Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.7% → 91.6%

Time: 21.9s

Alternatives: 13

Speedup: 24.8×

• 34.7% 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: 34.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e-256)
   (*
    2.0
    (*
     (/ (cos k) t)
     (/ (- (* l (/ l (* k k))) (* l (* l -0.3333333333333333))) (* k k))))
   (* 2.0 (/ (cos k) (/ (* t (pow (sin k) 2.0)) (* (/ l k) (/ l k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-256) {
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * (cos(k) / ((t * pow(sin(k), 2.0)) / ((l / k) * (l / k))));
	}
	return tmp;
}

Fortran

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) <= 2d-256) then
        tmp = 2.0d0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * (-0.3333333333333333d0)))) / (k * k)))
    else
        tmp = 2.0d0 * (cos(k) / ((t * (sin(k) ** 2.0d0)) / ((l / k) * (l / k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-256) {
		tmp = 2.0 * ((Math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * (Math.cos(k) / ((t * Math.pow(Math.sin(k), 2.0)) / ((l / k) * (l / k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e-256:
		tmp = 2.0 * ((math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)))
	else:
		tmp = 2.0 * (math.cos(k) / ((t * math.pow(math.sin(k), 2.0)) / ((l / k) * (l / k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-256)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(Float64(Float64(l * Float64(l / Float64(k * k))) - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(t * (sin(k) ^ 2.0)) / Float64(Float64(l / k) * Float64(l / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e-256)
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	else
		tmp = 2.0 * (cos(k) / ((t * (sin(k) ^ 2.0)) / ((l / k) * (l / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-256], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999995e-256

   1. Initial program 20.1%

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

      1. associate-*l* 20.1%

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

      2. associate-*l* 20.1%

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

      3. associate-/r* 20.1%

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

      4. associate-/r/ 20.1%

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

      5. *-commutative 20.1%

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

      6. times-frac 20.1%

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

      7. +-commutative 20.1%

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

      8. associate--l+ 32.2%

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

      9. metadata-eval 32.2%

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

      10. +-rgt-identity 32.2%

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

      11. times-frac 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in t around 0 54.9%

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

      1. unpow2 54.9%

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

      2. times-frac 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-/r* 60.1%

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

      6. associate-/l* 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in k around 0 56.4%

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

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. unpow2 56.4%

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

      4. associate-*r/ 56.4%

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

      5. unpow2 56.4%

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

   9. Simplified 56.4%

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

   10. Taylor expanded in t around -inf 56.5%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 56.5%

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

       2. times-frac 60.2%

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

       3. distribute-rgt-neg-in 60.2%

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

       4. mul-1-neg 60.2%

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

       5. unsub-neg 60.2%

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

       6. *-commutative 60.2%

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

       7. unpow2 60.2%

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

       8. associate-*l* 60.2%

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

       9. unpow2 60.2%

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

       10. associate-*r/ 88.3%

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

       11. unpow2 88.3%

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

       12. unpow2 88.3%

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

   12. Simplified 88.3%

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

   if 1.99999999999999995e-256 < (*.f64 l l)

   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-*l* 42.1%

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

      2. associate-*l* 42.1%

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

      3. associate-/r* 42.1%

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

      4. associate-/r/ 42.1%

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

      5. *-commutative 42.1%

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

      6. times-frac 42.1%

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

      7. +-commutative 42.1%

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

      8. associate--l+ 48.4%

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

      9. metadata-eval 48.4%

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

      10. +-rgt-identity 48.4%

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

      11. times-frac 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in t around 0 74.5%

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

      1. unpow2 74.5%

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

      2. times-frac 76.5%

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

      3. unpow2 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r* 75.4%

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

      6. associate-/l* 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in l around 0 76.5%

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

      1. unpow2 76.5%

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

      2. associate-/l* 82.4%

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

      3. associate-*r/ 81.8%

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

   9. Simplified 81.8%

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

       1. associate-*r/ 84.7%

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

   11. Applied egg-rr 84.7%

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

   12. Taylor expanded in k around inf 74.5%

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

       1. associate-/l* 74.5%

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

       2. unpow2 74.5%

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

       3. *-commutative 74.5%

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

       4. associate-/l* 77.5%

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

       5. *-commutative 77.5%

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

       6. unpow2 77.5%

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

       7. times-frac 93.4%

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

   14. Simplified 93.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \end{array} \]

Alternative 2: 83.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (pow (sin k) 2.0)) (/ (/ (cos k) (* k k)) (/ t l)))))

