10. Applied egg-rr 89.2%

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

       1. associate-/l* 89.2%

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

       2. associate-*r* 89.2%

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

       3. *-commutative 89.2%

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

       4. *-commutative 89.2%

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

       5. mul-1-neg 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in k around 0 72.6%

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

       1. unpow2 72.6%

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

       2. associate-*l* 78.1%

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

   15. Simplified 78.1%

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

   if 1.16e-97 < k < 9.5000000000000001e150

   1. Initial program 29.1%

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

   2. Taylor expanded in t around 0 81.5%

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

      1. unpow2 81.5%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

      4. times-frac 94.5%

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

      5. unpow2 94.5%

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

      6. *-commutative 94.5%

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

   4. Simplified 94.5%

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

      1. associate-*l/ 96.0%

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

      2. associate-/l* 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 9.5000000000000001e150 < k

   1. Initial program 40.0%

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

   3. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 15.6%

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

      3. *-commutative 15.6%

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

      4. associate-*l* 15.6%

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

   5. Simplified 15.6%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 95.6%

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

      9. *-commutative 95.6%

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

      10. associate-*l* 95.6%

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

   8. Simplified 95.6%

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

      1. associate-*r/ 95.7%

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

      2. pow2 95.7%

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

   10. Applied egg-rr 95.7%

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

       1. associate-/l* 95.7%

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

       2. associate-*r* 95.7%

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

       3. *-commutative 95.7%

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

       4. *-commutative 95.7%

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

       5. mul-1-neg 95.7%

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

   12. Simplified 95.7%

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

       1. pow2 95.7%

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

       2. associate-*r/ 93.7%

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

       3. associate-/l* 95.7%

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

   14. Applied egg-rr 95.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{{\sin k}^{2}}{\frac{\ell}{t}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell}}}{-\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 2: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{t_1}{\frac{\ell}{t}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(-t_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 2.4e-97)
     (* -2.0 (/ (pow (/ l k) 2.0) (/ (* k (* k (- t))) (cos k))))
     (if (<= k 5.6e+149)
       (/ 2.0 (/ (* (* k k) (/ t_1 (/ l t))) (* l (cos k))))
       (* -2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (- t_1)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 2.4e-97) {
		tmp = -2.0 * (pow((l / k), 2.0) / ((k * (k * -t)) / cos(k)));
	} else if (k <= 5.6e+149) {
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)));
	} else {
		tmp = -2.0 * (((l / k) * (l / k)) * (cos(k) / (t * -t_1)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 2.4d-97) then
        tmp = (-2.0d0) * (((l / k) ** 2.0d0) / ((k * (k * -t)) / cos(k)))
    else if (k <= 5.6d+149) then
        tmp = 2.0d0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)))
    else
        tmp = (-2.0d0) * (((l / k) * (l / k)) * (cos(k) / (t * -t_1)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 2.4e-97) {
		tmp = -2.0 * (Math.pow((l / k), 2.0) / ((k * (k * -t)) / Math.cos(k)));
	} else if (k <= 5.6e+149) {
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * Math.cos(k)));
	} else {
		tmp = -2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * -t_1)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 2.4e-97:
		tmp = -2.0 * (math.pow((l / k), 2.0) / ((k * (k * -t)) / math.cos(k)))
	elif k <= 5.6e+149:
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * math.cos(k)))
	else:
		tmp = -2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * -t_1)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 2.4e-97)
		tmp = Float64(-2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(Float64(k * Float64(k * Float64(-t))) / cos(k))));
	elseif (k <= 5.6e+149)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_1 / Float64(l / t))) / Float64(l * cos(k))));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * Float64(-t_1)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 2.4e-97)
		tmp = -2.0 * (((l / k) ^ 2.0) / ((k * (k * -t)) / cos(k)));
	elseif (k <= 5.6e+149)
		tmp = 2.0 / (((k * k) * (t_1 / (l / t))) / (l * cos(k)));
	else
		tmp = -2.0 * (((l / k) * (l / k)) * (cos(k) / (t * -t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 2.4e-97], N[(-2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.6e+149], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$1 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * (-t$95$1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.4e-97

