Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 90.5%

Time: 8.7s

Alternatives: 8

Speedup: N/A×

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 90.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot l\_m\\ t_2 := {\sin k\_m}^{2}\\ \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot t\right)}{\cos k\_m} \cdot {\left(e^{2}\right)}^{\left(\log \sin k\_m - \log l\_m\right)}}\\ \mathbf{elif}\;k\_m \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{\frac{2}{k\_m}}{k\_m}}{t\_2} \cdot \frac{l\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k\_m} \cdot \left(-1 \cdot \left(t\_1 \cdot l\_m\right)\right)}{k\_m \cdot \left(t\_2 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l_m)) (t_2 (pow (sin k_m) 2.0)))
   (if (<= k_m 8.2e-69)
     (/
      2.0
      (*
       (/ (* k_m (* k_m t)) (cos k_m))
       (pow (exp 2.0) (- (log (sin k_m)) (log l_m)))))
     (if (<= k_m 2.1e+150)
       (* (/ (* t_1 (/ (/ 2.0 k_m) k_m)) t_2) (/ l_m t))
       (/ (* (/ -2.0 k_m) (* -1.0 (* t_1 l_m))) (* k_m (* t_2 t)))))))

C

l_m = fabs(l);
k_m = fabs(k);
double code(double t, double l_m, double k_m) {
	double t_1 = cos(k_m) * l_m;
	double t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.2e-69) {
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * pow(exp(2.0), (log(sin(k_m)) - log(l_m))));
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_1 * ((2.0 / k_m) / k_m)) / t_2) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_1 * l_m))) / (k_m * (t_2 * t));
	}
	return tmp;
}

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(k_m) * l_m
    t_2 = sin(k_m) ** 2.0d0
    if (k_m <= 8.2d-69) then
        tmp = 2.0d0 / (((k_m * (k_m * t)) / cos(k_m)) * (exp(2.0d0) ** (log(sin(k_m)) - log(l_m))))
    else if (k_m <= 2.1d+150) then
        tmp = ((t_1 * ((2.0d0 / k_m) / k_m)) / t_2) * (l_m / t)
    else
        tmp = (((-2.0d0) / k_m) * ((-1.0d0) * (t_1 * l_m))) / (k_m * (t_2 * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
public static double code(double t, double l_m, double k_m) {
	double t_1 = Math.cos(k_m) * l_m;
	double t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.2e-69) {
		tmp = 2.0 / (((k_m * (k_m * t)) / Math.cos(k_m)) * Math.pow(Math.exp(2.0), (Math.log(Math.sin(k_m)) - Math.log(l_m))));
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_1 * ((2.0 / k_m) / k_m)) / t_2) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_1 * l_m))) / (k_m * (t_2 * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
def code(t, l_m, k_m):
	t_1 = math.cos(k_m) * l_m
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 8.2e-69:
		tmp = 2.0 / (((k_m * (k_m * t)) / math.cos(k_m)) * math.pow(math.exp(2.0), (math.log(math.sin(k_m)) - math.log(l_m))))
	elif k_m <= 2.1e+150:
		tmp = ((t_1 * ((2.0 / k_m) / k_m)) / t_2) * (l_m / t)
	else:
		tmp = ((-2.0 / k_m) * (-1.0 * (t_1 * l_m))) / (k_m * (t_2 * t))
	return tmp