C

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

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 * ((l / (sin(k) ** 2.0d0)) * ((cos(k) / (k * k)) / (t / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in k around inf 68.7%

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

    1. times-frac 70.5%

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

    2. unpow2 70.5%

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

    3. *-commutative 70.5%

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

    4. associate-/r* 70.9%

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

    5. unpow2 70.9%

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

    6. associate-*r/ 76.9%

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

    7. associate-/l* 82.0%

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

    8. associate-/l* 81.0%

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

    9. associate-*r/ 82.0%

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

    10. associate-*r/ 85.4%

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

    11. *-commutative 85.4%

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

    12. times-frac 84.7%

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

12. Simplified 84.7%

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

13. Final simplification 84.7%

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

Alternative 3: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot t_1 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+249}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\cos k \cdot \left(t_1 \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))))
   (if (<= (* l l) 2e-256)
     (*
      2.0
      (*
       (/ (cos k) t)
       (/ (- (* l t_1) (* l (* l -0.3333333333333333))) (* k k))))
     (if (<= (* l l) 4e+249)
       (* (/ 2.0 (* (* k k) t)) (* (/ l (sin k)) (/ l (tan k))))
       (* 2.0 (* 0.3333333333333333 (* (cos k) (* t_1 (/ l t)))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if ((l * l) <= 2e-256) {
		tmp = 2.0 * ((cos(k) / t) * (((l * t_1) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else if ((l * l) <= 4e+249) {
		tmp = (2.0 / ((k * k) * t)) * ((l / sin(k)) * (l / tan(k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (cos(k) * (t_1 * (l / t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k * k)
    if ((l * l) <= 2d-256) then
        tmp = 2.0d0 * ((cos(k) / t) * (((l * t_1) - (l * (l * (-0.3333333333333333d0)))) / (k * k)))
    else if ((l * l) <= 4d+249) then
        tmp = (2.0d0 / ((k * k) * t)) * ((l / sin(k)) * (l / tan(k)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * (cos(k) * (t_1 * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if ((l * l) <= 2e-256) {
		tmp = 2.0 * ((Math.cos(k) / t) * (((l * t_1) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else if ((l * l) <= 4e+249) {
		tmp = (2.0 / ((k * k) * t)) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (Math.cos(k) * (t_1 * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (k * k)
	tmp = 0
	if (l * l) <= 2e-256:
		tmp = 2.0 * ((math.cos(k) / t) * (((l * t_1) - (l * (l * -0.3333333333333333))) / (k * k)))
	elif (l * l) <= 4e+249:
		tmp = (2.0 / ((k * k) * t)) * ((l / math.sin(k)) * (l / math.tan(k)))
	else:
		tmp = 2.0 * (0.3333333333333333 * (math.cos(k) * (t_1 * (l / t))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (Float64(l * l) <= 2e-256)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(Float64(Float64(l * t_1) - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * k))));
	elseif (Float64(l * l) <= 4e+249)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t)) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(cos(k) * Float64(t_1 * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k * k);
	tmp = 0.0;
	if ((l * l) <= 2e-256)
		tmp = 2.0 * ((cos(k) / t) * (((l * t_1) - (l * (l * -0.3333333333333333))) / (k * k)));
	elseif ((l * l) <= 4e+249)
		tmp = (2.0 / ((k * k) * t)) * ((l / sin(k)) * (l / tan(k)));
	else
		tmp = 2.0 * (0.3333333333333333 * (cos(k) * (t_1 * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e-256], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l * t$95$1), $MachinePrecision] - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 4e+249], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[Cos[k], $MachinePrecision] * N[(t$95$1 * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999995e-256

   1. Initial program 20.1%

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

      1. associate-*l* 20.1%

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

      2. associate-*l* 20.1%

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

      3. associate-/r* 20.1%

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

      4. associate-/r/ 20.1%

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

      5. *-commutative 20.1%

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

      6. times-frac 20.1%

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

      7. +-commutative 20.1%

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

      8. associate--l+ 32.2%

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

      9. metadata-eval 32.2%

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

      10. +-rgt-identity 32.2%

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

      11. times-frac 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in t around 0 54.9%

