Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 93.1%

Time: 24.6s

Alternatives: 13

Speedup: 3.9×

• 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: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 8 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \left(\frac{t}{\cos k} \cdot t_1\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \end{array} \]

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 8e+100)
     (/ 2.0 (/ (* (/ (pow k 2.0) l) (* (/ t (cos k)) t_1)) l))
     (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) 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 <= 8e+100) {
		tmp = 2.0 / (((pow(k, 2.0) / l) * ((t / cos(k)) * t_1)) / l);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / 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 <= 8d+100) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * ((t / cos(k)) * t_1)) / l)
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / 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 <= 8e+100) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l) * ((t / Math.cos(k)) * t_1)) / l);
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / 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 <= 8e+100:
		tmp = 2.0 / (((math.pow(k, 2.0) / l) * ((t / math.cos(k)) * t_1)) / l)
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 8e+100)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l) * Float64(Float64(t / cos(k)) * t_1)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / 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 <= 8e+100)
		tmp = 2.0 / ((((k ^ 2.0) / l) * ((t / cos(k)) * t_1)) / l);
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / 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, 8e+100], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000013e100

   1. Initial program 34.5%

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

   3. Simplified 41.2%

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

      1. associate-*l/ 40.7%

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

      2. associate-/r* 46.1%

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

      3. associate-*r* 46.1%

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

      4. *-commutative 46.1%

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

   5. Applied egg-rr 46.1%

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

   6. Taylor expanded in t around 0 83.2%

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

      1. times-frac 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in k around inf 83.2%

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

       1. associate-/l* 88.3%

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

       2. times-frac 85.7%

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

       3. associate-/r/ 88.3%

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

       4. associate-/l* 83.1%

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

       5. associate-*l/ 88.3%

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

       6. associate-/r/ 88.3%

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

   11. Simplified 88.3%

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

   if 8.00000000000000013e100 < k

   1. Initial program 42.1%

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

      1. associate-/r* 41.9%

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

      2. associate-*l/ 41.9%

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

      3. associate--l+ 41.9%

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

   3. Simplified 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. times-frac 41.9%

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

      4. associate-+r- 41.9%

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

      5. add-exp-log 41.7%

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

      6. log1p-udef 41.7%

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

      7. expm1-udef 41.7%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

      8. expm1-log1p-u 41.9%

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

      9. sqrt-pow1 41.8%

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

      10. metadata-eval 41.8%

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

      11. pow1 41.8%

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

      12. clear-num 41.8%

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

   5. Applied egg-rr 51.7%

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

      1. associate-/l* 51.7%

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

      2. associate-/l/ 51.7%

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

      3. *-commutative 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in t around 0 66.1%

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

      1. associate-*r* 66.1%

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

      2. times-frac 66.1%

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

      3. associate-/r* 68.1%

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

      4. unpow2 68.1%

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

      5. unpow2 68.1%

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

      6. times-frac 92.7%

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

      7. unpow2 92.7%

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

   10. Simplified 92.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \left(\frac{t}{\cos k} \cdot {\sin k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-266}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{t_1}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \end{array} \]

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 (<= (* l l) 1e-266)
     (/ 2.0 (/ (* (* t (/ (pow k 2.0) (cos k))) (/ t_1 l)) l))
     (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) 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 ((l * l) <= 1e-266) {
		tmp = 2.0 / (((t * (pow(k, 2.0) / cos(k))) * (t_1 / l)) / l);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / 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 ((l * l) <= 1d-266) then
        tmp = 2.0d0 / (((t * ((k ** 2.0d0) / cos(k))) * (t_1 / l)) / l)
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / 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 ((l * l) <= 1e-266) {
		tmp = 2.0 / (((t * (Math.pow(k, 2.0) / Math.cos(k))) * (t_1 / l)) / l);
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / 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 (l * l) <= 1e-266:
		tmp = 2.0 / (((t * (math.pow(k, 2.0) / math.cos(k))) * (t_1 / l)) / l)
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e-266)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64((k ^ 2.0) / cos(k))) * Float64(t_1 / l)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / 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 ((l * l) <= 1e-266)
		tmp = 2.0 / (((t * ((k ^ 2.0) / cos(k))) * (t_1 / l)) / l);
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / 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[N[(l * l), $MachinePrecision], 1e-266], N[(2.0 / N[(N[(N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999998e-267