   1. Initial program 28.3%

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

   3. Applied egg-rr 16.6%

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

      1. expm1-def 17.2%

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

      2. expm1-log1p 17.4%

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

      3. *-commutative 17.4%

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

      4. associate-*l* 17.3%

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

   5. Simplified 17.3%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 88.7%

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

      9. *-commutative 88.7%

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

      10. associate-*l* 88.7%

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

   8. Simplified 88.7%

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

      1. associate-*r/ 89.2%

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

      2. pow2 89.2%

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

   10. Applied egg-rr 89.2%

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

       1. associate-/l* 89.2%

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

       2. associate-*r* 89.2%

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

       3. *-commutative 89.2%

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

       4. *-commutative 89.2%

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

       5. mul-1-neg 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in k around 0 72.6%

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

       1. unpow2 72.6%

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

       2. associate-*l* 78.1%

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

   15. Simplified 78.1%

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

   if 2.4e-97 < k < 5.5999999999999998e149

   1. Initial program 29.1%

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

   2. Taylor expanded in t around 0 81.5%

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

      1. unpow2 81.5%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

      4. times-frac 94.5%

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

      5. unpow2 94.5%

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

      6. *-commutative 94.5%

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

   4. Simplified 94.5%

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

      1. associate-*l/ 96.0%

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

      2. associate-/l* 97.7%

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

   6. Applied egg-rr 97.7%

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

   if 5.5999999999999998e149 < k

   1. Initial program 40.0%

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

   3. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 15.6%

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

      3. *-commutative 15.6%

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

      4. associate-*l* 15.6%

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

   5. Simplified 15.6%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 95.6%

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

      9. *-commutative 95.6%

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

      10. associate-*l* 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in t around 0 95.6%

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

       1. mul-1-neg 95.6%

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

       2. distribute-rgt-neg-in 95.6%

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

   11. Simplified 95.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{{\sin k}^{2}}{\frac{\ell}{t}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(-{\sin k}^{2}\right)}\right)\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e-7)
   (* -2.0 (/ (pow (/ l k) 2.0) (/ (* k (* k (- t))) (cos k))))
   (if (<= k 1.55e+154)
     (* 2.0 (/ (* (cos k) (* l l)) (* t (* (* k k) (pow (sin k) 2.0)))))
     (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-7) {
		tmp = -2.0 * (pow((l / k), 2.0) / ((k * (k * -t)) / cos(k)));
	} else if (k <= 1.55e+154) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (t * ((k * k) * pow(sin(k), 2.0))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-7) then
        tmp = (-2.0d0) * (((l / k) ** 2.0d0) / ((k * (k * -t)) / cos(k)))
    else if (k <= 1.55d+154) then
        tmp = 2.0d0 * ((cos(k) * (l * l)) / (t * ((k * k) * (sin(k) ** 2.0d0))))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e-7) {
		tmp = -2.0 * (Math.pow((l / k), 2.0) / ((k * (k * -t)) / Math.cos(k)));
	} else if (k <= 1.55e+154) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (t * ((k * k) * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e-7:
		tmp = -2.0 * (math.pow((l / k), 2.0) / ((k * (k * -t)) / math.cos(k)))
	elif k <= 1.55e+154:
		tmp = 2.0 * ((math.cos(k) * (l * l)) / (t * ((k * k) * math.pow(math.sin(k), 2.0))))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e-7)
		tmp = Float64(-2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(Float64(k * Float64(k * Float64(-t))) / cos(k))));
	elseif (k <= 1.55e+154)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64(t * Float64(Float64(k * k) * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e-7)
		tmp = -2.0 * (((l / k) ^ 2.0) / ((k * (k * -t)) / cos(k)));
	elseif (k <= 1.55e+154)
		tmp = 2.0 * ((cos(k) * (l * l)) / (t * ((k * k) * (sin(k) ^ 2.0))));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e-7], N[(-2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+154], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(k * k), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4999999999999999e-7