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

      1. unpow2 54.9%

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

      2. times-frac 56.4%

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

      3. unpow2 56.4%

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

      4. *-commutative 56.4%

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

      5. associate-/r* 60.1%

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

      6. associate-/l* 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in k around 0 56.4%

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

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. unpow2 56.4%

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

      4. associate-*r/ 56.4%

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

      5. unpow2 56.4%

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

   9. Simplified 56.4%

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

   10. Taylor expanded in t around -inf 56.5%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 56.5%

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

       2. times-frac 60.2%

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

       3. distribute-rgt-neg-in 60.2%

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

       4. mul-1-neg 60.2%

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

       5. unsub-neg 60.2%

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

       6. *-commutative 60.2%

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

       7. unpow2 60.2%

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

       8. associate-*l* 60.2%

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

       9. unpow2 60.2%

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

       10. associate-*r/ 88.3%

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

       11. unpow2 88.3%

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

       12. unpow2 88.3%

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

   12. Simplified 88.3%

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

   if 1.99999999999999995e-256 < (*.f64 l l) < 3.9999999999999997e249

   1. Initial program 48.4%

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

      1. associate-*l* 48.4%

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

      2. associate-*l* 48.4%

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

      3. associate-/r* 48.4%

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

      4. associate-/r/ 48.4%

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

      5. *-commutative 48.4%

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

      6. times-frac 48.5%

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

      7. +-commutative 48.5%

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

      8. associate--l+ 58.0%

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

      9. metadata-eval 58.0%

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

      10. +-rgt-identity 58.0%

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

      11. times-frac 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in t around 0 87.8%

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

      1. unpow2 87.8%

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

   6. Simplified 87.8%

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

   if 3.9999999999999997e249 < (*.f64 l l)

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. associate-*l* 34.9%

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

      3. associate-/r* 34.9%

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

      4. associate-/r/ 34.9%

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

      5. *-commutative 34.9%

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

      6. times-frac 34.8%

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

      7. +-commutative 34.8%

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

      8. associate--l+ 37.4%

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

      9. metadata-eval 37.4%

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

      10. +-rgt-identity 37.4%

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

      11. times-frac 37.4%

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

   3. Simplified 37.4%

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

   4. Taylor expanded in t around 0 61.4%

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

      1. unpow2 61.4%

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

      2. times-frac 64.9%

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

      3. unpow2 64.9%

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

      4. *-commutative 64.9%

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

      5. associate-/r* 64.9%

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

      6. associate-/l* 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

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

      2. *-commutative 60.7%

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

      3. unpow2 60.7%

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

      4. associate-*r/ 60.7%

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

      5. unpow2 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in k around inf 60.4%

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

       1. unpow2 60.4%

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

       2. associate-*r/ 60.4%

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

       3. unpow2 60.4%

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

       4. unpow2 60.4%

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

       5. times-frac 67.5%

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

       6. unpow2 67.5%

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

       7. *-commutative 67.5%

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

   12. Simplified 67.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-256}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+249}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 4: 73.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-57}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right) + \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-57)
   (*
    2.0
    (*
     (/ (cos k) t)
     (/ (- (* l (/ l (* k k))) (* l (* l -0.3333333333333333))) (* k k))))
   (*
    2.0
    (*
     (/ (cos k) (* k k))
     (+ (* 0.3333333333333333 (* l (/ l t))) (/ l (/ (* k k) (/ l t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-57) {
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((0.3333333333333333 * (l * (l / t))) + (l / ((k * k) / (l / t)))));
	}
	return tmp;
}

Fortran

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-57) then
        tmp = 2.0d0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * (-0.3333333333333333d0)))) / (k * k)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((0.3333333333333333d0 * (l * (l / t))) + (l / ((k * k) / (l / t)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-57) {
		tmp = 2.0 * ((Math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((0.3333333333333333 * (l * (l / t))) + (l / ((k * k) / (l / t)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-57:
		tmp = 2.0 * ((math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((0.3333333333333333 * (l * (l / t))) + (l / ((k * k) / (l / t)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-57)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(Float64(Float64(l * Float64(l / Float64(k * k))) - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(0.3333333333333333 * Float64(l * Float64(l / t))) + Float64(l / Float64(Float64(k * k) / Float64(l / t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-57)
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((0.3333333333333333 * (l * (l / t))) + (l / ((k * k) / (l / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-57], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(0.3333333333333333 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999955e-58