Julia

l_m = abs(l)
k_m = abs(k)
function code(t, l_m, k_m)
	t_1 = Float64(cos(k_m) * l_m)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 8.2e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * t)) / cos(k_m)) * (exp(2.0) ^ Float64(log(sin(k_m)) - log(l_m)))));
	elseif (k_m <= 2.1e+150)
		tmp = Float64(Float64(Float64(t_1 * Float64(Float64(2.0 / k_m) / k_m)) / t_2) * Float64(l_m / t));
	else
		tmp = Float64(Float64(Float64(-2.0 / k_m) * Float64(-1.0 * Float64(t_1 * l_m))) / Float64(k_m * Float64(t_2 * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
k_m = abs(k);
function tmp_2 = code(t, l_m, k_m)
	t_1 = cos(k_m) * l_m;
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 8.2e-69)
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * (exp(2.0) ^ (log(sin(k_m)) - log(l_m))));
	elseif (k_m <= 2.1e+150)
		tmp = ((t_1 * ((2.0 / k_m) / k_m)) / t_2) * (l_m / t);
	else
		tmp = ((-2.0 / k_m) * (-1.0 * (t_1 * l_m))) / (k_m * (t_2 * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 8.2e-69], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[Log[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision] - N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.1e+150], N[(N[(N[(t$95$1 * N[(N[(2.0 / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / k$95$m), $MachinePrecision] * N[(-1.0 * N[(t$95$1 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.1999999999999998e-69

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 16.0

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

   9. Applied rewrites 16.0%

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

       1. lift-exp.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-log.f64 N/A

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

       8. distribute-rgt-out-- N/A

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

       9. exp-prod N/A

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

       10. lower-pow.f64 N/A

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

       11. lower-exp.f64 N/A

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

       12. lower--.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-log.f64 N/A

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

       15. lift-log.f64 16.0

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

   11. Applied rewrites 16.0%

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

   if 8.1999999999999998e-69 < k < 2.09999999999999998e150

   1. Initial program 26.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 90.7

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

   9. Applied rewrites 90.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. lift-sin.f64 N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 99.1%

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

   if 2.09999999999999998e150 < k

   1. Initial program 36.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 56.9

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

   5. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. associate-/l* N/A

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

      13. associate-/r* N/A

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

      14. frac-times N/A

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

      15. lower-/.f64 N/A

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

   9. Applied rewrites 76.4%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot {\left(e^{2}\right)}^{\left(\log \sin k - \log \ell\right)}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k}}{{\sin k}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k} \cdot \left(-1 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)\right)}{k \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot l\_m\\ \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot t\right)}{\cos k\_m} \cdot e^{\log \sin k\_m \cdot 2 - \log l\_m \cdot 2}}\\ \mathbf{elif}\;k\_m \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_2 \cdot \frac{\frac{2}{k\_m}}{k\_m}}{t\_1} \cdot \frac{l\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k\_m} \cdot \left(-1 \cdot \left(t\_2 \cdot l\_m\right)\right)}{k\_m \cdot \left(t\_1 \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l_m)))
   (if (<= k_m 8.2e-69)
     (/
      2.0
      (*
       (/ (* k_m (* k_m t)) (cos k_m))
       (exp (- (* (log (sin k_m)) 2.0) (* (log l_m) 2.0)))))
     (if (<= k_m 2.1e+150)
       (* (/ (* t_2 (/ (/ 2.0 k_m) k_m)) t_1) (/ l_m t))
       (/ (* (/ -2.0 k_m) (* -1.0 (* t_2 l_m))) (* k_m (* t_1 t)))))))

C

l_m = fabs(l);
k_m = fabs(k);
double code(double t, double l_m, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l_m;
	double tmp;
	if (k_m <= 8.2e-69) {
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * exp(((log(sin(k_m)) * 2.0) - (log(l_m) * 2.0))));
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_2 * l_m))) / (k_m * (t_1 * t));
	}
	return tmp;
}