   1. Initial program 28.8%

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

   3. Applied egg-rr 17.4%

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

      1. expm1-def 19.1%

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

      2. expm1-log1p 19.2%

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

      3. *-commutative 19.2%

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

      4. associate-*l* 19.2%

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

   5. Simplified 19.2%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 88.2%

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

      9. *-commutative 88.2%

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

      10. associate-*l* 88.2%

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

   8. Simplified 88.2%

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

      1. associate-*r/ 88.6%

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

      2. pow2 88.6%

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

   10. Applied egg-rr 88.6%

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

       1. associate-/l* 88.6%

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

       2. associate-*r* 88.6%

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

       3. *-commutative 88.6%

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

       4. *-commutative 88.6%

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

       5. mul-1-neg 88.6%

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

   12. Simplified 88.6%

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

   13. Taylor expanded in k around 0 73.6%

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

       1. unpow2 73.6%

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

       2. associate-*l* 78.6%

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

   15. Simplified 78.6%

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

   if 1.4999999999999999e-7 < k < 1.5500000000000001e154

   1. Initial program 27.3%

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

   3. Applied egg-rr 7.9%

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

      1. expm1-def 16.9%

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

      2. expm1-log1p 16.9%

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

      3. *-commutative 16.9%

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

      4. associate-*l* 17.0%

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

   5. Simplified 17.0%

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

   6. Taylor expanded in t around 0 85.4%

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

      1. unpow2 85.4%

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

      2. associate-*r* 85.5%

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

      3. unpow2 85.5%

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

   8. Simplified 85.5%

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

   if 1.5500000000000001e154 < k

   1. Initial program 40.0%

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

      1. associate-/r* 40.0%

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

      2. *-commutative 40.0%

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

      3. associate-/r* 44.9%

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

      4. associate-*r/ 44.9%

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

      5. associate-/l* 44.9%

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

      6. +-commutative 44.9%

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

      7. unpow2 44.9%

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

      8. sqr-neg 44.9%

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

      9. distribute-frac-neg 44.9%

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

      10. distribute-frac-neg 44.9%

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

      11. unpow2 44.9%

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

      12. associate--l+ 58.3%

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

      13. metadata-eval 58.3%

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

      14. +-rgt-identity 58.3%

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

      15. unpow2 58.3%

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

      16. distribute-frac-neg 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in k around 0 47.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 47.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 47.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. associate-/l* 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. unpow2 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. distribute-rgt-out 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. metadata-eval 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 51.6%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 65.0%