   1. Initial program 29.3%

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

      1. associate-*l* 29.2%

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

      2. associate-*l* 29.2%

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

      3. associate-/r* 29.2%

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

      4. associate-/r/ 29.2%

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

      5. *-commutative 29.2%

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

      6. times-frac 30.1%

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

      7. +-commutative 30.1%

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

      8. associate--l+ 41.2%

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

      9. metadata-eval 41.2%

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

      10. +-rgt-identity 41.2%

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

      11. times-frac 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in t around 0 66.2%

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

      1. unpow2 66.2%

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

      2. times-frac 68.5%

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

      3. unpow2 68.5%

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

      4. *-commutative 68.5%

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

      5. associate-/r* 69.2%

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

      6. associate-/l* 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in k around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-commutative 64.3%

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

      3. unpow2 64.3%

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

      4. associate-*r/ 64.3%

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

      5. unpow2 64.3%

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

   9. Simplified 64.3%

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

   10. Taylor expanded in t around -inf 64.4%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 64.4%

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

       2. times-frac 66.7%

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

       3. distribute-rgt-neg-in 66.7%

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

       4. mul-1-neg 66.7%

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

       5. unsub-neg 66.7%

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

       6. *-commutative 66.7%

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

       7. unpow2 66.7%

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

       8. associate-*l* 66.7%

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

       9. unpow2 66.7%

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

       10. associate-*r/ 84.8%

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

       11. unpow2 84.8%

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

       12. unpow2 84.8%

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

   12. Simplified 84.8%

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

   if 9.99999999999999955e-58 < (*.f64 l l)

   1. Initial program 41.0%

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

      1. associate-*l* 41.0%

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

      2. associate-*l* 41.0%

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

      3. associate-/r* 41.0%

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

      4. associate-/r/ 40.9%

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

      5. *-commutative 40.9%

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

      6. times-frac 40.2%

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

      7. +-commutative 40.2%

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

      8. associate--l+ 45.6%

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

      9. metadata-eval 45.6%

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

      10. +-rgt-identity 45.6%

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

      11. times-frac 45.6%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in t around 0 70.9%

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

      1. unpow2 70.9%

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

      2. times-frac 72.3%

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

      3. unpow2 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-/r* 72.2%

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

      6. associate-/l* 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in l around 0 72.3%

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

      1. unpow2 72.3%

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

      2. associate-/l* 79.9%

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

      3. associate-*r/ 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in k around 0 64.3%

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

       1. +-commutative 64.3%

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

       2. *-commutative 64.3%

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

       3. unpow2 64.3%

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

       4. associate-*r/ 64.3%

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

       5. unpow2 64.3%

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

       6. unpow2 64.3%

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

       7. associate-/l* 68.0%

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

       8. unpow2 68.0%

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

       9. associate-/l* 68.0%

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

       10. unpow2 68.0%

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

   12. Simplified 68.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-57}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right) + \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right)\right)\\ \end{array} \]

Alternative 5: 72.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+236}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+236)
   (*
    2.0
    (*
     (/ (cos k) t)
     (/ (- (* l (/ l (* k k))) (* l (* l -0.3333333333333333))) (* k k))))
   (* 2.0 (* 0.3333333333333333 (* (/ (cos k) (* k k)) (* l (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+236) {
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Fortran

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+236) then
        tmp = 2.0d0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * (-0.3333333333333333d0)))) / (k * k)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * ((cos(k) / (k * k)) * (l * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+236) {
		tmp = 2.0 * ((Math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((Math.cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+236:
		tmp = 2.0 * ((math.cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)))
	else:
		tmp = 2.0 * (0.3333333333333333 * ((math.cos(k) / (k * k)) * (l * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+236)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64(Float64(Float64(l * Float64(l / Float64(k * k))) - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+236)
		tmp = 2.0 * ((cos(k) / t) * (((l * (l / (k * k))) - (l * (l * -0.3333333333333333))) / (k * k)));
	else
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+236], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999997e236