   1. Initial program 30.1%

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

   3. Simplified 40.1%

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

      1. associate-*l/ 40.1%

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

      2. associate-/r* 52.9%

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

      3. associate-*r* 52.9%

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

      4. *-commutative 52.9%

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

   5. Applied egg-rr 52.9%

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

   6. Taylor expanded in t around 0 79.5%

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

      1. associate-*r* 79.5%

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

      2. *-commutative 79.5%

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

      3. times-frac 85.4%

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

      4. associate-/l* 85.4%

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

      5. associate-/r/ 85.4%

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

   8. Simplified 85.4%

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

   if 9.9999999999999998e-267 < (*.f64 l l)

   1. Initial program 38.4%

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

      1. associate-/r* 38.3%

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

      2. associate-*l/ 38.3%

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

      3. associate--l+ 38.3%

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

   3. Simplified 38.3%

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

      1. *-un-lft-identity 38.3%

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

      2. add-sqr-sqrt 38.3%

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

      3. times-frac 38.3%

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

      4. associate-+r- 38.3%

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

      5. add-exp-log 38.2%

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

      6. log1p-udef 38.2%

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

      7. expm1-udef 38.2%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

      8. expm1-log1p-u 38.3%

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

      9. sqrt-pow1 28.1%

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

      10. metadata-eval 28.1%

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

      11. pow1 28.1%

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

      12. clear-num 28.1%

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

   5. Applied egg-rr 53.9%

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

      1. associate-/l* 53.9%

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

      2. associate-/l/ 53.9%

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

      3. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in t around 0 78.9%

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

      1. associate-*r* 78.9%

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

      2. times-frac 79.8%

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

      3. associate-/r* 81.5%

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

      4. unpow2 81.5%

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

      5. unpow2 81.5%

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

      6. times-frac 95.1%

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

      7. unpow2 95.1%

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

   10. Simplified 95.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-266}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\left(\frac{\ell}{\sin k}\right)}^{2} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-110)
   (/ 2.0 (/ (pow k 2.0) (* (pow (/ l (sin k)) 2.0) (/ (cos k) t))))
   (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-110) {
		tmp = 2.0 / (pow(k, 2.0) / (pow((l / sin(k)), 2.0) * (cos(k) / t)));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-110) then
        tmp = 2.0d0 / ((k ** 2.0d0) / (((l / sin(k)) ** 2.0d0) * (cos(k) / t)))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-110) {
		tmp = 2.0 / (Math.pow(k, 2.0) / (Math.pow((l / Math.sin(k)), 2.0) * (Math.cos(k) / t)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-110:
		tmp = 2.0 / (math.pow(k, 2.0) / (math.pow((l / math.sin(k)), 2.0) * (math.cos(k) / t)))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-110)
		tmp = Float64(2.0 / Float64((k ^ 2.0) / Float64((Float64(l / sin(k)) ^ 2.0) * Float64(cos(k) / t))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-110)
		tmp = 2.0 / ((k ^ 2.0) / (((l / sin(k)) ^ 2.0) * (cos(k) / t)));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-110], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5e-110

   1. Initial program 33.0%

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

   3. Simplified 43.5%

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

      1. associate-*l/ 43.5%

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

      2. associate-/r* 53.4%

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

      3. associate-*r* 53.4%

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

      4. *-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

   6. Taylor expanded in t around 0 62.3%

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

      1. associate-/l* 64.7%

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

      2. *-commutative 64.7%

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

      3. times-frac 66.6%

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

      4. unpow2 66.6%

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

      5. unpow2 66.6%

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

      6. times-frac 83.9%

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

      7. *-lft-identity 83.9%

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

      8. associate-*l/ 83.9%

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

      9. *-lft-identity 83.9%

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

      10. associate-*l/ 83.8%

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

      11. unpow2 83.8%

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

      12. associate-*l/ 83.9%

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

      13. *-lft-identity 83.9%

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

   8. Simplified 83.9%

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

   if 5e-110 < (*.f64 l l)