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    t_2 = cos(k_m) * l_m
    if (k_m <= 8.2d-69) then
        tmp = 2.0d0 / (((k_m * (k_m * t)) / cos(k_m)) * exp(((log(sin(k_m)) * 2.0d0) - (log(l_m) * 2.0d0))))
    else if (k_m <= 2.1d+150) then
        tmp = ((t_2 * ((2.0d0 / k_m) / k_m)) / t_1) * (l_m / t)
    else
        tmp = (((-2.0d0) / k_m) * ((-1.0d0) * (t_2 * l_m))) / (k_m * (t_1 * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
public static double code(double t, double l_m, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double t_2 = Math.cos(k_m) * l_m;
	double tmp;
	if (k_m <= 8.2e-69) {
		tmp = 2.0 / (((k_m * (k_m * t)) / Math.cos(k_m)) * Math.exp(((Math.log(Math.sin(k_m)) * 2.0) - (Math.log(l_m) * 2.0))));
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_2 * l_m))) / (k_m * (t_1 * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
def code(t, l_m, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	t_2 = math.cos(k_m) * l_m
	tmp = 0
	if k_m <= 8.2e-69:
		tmp = 2.0 / (((k_m * (k_m * t)) / math.cos(k_m)) * math.exp(((math.log(math.sin(k_m)) * 2.0) - (math.log(l_m) * 2.0))))
	elif k_m <= 2.1e+150:
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t)
	else:
		tmp = ((-2.0 / k_m) * (-1.0 * (t_2 * l_m))) / (k_m * (t_1 * t))
	return tmp

Julia

l_m = abs(l)
k_m = abs(k)
function code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l_m)
	tmp = 0.0
	if (k_m <= 8.2e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m * t)) / cos(k_m)) * exp(Float64(Float64(log(sin(k_m)) * 2.0) - Float64(log(l_m) * 2.0)))));
	elseif (k_m <= 2.1e+150)
		tmp = Float64(Float64(Float64(t_2 * Float64(Float64(2.0 / k_m) / k_m)) / t_1) * Float64(l_m / t));
	else
		tmp = Float64(Float64(Float64(-2.0 / k_m) * Float64(-1.0 * Float64(t_2 * l_m))) / Float64(k_m * Float64(t_1 * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
k_m = abs(k);
function tmp_2 = code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0;
	t_2 = cos(k_m) * l_m;
	tmp = 0.0;
	if (k_m <= 8.2e-69)
		tmp = 2.0 / (((k_m * (k_m * t)) / cos(k_m)) * exp(((log(sin(k_m)) * 2.0) - (log(l_m) * 2.0))));
	elseif (k_m <= 2.1e+150)
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	else
		tmp = ((-2.0 / k_m) * (-1.0 * (t_2 * l_m))) / (k_m * (t_1 * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 8.2e-69], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.1e+150], N[(N[(N[(t$95$2 * N[(N[(2.0 / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / k$95$m), $MachinePrecision] * N[(-1.0 * N[(t$95$2 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.1999999999999998e-69

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-log.f64 16.0

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

   9. Applied rewrites 16.0%

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

   if 8.1999999999999998e-69 < k < 2.09999999999999998e150

   1. Initial program 26.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 90.7

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

   9. Applied rewrites 90.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. lift-sin.f64 N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 99.1%

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

   if 2.09999999999999998e150 < k

   1. Initial program 36.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 56.9

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

   5. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. associate-/l* N/A

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

      13. associate-/r* N/A

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

      14. frac-times N/A

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

      15. lower-/.f64 N/A

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

   9. Applied rewrites 76.4%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot e^{\log \sin k \cdot 2 - \log \ell \cdot 2}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k}}{{\sin k}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k} \cdot \left(-1 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)\right)}{k \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := t\_1 \cdot t\\ t_3 := \cos k\_m \cdot l\_m\\ \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{2}{k\_m \cdot k\_m} \cdot t\_3\right) \cdot l\_m}{t\_2}\\ \mathbf{elif}\;k\_m \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_3 \cdot \frac{\frac{2}{k\_m}}{k\_m}}{t\_1} \cdot \frac{l\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k\_m} \cdot \left(-1 \cdot \left(t\_3 \cdot l\_m\right)\right)}{k\_m \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* t_1 t)) (t_3 (* (cos k_m) l_m)))
   (if (<= k_m 3.6e-66)
     (/ (* (* (/ 2.0 (* k_m k_m)) t_3) l_m) t_2)
     (if (<= k_m 2.1e+150)
       (* (/ (* t_3 (/ (/ 2.0 k_m) k_m)) t_1) (/ l_m t))
       (/ (* (/ -2.0 k_m) (* -1.0 (* t_3 l_m))) (* k_m t_2))))))