   1. Initial program 35.7%

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

      1. associate-*l* 35.7%

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

      2. associate-*l* 35.7%

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

      3. associate-/r* 35.7%

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

      4. associate-/r/ 35.7%

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

      5. *-commutative 35.7%

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

      6. times-frac 35.8%

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

      7. +-commutative 35.8%

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

      8. associate--l+ 46.5%

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

      9. metadata-eval 46.5%

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

      10. +-rgt-identity 46.5%

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

      11. times-frac 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in t around 0 72.5%

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

      1. unpow2 72.5%

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

      2. times-frac 73.0%

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

      3. unpow2 73.0%

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

      4. *-commutative 73.0%

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

      5. associate-/r* 73.5%

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

      6. associate-/l* 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in k around 0 66.1%

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

      1. unpow2 66.1%

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

      2. *-commutative 66.1%

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

      3. unpow2 66.1%

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

      4. associate-*r/ 66.1%

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

      5. unpow2 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in t around -inf 66.4%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-0.3333333333333333 \cdot {\ell}^{2} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \cos k}{{k}^{2} \cdot t}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 66.4%

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

       2. times-frac 67.0%

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

       3. distribute-rgt-neg-in 67.0%

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

       4. mul-1-neg 67.0%

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

       5. unsub-neg 67.0%

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

       6. *-commutative 67.0%

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

       7. unpow2 67.0%

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

       8. associate-*l* 67.0%

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

       9. unpow2 67.0%

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

       10. associate-*r/ 79.5%

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

       11. unpow2 79.5%

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

       12. unpow2 79.5%

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

   12. Simplified 79.5%

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

   if 4.9999999999999997e236 < (*.f64 l l)

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-*l* 35.2%

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

      3. associate-/r* 35.2%

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

      4. associate-/r/ 35.2%

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

      5. *-commutative 35.2%

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

      6. times-frac 35.2%

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

      7. +-commutative 35.2%

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

      8. associate--l+ 37.7%

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

      9. metadata-eval 37.7%

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

      10. +-rgt-identity 37.7%

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

      11. times-frac 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around 0 61.2%

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

      1. unpow2 61.2%

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

      2. times-frac 65.7%

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

      3. unpow2 65.7%

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

      4. *-commutative 65.7%

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

      5. associate-/r* 65.7%

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

      6. associate-/l* 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

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

      2. *-commutative 60.7%

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

      3. unpow2 60.7%

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

      4. associate-*r/ 60.7%

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

      5. unpow2 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in k around inf 59.2%

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

       1. times-frac 60.9%

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

       2. unpow2 60.9%

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

       3. unpow2 60.9%

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

       4. associate-*r/ 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+236}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{\ell \cdot \frac{\ell}{k \cdot k} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 6: 71.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+204}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{1}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+204)
   (* 2.0 (* (/ l (/ (* k k) (/ l t))) (/ 1.0 (* k k))))
   (* 2.0 (* 0.3333333333333333 (* (cos k) (* (/ l (* k k)) (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+204) {
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (cos(k) * ((l / (k * k)) * (l / t))));
	}
	return tmp;
}

Fortran

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) <= 2d+204) then
        tmp = 2.0d0 * ((l / ((k * k) / (l / t))) * (1.0d0 / (k * k)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * (cos(k) * ((l / (k * k)) * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+204) {
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * (Math.cos(k) * ((l / (k * k)) * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+204:
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)))
	else:
		tmp = 2.0 * (0.3333333333333333 * (math.cos(k) * ((l / (k * k)) * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+204)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(Float64(k * k) / Float64(l / t))) * Float64(1.0 / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(cos(k) * Float64(Float64(l / Float64(k * k)) * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+204)
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	else
		tmp = 2.0 * (0.3333333333333333 * (cos(k) * ((l / (k * k)) * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+204], N[(2.0 * N[(N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999998e204