   1. Initial program 37.7%

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

      1. associate-/r* 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. associate-*l/ 37.6%

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

      3. associate--l+ 37.6%

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

   3. Simplified 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. add-sqr-sqrt 37.6%

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

      3. times-frac 37.6%

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

      4. associate-+r- 37.6%

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

      5. add-exp-log 37.5%

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

      6. log1p-udef 37.5%

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

      7. expm1-udef 37.5%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

      8. expm1-log1p-u 37.6%

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

      9. sqrt-pow1 26.6%

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

      10. metadata-eval 26.6%

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

      11. pow1 26.6%

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

      12. clear-num 26.6%

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

   5. Applied egg-rr 52.4%

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

      1. associate-/l* 52.4%

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

      2. associate-/l/ 52.4%

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

      3. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in t around 0 77.4%

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

      1. associate-*r* 77.4%

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

      2. times-frac 77.5%

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

      3. associate-/r* 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. times-frac 95.8%

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

      7. unpow2 95.8%

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

   10. Simplified 95.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\left(\frac{\ell}{\sin k}\right)}^{2} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. associate-/r* 35.8%

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

   2. associate-*l/ 35.7%

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

   3. associate--l+ 35.7%

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

3. Simplified 35.7%

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

4. Taylor expanded in t around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*r* 71.3%

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

   3. times-frac 72.0%

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

6. Simplified 72.0%

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

   1. add-sqr-sqrt 72.0%

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

   2. pow2 72.0%

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

   3. div-inv 72.0%

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

   4. sqrt-prod 72.0%

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

   5. pow2 72.0%

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

   6. sqrt-prod 41.6%

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

   7. add-sqr-sqrt 79.1%

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

   8. pow-flip 79.1%

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

   9. metadata-eval 79.1%

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

8. Applied egg-rr 79.1%

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

   1. expm1-log1p-u 47.9%

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

   2. expm1-udef 40.6%

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

   3. div-inv 40.6%

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

   4. associate-/r* 40.6%

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

   5. pow-flip 40.6%

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

   6. metadata-eval 40.6%

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

   7. *-commutative 40.6%

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

   8. unpow-prod-down 39.4%

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

   9. pow2 39.4%

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

   10. add-sqr-sqrt 39.4%

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

10. Applied egg-rr 39.4%

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

    1. expm1-def 45.6%

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

    2. expm1-log1p 72.0%

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

    3. associate-*l* 72.0%

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

    4. metadata-eval 72.0%

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

    5. pow-sqr 72.0%

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

    6. unpow-1 72.0%

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

    7. unpow-1 72.0%

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

    8. unpow2 72.0%

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

    9. swap-sqr 79.1%

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

    10. unpow2 79.1%

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

    11. associate-*l/ 79.1%

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

    12. *-lft-identity 79.1%

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

12. Simplified 79.1%

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

13. Final simplification 79.1%

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

Alternative 5: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. associate-/r* 35.8%

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

   2. associate-*l/ 35.7%

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

   3. associate--l+ 35.7%

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

3. Simplified 35.7%

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

4. Taylor expanded in t around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*r* 71.3%

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

   3. times-frac 72.0%

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

6. Simplified 72.0%

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

   1. add-sqr-sqrt 72.0%

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

   2. pow2 72.0%

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

   3. div-inv 72.0%

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

   4. sqrt-prod 72.0%

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

   5. pow2 72.0%

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

   6. sqrt-prod 41.6%

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

   7. add-sqr-sqrt 79.1%

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

   8. pow-flip 79.1%

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

   9. metadata-eval 79.1%

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

8. Applied egg-rr 79.1%

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

9. Taylor expanded in l around 0 72.0%

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

    1. unpow2 72.0%

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

    2. unpow2 72.0%

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

    3. times-frac 79.2%

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

    4. *-lft-identity 79.2%

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

    5. associate-*l/ 79.2%

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

    6. *-lft-identity 79.2%

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

    7. associate-*l/ 79.1%

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

    8. unpow2 79.1%

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

    9. associate-*l/ 79.2%

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

    10. *-lft-identity 79.2%

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

11. Simplified 79.2%

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

12. Final simplification 79.2%

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

Alternative 6: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. associate-/r* 35.8%