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

      1. associate-/r* 65.0%

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

      2. unpow2 65.0%

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

      3. unpow2 65.0%

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

   9. Simplified 65.0%

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

       1. frac-times 73.9%

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

   11. Applied egg-rr 73.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 4: 90.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(t \cdot {\sin k}^{2}\right) \cdot \frac{k \cdot \frac{k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-158)
   (* -2.0 (/ (pow (/ l k) 2.0) (/ (* k (* k (- t))) (cos k))))
   (* l (/ 2.0 (* (* t (pow (sin k) 2.0)) (/ (* k (/ k l)) (cos k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-158) {
		tmp = -2.0 * (pow((l / k), 2.0) / ((k * (k * -t)) / cos(k)));
	} else {
		tmp = l * (2.0 / ((t * pow(sin(k), 2.0)) * ((k * (k / l)) / cos(k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-158) then
        tmp = (-2.0d0) * (((l / k) ** 2.0d0) / ((k * (k * -t)) / cos(k)))
    else
        tmp = l * (2.0d0 / ((t * (sin(k) ** 2.0d0)) * ((k * (k / l)) / cos(k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-158) {
		tmp = -2.0 * (Math.pow((l / k), 2.0) / ((k * (k * -t)) / Math.cos(k)));
	} else {
		tmp = l * (2.0 / ((t * Math.pow(Math.sin(k), 2.0)) * ((k * (k / l)) / Math.cos(k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-158:
		tmp = -2.0 * (math.pow((l / k), 2.0) / ((k * (k * -t)) / math.cos(k)))
	else:
		tmp = l * (2.0 / ((t * math.pow(math.sin(k), 2.0)) * ((k * (k / l)) / math.cos(k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-158)
		tmp = Float64(-2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(Float64(k * Float64(k * Float64(-t))) / cos(k))));
	else
		tmp = Float64(l * Float64(2.0 / Float64(Float64(t * (sin(k) ^ 2.0)) * Float64(Float64(k * Float64(k / l)) / cos(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-158)
		tmp = -2.0 * (((l / k) ^ 2.0) / ((k * (k * -t)) / cos(k)));
	else
		tmp = l * (2.0 / ((t * (sin(k) ^ 2.0)) * ((k * (k / l)) / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-158], N[(-2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999972e-158

   1. Initial program 26.7%

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

   3. Applied egg-rr 15.9%

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

      1. expm1-def 16.5%

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

      2. expm1-log1p 16.7%

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

      3. *-commutative 16.7%

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

      4. associate-*l* 16.8%

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

   5. Simplified 16.8%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*l* 87.7%

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

   8. Simplified 87.7%

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

      1. associate-*r/ 88.3%

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

      2. pow2 88.3%

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

   10. Applied egg-rr 88.3%

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

       1. associate-/l* 88.3%

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

       2. associate-*r* 88.3%

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

       3. *-commutative 88.3%

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

       4. *-commutative 88.3%

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

       5. mul-1-neg 88.3%

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

   12. Simplified 88.3%

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

   13. Taylor expanded in k around 0 70.1%

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

       1. unpow2 70.1%

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

       2. associate-*l* 76.1%

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

   15. Simplified 76.1%

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

   if 4.99999999999999972e-158 < k

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0 74.7%

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

      1. unpow2 74.7%

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

      2. associate-*r* 74.8%

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

      3. *-commutative 74.8%

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

      4. times-frac 86.5%

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

      5. unpow2 86.5%

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

      6. *-commutative 86.5%

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

   4. Simplified 86.5%

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

      1. associate-*r/ 86.5%

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

      2. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. associate-/r/ 91.6%

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

      2. associate-*l/ 91.5%

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

      3. *-commutative 91.5%

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

   8. Applied egg-rr 91.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(t \cdot {\sin k}^{2}\right) \cdot \frac{k \cdot \frac{k}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 5: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e-158)
   (* -2.0 (/ (pow (/ l k) 2.0) (/ (* k (* k (- t))) (cos k))))
   (if (<= k 1.5e+46)
     (/ 2.0 (* (/ (* k k) (* l (cos k))) (* t (* k (/ k l)))))
     (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-158) {
		tmp = -2.0 * (pow((l / k), 2.0) / ((k * (k * -t)) / cos(k)));
	} else if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.2d-158) then
        tmp = (-2.0d0) * (((l / k) ** 2.0d0) / ((k * (k * -t)) / cos(k)))
    else if (k <= 1.5d+46) then
        tmp = 2.0d0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-158) {
		tmp = -2.0 * (Math.pow((l / k), 2.0) / ((k * (k * -t)) / Math.cos(k)));
	} else if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * Math.cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 4.2e-158:
		tmp = -2.0 * (math.pow((l / k), 2.0) / ((k * (k * -t)) / math.cos(k)))
	elif k <= 1.5e+46:
		tmp = 2.0 / (((k * k) / (l * math.cos(k))) * (t * (k * (k / l))))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e-158)
		tmp = Float64(-2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(Float64(k * Float64(k * Float64(-t))) / cos(k))));
	elseif (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * cos(k))) * Float64(t * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.2e-158)
		tmp = -2.0 * (((l / k) ^ 2.0) / ((k * (k * -t)) / cos(k)));
	elseif (k <= 1.5e+46)
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.2e-158], N[(-2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.19999999999999983e-158