C

l_m = fabs(l);
k_m = fabs(k);
double code(double t, double l_m, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = t_1 * t;
	double t_3 = cos(k_m) * l_m;
	double tmp;
	if (k_m <= 3.6e-66) {
		tmp = (((2.0 / (k_m * k_m)) * t_3) * l_m) / t_2;
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_3 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_3 * l_m))) / (k_m * t_2);
	}
	return tmp;
}

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    t_2 = t_1 * t
    t_3 = cos(k_m) * l_m
    if (k_m <= 3.6d-66) then
        tmp = (((2.0d0 / (k_m * k_m)) * t_3) * l_m) / t_2
    else if (k_m <= 2.1d+150) then
        tmp = ((t_3 * ((2.0d0 / k_m) / k_m)) / t_1) * (l_m / t)
    else
        tmp = (((-2.0d0) / k_m) * ((-1.0d0) * (t_3 * l_m))) / (k_m * t_2)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
public static double code(double t, double l_m, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double t_2 = t_1 * t;
	double t_3 = Math.cos(k_m) * l_m;
	double tmp;
	if (k_m <= 3.6e-66) {
		tmp = (((2.0 / (k_m * k_m)) * t_3) * l_m) / t_2;
	} else if (k_m <= 2.1e+150) {
		tmp = ((t_3 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	} else {
		tmp = ((-2.0 / k_m) * (-1.0 * (t_3 * l_m))) / (k_m * t_2);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
def code(t, l_m, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	t_2 = t_1 * t
	t_3 = math.cos(k_m) * l_m
	tmp = 0
	if k_m <= 3.6e-66:
		tmp = (((2.0 / (k_m * k_m)) * t_3) * l_m) / t_2
	elif k_m <= 2.1e+150:
		tmp = ((t_3 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t)
	else:
		tmp = ((-2.0 / k_m) * (-1.0 * (t_3 * l_m))) / (k_m * t_2)
	return tmp

Julia

l_m = abs(l)
k_m = abs(k)
function code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(t_1 * t)
	t_3 = Float64(cos(k_m) * l_m)
	tmp = 0.0
	if (k_m <= 3.6e-66)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k_m * k_m)) * t_3) * l_m) / t_2);
	elseif (k_m <= 2.1e+150)
		tmp = Float64(Float64(Float64(t_3 * Float64(Float64(2.0 / k_m) / k_m)) / t_1) * Float64(l_m / t));
	else
		tmp = Float64(Float64(Float64(-2.0 / k_m) * Float64(-1.0 * Float64(t_3 * l_m))) / Float64(k_m * t_2));
	end
	return tmp
end

MATLAB

l_m = abs(l);
k_m = abs(k);
function tmp_2 = code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0;
	t_2 = t_1 * t;
	t_3 = cos(k_m) * l_m;
	tmp = 0.0;
	if (k_m <= 3.6e-66)
		tmp = (((2.0 / (k_m * k_m)) * t_3) * l_m) / t_2;
	elseif (k_m <= 2.1e+150)
		tmp = ((t_3 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	else
		tmp = ((-2.0 / k_m) * (-1.0 * (t_3 * l_m))) / (k_m * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 3.6e-66], N[(N[(N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[k$95$m, 2.1e+150], N[(N[(N[(t$95$3 * N[(N[(2.0 / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / k$95$m), $MachinePrecision] * N[(-1.0 * N[(t$95$3 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.60000000000000012e-66

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

   if 3.60000000000000012e-66 < k < 2.09999999999999998e150

   1. Initial program 26.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 90.7

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

   9. Applied rewrites 90.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. lift-sin.f64 N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 99.1%