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

   2. associate-*l/ 35.7%

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

   3. associate--l+ 35.7%

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

3. Simplified 35.7%

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

   1. *-un-lft-identity 35.7%

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

   2. add-sqr-sqrt 35.7%

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

   3. times-frac 35.7%

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

   4. associate-+r- 35.7%

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

   5. add-exp-log 35.6%

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

   6. log1p-udef 35.6%

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

   7. expm1-udef 35.6%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \frac{\frac{2}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k}}{\sqrt{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]

   8. expm1-log1p-u 35.7%

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

   9. sqrt-pow1 28.7%

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

   10. metadata-eval 28.7%

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

   11. pow1 28.7%

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

   12. clear-num 28.7%

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

5. Applied egg-rr 49.7%

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

   1. associate-/l* 49.7%

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

   2. associate-/l/ 49.7%

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

   3. *-commutative 49.7%

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

7. Simplified 49.7%

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

8. Taylor expanded in t around 0 71.3%

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

   1. associate-*r* 71.3%

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

   2. times-frac 72.3%

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

   3. associate-/r* 73.0%

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

   4. unpow2 73.0%

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

   5. unpow2 73.0%

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

   6. times-frac 89.2%

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

   7. unpow2 89.2%

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

10. Simplified 89.2%

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

11. Final simplification 89.2%

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

Alternative 7: 72.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-209}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{{k}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-209)
   (/ 2.0 (* t (pow (/ (pow k 2.0) l) 2.0)))
   (* 2.0 (/ (* (cos k) (pow (/ l k) 2.0)) (* (pow k 2.0) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-209) {
		tmp = 2.0 / (t * pow((pow(k, 2.0) / l), 2.0));
	} else {
		tmp = 2.0 * ((cos(k) * pow((l / k), 2.0)) / (pow(k, 2.0) * 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 ((l * l) <= 1d-209) then
        tmp = 2.0d0 / (t * (((k ** 2.0d0) / l) ** 2.0d0))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) ** 2.0d0)) / ((k ** 2.0d0) * 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 ((l * l) <= 1e-209) {
		tmp = 2.0 / (t * Math.pow((Math.pow(k, 2.0) / l), 2.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow((l / k), 2.0)) / (Math.pow(k, 2.0) * t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-209:
		tmp = 2.0 / (t * math.pow((math.pow(k, 2.0) / l), 2.0))
	else:
		tmp = 2.0 * ((math.cos(k) * math.pow((l / k), 2.0)) / (math.pow(k, 2.0) * t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-209)
		tmp = Float64(2.0 / Float64(t * (Float64((k ^ 2.0) / l) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (Float64(l / k) ^ 2.0)) / Float64((k ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-209)
		tmp = 2.0 / (t * (((k ^ 2.0) / l) ^ 2.0));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) ^ 2.0)) / ((k ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-209], N[(2.0 / N[(t * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-209