   1. Initial program 26.7%

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

   3. Applied egg-rr 15.9%

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

      1. expm1-def 16.5%

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

      2. expm1-log1p 16.7%

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

      3. *-commutative 16.7%

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

      4. associate-*l* 16.8%

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

   5. Simplified 16.8%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*l* 87.7%

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

   8. Simplified 87.7%

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

      1. associate-*r/ 88.3%

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

      2. pow2 88.3%

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

   10. Applied egg-rr 88.3%

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

       1. associate-/l* 88.3%

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

       2. associate-*r* 88.3%

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

       3. *-commutative 88.3%

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

       4. *-commutative 88.3%

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

       5. mul-1-neg 88.3%

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

   12. Simplified 88.3%

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

   13. Taylor expanded in k around 0 70.1%

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

       1. unpow2 70.1%

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

       2. associate-*l* 76.1%

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

   15. Simplified 76.1%

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

   if 4.19999999999999983e-158 < k < 1.50000000000000012e46

   1. Initial program 31.4%

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

   2. Taylor expanded in t around 0 78.1%

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

      1. unpow2 78.1%

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

      2. associate-*r* 78.1%

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

      3. *-commutative 78.1%

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

      4. times-frac 95.1%

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

      5. unpow2 95.1%

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

      6. *-commutative 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in k around 0 74.2%

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

      1. associate-/l* 72.2%

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

      2. associate-/r/ 78.5%

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

      3. unpow2 78.5%

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

      4. associate-*r/ 78.7%

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

   7. Simplified 78.7%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

      1. associate-/r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/r* 40.7%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.7%

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

      6. +-commutative 40.7%

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

      7. unpow2 40.7%

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

      8. sqr-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. distribute-frac-neg 40.7%

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

      11. unpow2 40.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

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

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around 0 47.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 47.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 47.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. associate-/l* 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. distribute-rgt-out 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. metadata-eval 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 61.9%

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

      1. associate-/r* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

   9. Simplified 62.0%

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

       1. frac-times 67.9%

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

   11. Applied egg-rr 67.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{k \cdot \left(k \cdot \left(-t\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 6: 75.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq 3.05 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(-t\right)\right)}\right)\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{t_1}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k))))
   (if (<= k 3.05e-158)
     (* -2.0 (* t_1 (/ (cos k) (* k (* k (- t))))))
     (if (<= k 1.5e+46)
       (/ 2.0 (* (/ (* k k) (* l (cos k))) (* t (* k (/ k l)))))
       (* -0.3333333333333333 (/ t_1 t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (k <= 3.05e-158) {
		tmp = -2.0 * (t_1 * (cos(k) / (k * (k * -t))));
	} else if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (t_1 / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) * (l / k)
    if (k <= 3.05d-158) then
        tmp = (-2.0d0) * (t_1 * (cos(k) / (k * (k * -t))))
    else if (k <= 1.5d+46) then
        tmp = 2.0d0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))))
    else
        tmp = (-0.3333333333333333d0) * (t_1 / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (k <= 3.05e-158) {
		tmp = -2.0 * (t_1 * (Math.cos(k) / (k * (k * -t))));
	} else if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * Math.cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (t_1 / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) * (l / k)
	tmp = 0
	if k <= 3.05e-158:
		tmp = -2.0 * (t_1 * (math.cos(k) / (k * (k * -t))))
	elif k <= 1.5e+46:
		tmp = 2.0 / (((k * k) / (l * math.cos(k))) * (t * (k * (k / l))))
	else:
		tmp = -0.3333333333333333 * (t_1 / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	tmp = 0.0
	if (k <= 3.05e-158)
		tmp = Float64(-2.0 * Float64(t_1 * Float64(cos(k) / Float64(k * Float64(k * Float64(-t))))));
	elseif (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * cos(k))) * Float64(t * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(t_1 / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / k);
	tmp = 0.0;
	if (k <= 3.05e-158)
		tmp = -2.0 * (t_1 * (cos(k) / (k * (k * -t))));
	elseif (k <= 1.5e+46)
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	else
		tmp = -0.3333333333333333 * (t_1 / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.05e-158], N[(-2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(k * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.0499999999999999e-158