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

   if 2.09999999999999998e150 < k

   1. Initial program 36.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 56.9

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

   5. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. associate-/l* N/A

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

      13. associate-/r* N/A

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

      14. frac-times N/A

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

      15. lower-/.f64 N/A

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

   9. Applied rewrites 76.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{2}{k \cdot k} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \ell}{{\sin k}^{2} \cdot t}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k}}{{\sin k}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{k} \cdot \left(-1 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)\right)}{k \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (/
  2.0
  (/ (* k_m (* k_m (* t (/ (pow (sin k_m) 2.0) l_m)))) (* l_m (cos k_m)))))

C

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

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m * (k_m * (t * ((sin(k_m) ** 2.0d0) / l_m)))) / (l_m * cos(k_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(t * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 78.5

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

5. Applied rewrites 78.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. pow2 N/A

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

   13. frac-times N/A

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

   14. lower-/.f64 N/A

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

7. Applied rewrites 86.3%

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

8. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 88.2

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

10. Applied rewrites 88.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. lift-sin.f64 N/A

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

    7. associate-*l* N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. *-commutative N/A

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

    11. associate-/l* N/A

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

    12. lower-*.f64 N/A

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

    13. lift-sin.f64 N/A

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

    14. lift-pow.f64 N/A

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

    15. lift-/.f64 92.4

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

12. Applied rewrites 92.4%

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

Alternative 5: 86.1% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot l\_m\\ \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\frac{2}{k\_m \cdot k\_m} \cdot t\_2\right) \cdot l\_m}{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \frac{\frac{2}{k\_m}}{k\_m}}{t\_1} \cdot \frac{l\_m}{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l_m)))
   (if (<= k_m 3.6e-66)
     (/ (* (* (/ 2.0 (* k_m k_m)) t_2) l_m) (* t_1 t))
     (* (/ (* t_2 (/ (/ 2.0 k_m) k_m)) t_1) (/ l_m t)))))

C

l_m = fabs(l);
k_m = fabs(k);
double code(double t, double l_m, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l_m;
	double tmp;
	if (k_m <= 3.6e-66) {
		tmp = (((2.0 / (k_m * k_m)) * t_2) * l_m) / (t_1 * t);
	} else {
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	}
	return tmp;
}

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    t_2 = cos(k_m) * l_m
    if (k_m <= 3.6d-66) then
        tmp = (((2.0d0 / (k_m * k_m)) * t_2) * l_m) / (t_1 * t)
    else
        tmp = ((t_2 * ((2.0d0 / k_m) / k_m)) / t_1) * (l_m / t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
public static double code(double t, double l_m, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double t_2 = Math.cos(k_m) * l_m;
	double tmp;
	if (k_m <= 3.6e-66) {
		tmp = (((2.0 / (k_m * k_m)) * t_2) * l_m) / (t_1 * t);
	} else {
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
def code(t, l_m, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	t_2 = math.cos(k_m) * l_m
	tmp = 0
	if k_m <= 3.6e-66:
		tmp = (((2.0 / (k_m * k_m)) * t_2) * l_m) / (t_1 * t)
	else:
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t)
	return tmp

Julia

l_m = abs(l)
k_m = abs(k)
function code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l_m)
	tmp = 0.0
	if (k_m <= 3.6e-66)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k_m * k_m)) * t_2) * l_m) / Float64(t_1 * t));
	else
		tmp = Float64(Float64(Float64(t_2 * Float64(Float64(2.0 / k_m) / k_m)) / t_1) * Float64(l_m / t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
k_m = abs(k);
function tmp_2 = code(t, l_m, k_m)
	t_1 = sin(k_m) ^ 2.0;
	t_2 = cos(k_m) * l_m;
	tmp = 0.0;
	if (k_m <= 3.6e-66)
		tmp = (((2.0 / (k_m * k_m)) * t_2) * l_m) / (t_1 * t);
	else
		tmp = ((t_2 * ((2.0 / k_m) / k_m)) / t_1) * (l_m / t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 3.6e-66], N[(N[(N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * N[(N[(2.0 / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000012e-66