   1. Initial program 29.8%

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

   3. Simplified 40.0%

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

      1. associate-*l/ 40.0%

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

      2. associate-/r* 51.6%

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

      3. associate-*r* 51.6%

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

      4. *-commutative 51.6%

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

   5. Applied egg-rr 51.6%

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

   6. Taylor expanded in k around 0 69.7%

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

      1. associate-/l* 68.1%

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

      2. associate-/r/ 73.0%

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

   8. Simplified 73.0%

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

      1. expm1-log1p-u 43.2%

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

      2. expm1-udef 28.0%

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

      3. associate-/l* 26.7%

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

   10. Applied egg-rr 26.7%

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

       1. expm1-def 39.4%

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

       2. expm1-log1p 69.1%

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

       3. *-rgt-identity 69.1%

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

       4. associate-*r/ 69.1%

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

       5. *-commutative 69.1%

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

       6. associate-*r/ 68.1%

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

       7. associate-/l* 69.7%

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

       8. *-commutative 69.7%

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

       9. associate-*r/ 73.0%

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

       10. *-commutative 73.0%

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

       11. associate-*l* 70.9%

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

       12. associate-*r/ 70.9%

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

       13. *-rgt-identity 70.9%

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

       14. *-rgt-identity 70.9%

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

       15. associate-*r/ 70.9%

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

       16. *-commutative 70.9%

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

       17. metadata-eval 70.9%

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

       18. pow-sqr 70.8%

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

       19. associate-*r* 77.6%

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

       20. *-commutative 77.6%

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

       21. associate-*r/ 77.6%

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

       22. *-rgt-identity 77.6%

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

       23. associate-*r/ 80.0%

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

   12. Simplified 80.0%

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

   if 1e-209 < (*.f64 l l)

   1. Initial program 38.9%

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

      1. associate-/r* 38.9%

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

      2. associate-*l/ 38.9%

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

      3. associate--l+ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-*r* 79.0%

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

      3. times-frac 79.6%

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

   6. Simplified 79.6%

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

      1. add-sqr-sqrt 79.6%

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

      2. pow2 79.6%

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

      3. div-inv 79.6%

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

      4. sqrt-prod 79.5%

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

      5. pow2 79.5%

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

      6. sqrt-prod 43.6%

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

      7. add-sqr-sqrt 79.5%

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

      8. pow-flip 79.5%

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

      9. metadata-eval 79.5%

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

   8. Applied egg-rr 79.5%

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

   9. Taylor expanded in k around 0 70.9%

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

       1. associate-*l/ 70.9%

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

       2. un-div-inv 70.9%

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

   11. Applied egg-rr 70.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-209}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{{k}^{2} \cdot t}\\ \end{array} \]

Alternative 8: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{{k}^{2} \cdot t}{\cos k}}{\ell}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Simplified 42.9%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r* 47.1%

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

   3. associate-*r* 47.1%

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

   4. *-commutative 47.1%

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

5. Applied egg-rr 47.1%

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

6. Taylor expanded in t around 0 81.1%

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

   1. times-frac 86.1%

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

8. Simplified 86.1%

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

9. Taylor expanded in k around 0 73.1%

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

10. Final simplification 73.1%

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

Alternative 9: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 7.2e-102)
   (/ 2.0 (* t (pow (/ (pow k 2.0) l) 2.0)))
   (* 2.0 (* (/ (cos k) (* (pow k 2.0) t)) (* (/ l k) (/ l k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 7.2e-102) {
		tmp = 2.0 / (t * pow((pow(k, 2.0) / l), 2.0));
	} else {
		tmp = 2.0 * ((cos(k) / (pow(k, 2.0) * t)) * ((l / k) * (l / 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 (l <= 7.2d-102) then
        tmp = 2.0d0 / (t * (((k ** 2.0d0) / l) ** 2.0d0))
    else
        tmp = 2.0d0 * ((cos(k) / ((k ** 2.0d0) * t)) * ((l / k) * (l / 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 (l <= 7.2e-102) {
		tmp = 2.0 / (t * Math.pow((Math.pow(k, 2.0) / l), 2.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (Math.pow(k, 2.0) * t)) * ((l / k) * (l / k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if l <= 7.2e-102:
		tmp = 2.0 / (t * math.pow((math.pow(k, 2.0) / l), 2.0))
	else:
		tmp = 2.0 * ((math.cos(k) / (math.pow(k, 2.0) * t)) * ((l / k) * (l / k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (l <= 7.2e-102)
		tmp = Float64(2.0 / Float64(t * (Float64((k ^ 2.0) / l) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64((k ^ 2.0) * t)) * Float64(Float64(l / k) * Float64(l / k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 7.2e-102)
		tmp = 2.0 / (t * (((k ^ 2.0) / l) ^ 2.0));
	else
		tmp = 2.0 * ((cos(k) / ((k ^ 2.0) * t)) * ((l / k) * (l / k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 7.2e-102], N[(2.0 / N[(t * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.2e-102