   1. Initial program 26.7%

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

   3. Applied egg-rr 15.9%

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

      1. expm1-def 16.5%

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

      2. expm1-log1p 16.7%

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

      3. *-commutative 16.7%

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

      4. associate-*l* 16.8%

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

   5. Simplified 16.8%

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

   6. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. unpow2 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*l* 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in k around 0 69.5%

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

       1. mul-1-neg 69.5%

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

       2. unpow2 69.5%

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

       3. associate-*l* 75.5%

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

       4. distribute-rgt-neg-in 75.5%

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

   11. Simplified 75.5%

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

   if 3.0499999999999999e-158 < k < 1.50000000000000012e46

   1. Initial program 31.4%

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

   2. Taylor expanded in t around 0 78.1%

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

      1. unpow2 78.1%

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

      2. associate-*r* 78.1%

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

      3. *-commutative 78.1%

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

      4. times-frac 95.1%

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

      5. unpow2 95.1%

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

      6. *-commutative 95.1%

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

   4. Simplified 95.1%

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

   5. Taylor expanded in k around 0 74.2%

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

      1. associate-/l* 72.2%

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

      2. associate-/r/ 78.5%

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

      3. unpow2 78.5%

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

      4. associate-*r/ 78.7%

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

   7. Simplified 78.7%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

      1. associate-/r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/r* 40.7%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.7%

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

      6. +-commutative 40.7%

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

      7. unpow2 40.7%

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

      8. sqr-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. distribute-frac-neg 40.7%

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

      11. unpow2 40.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

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

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around 0 47.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 47.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 47.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. associate-/l* 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. distribute-rgt-out 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. metadata-eval 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 61.9%

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

      1. associate-/r* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

   9. Simplified 62.0%

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

       1. frac-times 67.9%

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

   11. Applied egg-rr 67.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(-t\right)\right)}\right)\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 7: 74.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+46)
   (/ 2.0 (* (/ (* k k) (* l (cos k))) (* t (* k (/ k l)))))
   (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d+46) then
        tmp = 2.0d0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / (((k * k) / (l * Math.cos(k))) * (t * (k * (k / l))));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e+46:
		tmp = 2.0 / (((k * k) / (l * math.cos(k))) * (t * (k * (k / l))))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * cos(k))) * Float64(t * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e+46)
		tmp = 2.0 / (((k * k) / (l * cos(k))) * (t * (k * (k / l))));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000012e46

   1. Initial program 27.8%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. unpow2 68.8%

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

      2. associate-*r* 68.8%

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

      3. *-commutative 68.8%

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

      4. times-frac 81.9%

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

      5. unpow2 81.9%

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

      6. *-commutative 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in k around 0 71.0%

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

      1. associate-/l* 70.8%

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

      2. associate-/r/ 72.2%

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

      3. unpow2 72.2%

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

      4. associate-*r/ 72.2%

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

   7. Simplified 72.2%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

      1. associate-/r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/r* 40.7%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.7%

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

      6. +-commutative 40.7%

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

      7. unpow2 40.7%

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

      8. sqr-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. distribute-frac-neg 40.7%

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

      11. unpow2 40.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

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

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around 0 47.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 47.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 47.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. associate-/l* 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. distribute-rgt-out 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. metadata-eval 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 61.9%