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

   if 3.60000000000000012e-66 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 74.0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. frac-times N/A

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

   7. Applied rewrites 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 78.6

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

   9. Applied rewrites 78.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-cos.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. lift-sin.f64 N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 84.0%

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

Alternative 6: 83.4% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (*
  (/ (* (* (cos k_m) l_m) (/ (/ 2.0 k_m) k_m)) (pow (sin k_m) 2.0))
  (/ l_m t)))

C

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

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = (((cos(k_m) * l_m) * ((2.0d0 / k_m) / k_m)) / (sin(k_m) ** 2.0d0)) * (l_m / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision] * N[(N[(2.0 / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 78.5

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

5. Applied rewrites 78.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. frac-times N/A

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

   10. pow2 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. frac-times N/A

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

7. Applied rewrites 77.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-*.f64 86.1

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

9. Applied rewrites 86.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-cos.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-sin.f64 N/A

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

    11. times-frac N/A

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

    12. lower-*.f64 N/A

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

11. Applied rewrites 86.0%

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

Alternative 7: 74.0% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (*
  (/ 2.0 (* (* k_m k_m) t))
  (/ (* (cos k_m) (* l_m l_m)) (pow (sin k_m) 2.0))))

C

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

Fortran

l_m =     private
k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = (2.0d0 / ((k_m * k_m) * t)) * ((cos(k_m) * (l_m * l_m)) / (sin(k_m) ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 78.5

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

5. Applied rewrites 78.5%

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

Alternative 8: 65.0% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)}{\mathsf{fma}\left(0.044444444444444446 \cdot \left(k\_m \cdot k\_m\right) - 0.3333333333333333, k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
(FPCore (t l_m k_m)
 :precision binary64
 (*
  (/ 2.0 (* (* k_m k_m) t))
  (/
   (* (cos k_m) (* l_m l_m))
   (*
    (fma
     (- (* 0.044444444444444446 (* k_m k_m)) 0.3333333333333333)
     (* k_m k_m)
     1.0)
    (* k_m k_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
double code(double t, double l_m, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((cos(k_m) * (l_m * l_m)) / (fma(((0.044444444444444446 * (k_m * k_m)) - 0.3333333333333333), (k_m * k_m), 1.0) * (k_m * k_m)));
}

Julia

l_m = abs(l)
k_m = abs(k)
function code(t, l_m, k_m)
	return Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(cos(k_m) * Float64(l_m * l_m)) / Float64(fma(Float64(Float64(0.044444444444444446 * Float64(k_m * k_m)) - 0.3333333333333333), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
code[t_, l$95$m_, k$95$m_] := N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 78.5

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

5. Applied rewrites 78.5%

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

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(1 + {k}^{2} \cdot \left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(1 + {k}^{2} \cdot \left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   3. +-commutative N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot \left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}\right) + 1\right) \cdot {k}^{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(\left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot {k}^{2} - \frac{1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot \left(k \cdot k\right) - \frac{1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot \left(k \cdot k\right) - \frac{1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}} \]

   10. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot \left(k \cdot k\right) - \frac{1}{3}, k \cdot k, 1\right) \cdot {k}^{2}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot \left(k \cdot k\right) - \frac{1}{3}, k \cdot k, 1\right) \cdot {k}^{2}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(\frac{2}{45} \cdot \left(k \cdot k\right) - \frac{1}{3}, k \cdot k, 1\right) \cdot \left(k \cdot k\right)} \]

   13. lift-*.f64 69.8

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(0.044444444444444446 \cdot \left(k \cdot k\right) - 0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)} \]

8. Applied rewrites 69.8%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\mathsf{fma}\left(0.044444444444444446 \cdot \left(k \cdot k\right) - 0.3333333333333333, k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))