   1. Initial program 34.2%

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

   3. Simplified 42.0%

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

      1. associate-*l/ 41.5%

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

      2. associate-/r* 48.8%

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

      3. associate-*r* 48.8%

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

      4. *-commutative 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in k around 0 68.5%

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

      1. associate-/l* 67.6%

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

      2. associate-/r/ 69.7%

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

   8. Simplified 69.7%

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

      1. expm1-log1p-u 49.7%

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

      2. expm1-udef 25.1%

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

      3. associate-/l* 24.4%

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

   10. Applied egg-rr 24.4%

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

       1. expm1-def 47.7%

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

       2. expm1-log1p 67.6%

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

       3. *-rgt-identity 67.6%

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

       4. associate-*r/ 67.6%

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

       5. *-commutative 67.6%

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

       6. associate-*r/ 67.6%

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

       7. associate-/l* 68.5%

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

       8. *-commutative 68.5%

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

       9. associate-*r/ 69.7%

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

       10. *-commutative 69.7%

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

       11. associate-*l* 68.0%

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

       12. associate-*r/ 68.1%

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

       13. *-rgt-identity 68.1%

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

       14. *-rgt-identity 68.1%

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

       15. associate-*r/ 68.0%

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

       16. *-commutative 68.0%

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

       17. metadata-eval 68.0%

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

       18. pow-sqr 68.0%

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

       19. associate-*r* 72.7%

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

       20. *-commutative 72.7%

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

       21. associate-*r/ 72.7%

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

       22. *-rgt-identity 72.7%

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

       23. associate-*r/ 74.8%

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

   12. Simplified 74.8%

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

   if 7.2e-102 < l

   1. Initial program 38.8%

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

      1. associate-/r* 38.7%

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

      2. associate-*l/ 38.7%

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

      3. associate--l+ 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in t around 0 83.3%

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

      1. *-commutative 83.3%

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

      2. associate-*r* 83.3%

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

      3. times-frac 83.4%

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

   6. Simplified 83.4%

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

      1. add-sqr-sqrt 83.3%

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

      2. pow2 83.3%

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

      3. div-inv 83.3%

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

      4. sqrt-prod 83.3%

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

      5. pow2 83.3%

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

      6. sqrt-prod 83.2%

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

      7. add-sqr-sqrt 83.3%

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

      8. pow-flip 83.2%

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

      9. metadata-eval 83.2%

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

   8. Applied egg-rr 83.2%

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

   9. Taylor expanded in k around 0 69.8%

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

       1. unpow2 69.8%

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

       2. un-div-inv 69.8%

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

       3. un-div-inv 69.8%

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

   11. Applied egg-rr 69.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \]

Alternative 10: 69.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Simplified 42.9%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r* 47.1%