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. associate-*l* 36.6%

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

      3. associate-/r* 36.6%

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

      4. associate-/r/ 36.6%

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

      5. *-commutative 36.6%

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

      6. times-frac 36.7%

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

      7. +-commutative 36.7%

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

      8. associate--l+ 47.9%

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

      9. metadata-eval 47.9%

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

      10. +-rgt-identity 47.9%

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

      11. times-frac 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in t around 0 73.1%

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

      1. unpow2 73.1%

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

      2. times-frac 74.2%

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

      3. unpow2 74.2%

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

      4. *-commutative 74.2%

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

      5. associate-/r* 74.7%

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

      6. associate-/l* 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in l around 0 74.2%

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

      1. unpow2 74.2%

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

      2. associate-/l* 84.1%

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

      3. associate-*r/ 85.6%

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

   9. Simplified 85.6%

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

   10. Taylor expanded in k around 0 75.0%

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

       1. associate-/l* 76.6%

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

       2. unpow2 76.6%

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

   12. Simplified 76.6%

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

   13. Taylor expanded in k around 0 76.4%

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

       1. unpow2 76.4%

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

   15. Simplified 76.4%

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

   if 1.99999999999999998e204 < (*.f64 l l)

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. associate-*l* 33.7%

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

      3. associate-/r* 33.7%

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

      4. associate-/r/ 33.7%

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

      5. *-commutative 33.7%

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

      6. times-frac 33.6%

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

      7. +-commutative 33.6%

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

      8. associate--l+ 36.0%

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

      9. metadata-eval 36.0%

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

      10. +-rgt-identity 36.0%

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

      11. times-frac 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in t around 0 61.0%

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

      1. unpow2 61.0%

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

      2. times-frac 64.1%

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

      3. unpow2 64.1%

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

      4. *-commutative 64.1%

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

      5. associate-/r* 64.1%

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

      6. associate-/l* 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in k around 0 59.4%

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

      1. unpow2 59.4%

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

      2. *-commutative 59.4%

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

      3. unpow2 59.4%

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

      4. associate-*r/ 59.4%

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

      5. unpow2 59.4%

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

   9. Simplified 59.4%

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

   10. Taylor expanded in k around inf 57.4%

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

       1. unpow2 57.4%

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

       2. associate-*r/ 57.4%

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

       3. unpow2 57.4%

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

       4. unpow2 57.4%

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

       5. times-frac 64.8%

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

       6. unpow2 64.8%

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

       7. *-commutative 64.8%

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

   12. Simplified 64.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+204}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{1}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 7: 70.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+220}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{1}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+220)
   (* 2.0 (* (/ l (/ (* k k) (/ l t))) (/ 1.0 (* k k))))
   (* 2.0 (* 0.3333333333333333 (* (/ (cos k) (* k k)) (* l (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+220) {
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Fortran

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+220) then
        tmp = 2.0d0 * ((l / ((k * k) / (l / t))) * (1.0d0 / (k * k)))
    else
        tmp = 2.0d0 * (0.3333333333333333d0 * ((cos(k) / (k * k)) * (l * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+220) {
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	} else {
		tmp = 2.0 * (0.3333333333333333 * ((Math.cos(k) / (k * k)) * (l * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+220:
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)))
	else:
		tmp = 2.0 * (0.3333333333333333 * ((math.cos(k) / (k * k)) * (l * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+220)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(Float64(k * k) / Float64(l / t))) * Float64(1.0 / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+220)
		tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
	else
		tmp = 2.0 * (0.3333333333333333 * ((cos(k) / (k * k)) * (l * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+220], N[(2.0 * N[(N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.3333333333333333 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000002e220

   1. Initial program 36.2%

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

      1. associate-*l* 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-/r* 36.2%

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

      4. associate-/r/ 36.1%

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

      5. *-commutative 36.1%

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

      6. times-frac 36.2%

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

      7. +-commutative 36.2%

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

      8. associate--l+ 47.1%

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

      9. metadata-eval 47.1%

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

      10. +-rgt-identity 47.1%

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

      11. times-frac 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in t around 0 72.7%

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

      1. unpow2 72.7%

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

      2. times-frac 73.8%

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

      3. unpow2 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-/r* 74.3%

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

      6. associate-/l* 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in l around 0 73.8%