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

      1. associate-/r* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

   9. Simplified 62.0%

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

       1. frac-times 67.9%

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

   11. Applied egg-rr 67.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 8: 71.1% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.5e+46)
   (/ 2.0 (* (* k (/ k l)) (/ (* k k) (/ l t))))
   (* -0.3333333333333333 (/ (* (/ l k) (/ l k)) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / ((k * (k / l)) * ((k * k) / (l / t)));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d+46) then
        tmp = 2.0d0 / ((k * (k / l)) * ((k * k) / (l / t)))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) * (l / k)) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.5e+46) {
		tmp = 2.0 / ((k * (k / l)) * ((k * k) / (l / t)));
	} else {
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.5e+46:
		tmp = 2.0 / ((k * (k / l)) * ((k * k) / (l / t)))
	else:
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.5e+46)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k / l)) * Float64(Float64(k * k) / Float64(l / t))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(l / k) * Float64(l / k)) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.5e+46)
		tmp = 2.0 / ((k * (k / l)) * ((k * k) / (l / t)));
	else
		tmp = -0.3333333333333333 * (((l / k) * (l / k)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.5e+46], N[(2.0 / N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000012e46

   1. Initial program 27.8%

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

   2. Taylor expanded in t around 0 68.8%

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

      1. unpow2 68.8%

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

      2. associate-*r* 68.8%

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

      3. *-commutative 68.8%

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

      4. times-frac 81.9%

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

      5. unpow2 81.9%

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

      6. *-commutative 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in k around 0 71.0%

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

      1. associate-/l* 70.8%

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

      2. unpow2 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in k around 0 69.5%

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

      1. unpow2 69.5%

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

      2. associate-*r/ 69.5%

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

   10. Simplified 69.5%

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

   if 1.50000000000000012e46 < k

   1. Initial program 37.6%

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

      1. associate-/r* 37.6%

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

      2. *-commutative 37.6%

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

      3. associate-/r* 40.7%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.7%

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

      6. +-commutative 40.7%

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

      7. unpow2 40.7%

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

      8. sqr-neg 40.7%

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

      9. distribute-frac-neg 40.7%

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

      10. distribute-frac-neg 40.7%

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

      11. unpow2 40.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

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

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around 0 47.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 47.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. unpow2 47.2%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. associate-/l* 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. distribute-rgt-out 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. metadata-eval 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 50.1%

         \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 50.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 61.9%

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

      1. associate-/r* 62.0%

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

      2. unpow2 62.0%

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

      3. unpow2 62.0%

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

   9. Simplified 62.0%

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

       1. frac-times 67.9%

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

   11. Applied egg-rr 67.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 9: 34.9% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-/r* 31.0%

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

   2. *-commutative 31.0%

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

   3. associate-/r* 36.0%

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

   4. associate-*r/ 37.1%

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

   5. associate-/l* 36.0%

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

   6. +-commutative 36.0%

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

   7. unpow2 36.0%

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

   8. sqr-neg 36.0%

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

   9. distribute-frac-neg 36.0%

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

   10. distribute-frac-neg 36.0%

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

   11. unpow2 36.0%

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

   12. associate--l+ 45.9%

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

   13. metadata-eval 45.9%

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

   14. +-rgt-identity 45.9%

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

   15. unpow2 45.9%

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

   16. distribute-frac-neg 45.9%

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

3. Simplified 45.9%

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

4. Taylor expanded in k around 0 28.5%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. unpow2 28.5%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. associate-/l* 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. unpow2 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   6. distribute-rgt-out 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   7. metadata-eval 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot \color{blue}{0.16666666666666666}}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   8. unpow2 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   9. *-commutative 30.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 30.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}{t \cdot 0.16666666666666666}}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in k around inf 33.2%

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

   1. associate-/r* 33.2%

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

   2. unpow2 33.2%

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

   3. unpow2 33.2%

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

9. Simplified 33.2%

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

    1. frac-times 36.6%

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

11. Applied egg-rr 36.6%

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

12. Final simplification 36.6%

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

Reproduce

herbie shell --seed 2023272 
(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))))