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

   3. associate-*r* 47.1%

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

   4. *-commutative 47.1%

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

5. Applied egg-rr 47.1%

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

6. Taylor expanded in k around 0 67.8%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 68.6%

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

8. Simplified 68.6%

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

   1. expm1-log1p-u 54.6%

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

   2. expm1-udef 27.3%

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

   3. associate-/l* 26.8%

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

10. Applied egg-rr 26.8%

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

    1. expm1-def 53.2%

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

    2. expm1-log1p 67.3%

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

    3. *-rgt-identity 67.3%

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

    4. associate-*r/ 67.2%

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

    5. *-commutative 67.2%

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

    6. associate-*r/ 67.2%

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

    7. associate-/l* 67.8%

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

    8. *-commutative 67.8%

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

    9. associate-*r/ 68.6%

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

    10. *-commutative 68.6%

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

    11. associate-*l* 67.4%

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

    12. associate-*r/ 67.4%

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

    13. *-rgt-identity 67.4%

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

    14. *-rgt-identity 67.4%

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

    15. associate-*r/ 67.4%

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

    16. *-commutative 67.4%

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

    17. metadata-eval 67.4%

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

    18. pow-sqr 67.3%

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

    19. associate-*r* 70.4%

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

    20. *-commutative 70.4%

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

    21. associate-*r/ 70.4%

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

    22. *-rgt-identity 70.4%

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

    23. associate-*r/ 71.8%

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

12. Simplified 71.8%

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

13. Final simplification 71.8%

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

Alternative 11: 66.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \frac{2}{t \cdot \frac{{k}^{4}}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* l (/ 2.0 (* t (/ (pow k 4.0) l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * (2.0 / (t * (pow(k, 4.0) / l)));
}

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 = l * (2.0d0 / (t * ((k ** 4.0d0) / l)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * (2.0 / (t * (Math.pow(k, 4.0) / l)));
}

Python

k = abs(k)
def code(t, l, k):
	return l * (2.0 / (t * (math.pow(k, 4.0) / l)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(2.0 / Float64(t * Float64((k ^ 4.0) / l))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * (2.0 / (t * ((k ^ 4.0) / l)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Simplified 42.9%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r* 47.1%

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

   3. associate-*r* 47.1%

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

   4. *-commutative 47.1%

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

5. Applied egg-rr 47.1%

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

6. Taylor expanded in k around 0 67.8%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 68.6%

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

8. Simplified 68.6%

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

   1. associate-/r/ 68.6%

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

   2. *-commutative 68.6%

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

10. Applied egg-rr 68.6%

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

11. Final simplification 68.6%

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

Alternative 12: 66.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\frac{t \cdot \frac{{k}^{4}}{\ell}}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (/ 2.0 (/ (* t (/ (pow k 4.0) l)) l)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / ((t * (pow(k, 4.0) / l)) / l);
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((t * ((k ** 4.0d0) / l)) / l)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / ((t * (Math.pow(k, 4.0) / l)) / l);
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((t * (math.pow(k, 4.0) / l)) / l)

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t * Float64((k ^ 4.0) / l)) / l))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / ((t * ((k ^ 4.0) / l)) / l);
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

3. Simplified 42.9%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r* 47.1%

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

   3. associate-*r* 47.1%

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

   4. *-commutative 47.1%

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

5. Applied egg-rr 47.1%

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

6. Taylor expanded in k around 0 67.8%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 68.6%

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

8. Simplified 68.6%

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

9. Final simplification 68.6%

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

Alternative 13: 34.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333))

C

k = abs(k);
double code(double t, double l, double k) {
	return (pow((l / k), 2.0) / t) * -0.3333333333333333;
}

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 = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
}

Python

k = abs(k)
def code(t, l, k):
	return (math.pow((l / k), 2.0) / t) * -0.3333333333333333

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333)
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. associate-/r* 35.8%

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

   2. associate-*l/ 35.7%

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

   3. associate--l+ 35.7%

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

3. Simplified 35.7%

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

4. Taylor expanded in k around 0 31.0%

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

5. Taylor expanded in k around inf 25.4%

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

   1. add-cube-cbrt 25.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right)} \]

   2. pow3 25.4%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right)}^{3}} \]

7. Applied egg-rr 25.4%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 25.4%

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

   2. expm1-log1p-u 24.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

   3. expm1-udef 24.6%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} - 1\right)} \]

9. Applied egg-rr 24.6%

   \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 24.9%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)} \]

    2. expm1-log1p 25.4%

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

    3. associate-/r* 25.4%

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

    4. unpow2 25.4%

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

    5. unpow2 25.4%

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

    6. times-frac 26.7%

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

    7. unpow2 26.7%

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

11. Simplified 26.7%

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

12. Final simplification 26.7%

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

Reproduce

herbie shell --seed 2023313 
(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))))