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

      1. unpow2 73.8%

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

      2. associate-/l* 83.4%

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

      3. associate-*r/ 84.9%

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

   9. Simplified 84.9%

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

   10. Taylor expanded in k around 0 74.6%

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

       1. associate-/l* 76.1%

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

       2. unpow2 76.1%

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

   12. Simplified 76.1%

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

   13. Taylor expanded in k around 0 75.9%

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

       1. unpow2 75.9%

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

   15. Simplified 75.9%

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

   if 5.0000000000000002e220 < (*.f64 l l)

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l* 34.4%

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

      3. associate-/r* 34.4%

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

      4. associate-/r/ 34.4%

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

      5. *-commutative 34.4%

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

      6. times-frac 34.4%

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

      7. +-commutative 34.4%

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

      8. associate--l+ 36.9%

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

      9. metadata-eval 36.9%

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

      10. +-rgt-identity 36.9%

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

      11. times-frac 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around 0 61.0%

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

      1. unpow2 61.0%

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

      2. times-frac 64.3%

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

      3. unpow2 64.3%

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

      4. *-commutative 64.3%

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

      5. associate-/r* 64.3%

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

      6. associate-/l* 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in k around 0 59.4%

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

      1. unpow2 59.4%

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

      2. *-commutative 59.4%

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

      3. unpow2 59.4%

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

      4. associate-*r/ 59.4%

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

      5. unpow2 59.4%

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

   9. Simplified 59.4%

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

   10. Taylor expanded in k around inf 58.2%

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

       1. times-frac 59.7%

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

       2. unpow2 59.7%

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

       3. unpow2 59.7%

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

       4. associate-*r/ 65.2%

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

   12. Simplified 65.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+220}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{1}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 8: 71.4% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (cos k) (* k k)) (/ l (/ (* k k) (/ l t))))))

C

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

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 * ((cos(k) / (k * k)) * (l / ((k * k) / (l / t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in k around 0 70.6%

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

    1. associate-/l* 71.6%

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

    2. unpow2 71.6%

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

12. Simplified 71.6%

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

13. Final simplification 71.6%

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

Alternative 9: 71.5% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* (cos k) (* (/ l (* k k)) (/ l t))) (* k k))))

C

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

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 * ((cos(k) * ((l / (k * k)) * (l / t))) / (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in k around 0 70.6%

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

    1. associate-/l* 71.6%

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

    2. unpow2 71.6%

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

12. Simplified 71.6%

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

    1. associate-*l/ 71.8%

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

    2. associate-/r/ 71.8%

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

14. Applied egg-rr 71.8%

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

15. Final simplification 71.8%

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

Alternative 10: 71.2% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* l (/ (cos k) (* k k))) (/ (* k k) (/ l t)))))

C

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

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 * ((l * (cos(k) / (k * k))) / ((k * k) / (l / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

    1. associate-*r/ 85.4%

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

11. Applied egg-rr 85.4%

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

12. Taylor expanded in k around 0 70.9%

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

    1. associate-/l* 71.6%

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

    2. unpow2 71.6%

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

14. Simplified 72.6%

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

15. Final simplification 72.6%

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

Alternative 11: 69.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (/ (* k k) (/ l t))) (+ (/ 1.0 (* k k)) -0.5))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / ((k * k) / (l / t))) * ((1.0 / (k * k)) + -0.5));
}

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 * ((l / ((k * k) / (l / t))) * ((1.0d0 / (k * k)) + (-0.5d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / ((k * k) / (l / t))) * ((1.0 / (k * k)) + -0.5));
}

Python

def code(t, l, k):
	return 2.0 * ((l / ((k * k) / (l / t))) * ((1.0 / (k * k)) + -0.5))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(Float64(k * k) / Float64(l / t))) * Float64(Float64(1.0 / Float64(k * k)) + -0.5)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / ((k * k) / (l / t))) * ((1.0 / (k * k)) + -0.5));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in k around 0 70.6%

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

    1. associate-/l* 71.6%

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

    2. unpow2 71.6%

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

12. Simplified 71.6%

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

13. Taylor expanded in k around 0 70.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right) \]
14. Step-by-step derivation

    1. sub-neg 70.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right) \]

    2. unpow2 70.2%

       \[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right) \]

    3. metadata-eval 70.2%

       \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right) \]

15. Simplified 70.2%

    \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}}\right) \]

16. Final simplification 70.2%

    \[\leadsto 2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right) \]

Alternative 12: 69.3% accurate, 24.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (/ (* k k) (/ l t))) (/ 1.0 (* k k)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
}

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 * ((l / ((k * k) / (l / t))) * (1.0d0 / (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / Float64(Float64(k * k) / Float64(l / t))) * Float64(1.0 / Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / ((k * k) / (l / t))) * (1.0 / (k * k)));
end

Wolfram

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

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in l around 0 70.5%

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

   1. unpow2 70.5%

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

   2. associate-/l* 81.0%

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

   3. associate-*r/ 82.0%

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

9. Simplified 82.0%

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

10. Taylor expanded in k around 0 70.6%

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

    1. associate-/l* 71.6%

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

    2. unpow2 71.6%

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

12. Simplified 71.6%

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

13. Taylor expanded in k around 0 68.6%

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

    1. unpow2 68.6%

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

15. Simplified 68.6%

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

16. Final simplification 68.6%

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

Alternative 13: 31.9% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot 0.041666666666666664\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (* l (/ l t)) 0.041666666666666664)))

C

double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) * 0.041666666666666664);
}

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 * ((l * (l / t)) * 0.041666666666666664d0)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) * 0.041666666666666664);
}

Python

def code(t, l, k):
	return 2.0 * ((l * (l / t)) * 0.041666666666666664)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / t)) * 0.041666666666666664))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / t)) * 0.041666666666666664);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

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

   1. associate-*l* 35.6%

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

   2. associate-*l* 35.6%

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

   3. associate-/r* 35.6%

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

   4. associate-/r/ 35.6%

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

   5. *-commutative 35.6%

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

   6. times-frac 35.6%

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

   7. +-commutative 35.6%

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

   8. associate--l+ 43.6%

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

   9. metadata-eval 43.6%

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

   10. +-rgt-identity 43.6%

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

   11. times-frac 48.7%

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

3. Simplified 48.7%

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

4. Taylor expanded in t around 0 68.7%

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

   1. unpow2 68.7%

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

   2. times-frac 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-/r* 70.9%

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

   6. associate-/l* 76.9%

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

6. Simplified 76.9%

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

7. Taylor expanded in k around 0 42.6%

   \[\leadsto 2 \cdot \left(\color{blue}{\left(\left(\frac{1}{{k}^{2}} + 0.041666666666666664 \cdot {k}^{2}\right) - 0.5\right)} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]
8. Step-by-step derivation

   1. associate--l+ 42.6%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(0.041666666666666664 \cdot {k}^{2} - 0.5\right)\right)} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]

   2. unpow2 42.6%

      \[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(0.041666666666666664 \cdot {k}^{2} - 0.5\right)\right) \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]

   3. *-commutative 42.6%

      \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\color{blue}{{k}^{2} \cdot 0.041666666666666664} - 0.5\right)\right) \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]

   4. unpow2 42.6%

      \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\color{blue}{\left(k \cdot k\right)} \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]

9. Simplified 42.6%

   \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right)} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \]

10. Taylor expanded in k around 0 46.0%

    \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}\right) \]
11. Step-by-step derivation

    1. unpow2 46.0%

       \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}\right) \]

    2. *-commutative 46.0%

       \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{2}}}\right) \]

    3. unpow2 46.0%

       \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(k \cdot k\right)}}\right) \]

12. Simplified 46.0%

    \[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}}\right) \]

13. Taylor expanded in k around inf 36.5%

    \[\leadsto 2 \cdot \color{blue}{\left(0.041666666666666664 \cdot \frac{{\ell}^{2}}{t}\right)} \]
14. Step-by-step derivation

    1. unpow2 36.5%

       \[\leadsto 2 \cdot \left(0.041666666666666664 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

    2. associate-*r/ 30.7%

       \[\leadsto 2 \cdot \left(0.041666666666666664 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right) \]

15. Simplified 30.7%

    \[\leadsto 2 \cdot \color{blue}{\left(0.041666666666666664 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)} \]

16. Final simplification 30.7%

    \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot 0.041666666666666664\right) \]

Reproduce

herbie shell --seed 2023171 
(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))))