Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.5% → 95.0%

Time: 6.8s

Alternatives: 14

Speedup: 5.7×

• 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: 36.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 95.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{\frac{k\_m}{\ell}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.4e+72)
   (/ 2.0 (/ (* (/ (* (* (pow (sin k_m) 2.0) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    2.0
    (*
     (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) (/ k_m l))
     (/ (/ k_m l) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e+72) {
		tmp = 2.0 / (((((pow(sin(k_m), 2.0) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.4d+72) then
        tmp = 2.0d0 / ((((((sin(k_m) ** 2.0d0) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e+72) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k_m), 2.0) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.4e+72:
		tmp = 2.0 / (((((math.pow(math.sin(k_m), 2.0) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.4e+72)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * Float64(k_m / l)) * Float64(Float64(k_m / l) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.4e+72)
		tmp = 2.0 / ((((((sin(k_m) ^ 2.0) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.4e+72], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3999999999999998e72

   1. Initial program 38.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 78.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 92.6

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

   10. Applied rewrites 92.6%

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

   if 3.3999999999999998e72 < k

   1. Initial program 34.1%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 92.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. associate-/l* N/A

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{\frac{k}{\ell}}{\cos k}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{\ell} \cdot \left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right)}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.45e-8)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    2.0
    (/
     (* (/ k_m l) (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) (/ k_m l)))
     (cos k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / (((k_m / l) * (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l))) / cos(k_m));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d-8) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / (((k_m / l) * (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * (k_m / l))) / cos(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / (((k_m / l) * (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l))) / Math.cos(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.45e-8:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / (((k_m / l) * (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l))) / math.cos(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * Float64(k_m / l))) / cos(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e-8)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / (((k_m / l) * (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l))) / cos(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.45e-8], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e-8

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if 2.4500000000000001e-8 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 93.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 98.5%

       \[\leadsto \frac{2}{\frac{\frac{k}{\ell} \cdot \left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\ell}\right)}{\cos \color{blue}{k}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \frac{\frac{k\_m}{\ell}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.45e-8)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    2.0
    (*
     (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) (/ k_m l))
     (/ (/ k_m l) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d-8) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.45e-8:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * Float64(k_m / l)) * Float64(Float64(k_m / l) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e-8)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * (k_m / l)) * ((k_m / l) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.45e-8], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e-8

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if 2.4500000000000001e-8 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. associate-/l* N/A

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

   10. Applied rewrites 98.5%

       \[\leadsto \frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{\frac{k}{\ell}}{\cos k}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\cos k\_m \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.45e-8)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    2.0
    (/
     (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) (/ k_m l))
     (* (cos k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / (((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * (k_m / l)) / (cos(k_m) * l));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d-8) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / (((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * (k_m / l)) / (cos(k_m) * l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / (((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * (k_m / l)) / (Math.cos(k_m) * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.45e-8:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / (((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * (k_m / l)) / (math.cos(k_m) * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * Float64(k_m / l)) / Float64(cos(k_m) * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e-8)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / (((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * (k_m / l)) / (cos(k_m) * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.45e-8], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e-8

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if 2.4500000000000001e-8 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. associate-/l* N/A

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

   10. Applied rewrites 91.3%

       \[\leadsto \frac{2}{\frac{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\cos k \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.45e-8)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    (* (* (* (cos k_m) l) l) 2.0)
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d-8) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = (((cos(k_m) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e-8) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = (((Math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.45e-8:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = (((math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e-8)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.45e-8], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e-8

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if 2.4500000000000001e-8 < k

   1. Initial program 30.8%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 72.6%

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

   6. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{1}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e+26)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/
    2.0
    (/
     (* (/ (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) l) (/ k_m l))
     1.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+26) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / ((((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) / l) * (k_m / l)) / 1.0);
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d+26) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / ((((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) / l) * (k_m / l)) / 1.0d0)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+26) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / ((((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) / l) * (k_m / l)) / 1.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25e+26:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / ((((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) / l) * (k_m / l)) / 1.0)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e+26)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) / l) * Float64(k_m / l)) / 1.0));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25e+26)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / ((((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) / l) * (k_m / l)) / 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e+26], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e26

   1. Initial program 40.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 88.0

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

   11. Applied rewrites 88.0%

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

   if 1.25e26 < k

   1. Initial program 32.3%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 70.8

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

   4. Applied rewrites 70.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 59.8%

       \[\leadsto \frac{2}{\frac{\frac{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k}{\ell} \cdot \frac{k}{\ell}}{1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\ell \cdot \ell}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 9.8e+137)
   (/ 2.0 (/ (* (/ (* (* (* k_m k_m) t) k_m) l) (/ k_m l)) (cos k_m)))
   (/ 2.0 (/ (/ (* (* (* (- 0.5 0.5) t) k_m) k_m) (* l l)) (cos k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 9.8e+137) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	} else {
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / cos(k_m));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 9.8d+137) then
        tmp = 2.0d0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m))
    else
        tmp = 2.0d0 / ((((((0.5d0 - 0.5d0) * t) * k_m) * k_m) / (l * l)) / cos(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 9.8e+137) {
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / Math.cos(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 9.8e+137:
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / math.cos(k_m))
	else:
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / math.cos(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 9.8e+137)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) / l) * Float64(k_m / l)) / cos(k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m) / Float64(l * l)) / cos(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 9.8e+137)
		tmp = 2.0 / ((((((k_m * k_m) * t) * k_m) / l) * (k_m / l)) / cos(k_m));
	else
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / cos(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 9.8e+137], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.80000000000000065e137

   1. Initial program 36.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 75.8

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

   11. Applied rewrites 75.8%

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

   if 9.80000000000000065e137 < l

   1. Initial program 35.6%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 64.0

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

   4. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 67.1%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 65.4%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}{\cos k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\ell}}{\ell}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t}}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4e-156)
   (/ 2.0 (/ (/ (/ (* (* (* (- 0.5 0.5) t) k_m) k_m) l) l) (cos k_m)))
   (/ (/ (* (* l l) 2.0) (* (* k_m k_m) t)) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-156) {
		tmp = 2.0 / (((((((0.5 - 0.5) * t) * k_m) * k_m) / l) / l) / cos(k_m));
	} else {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-156) then
        tmp = 2.0d0 / (((((((0.5d0 - 0.5d0) * t) * k_m) * k_m) / l) / l) / cos(k_m))
    else
        tmp = (((l * l) * 2.0d0) / ((k_m * k_m) * t)) / (k_m * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-156) {
		tmp = 2.0 / (((((((0.5 - 0.5) * t) * k_m) * k_m) / l) / l) / Math.cos(k_m));
	} else {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4e-156:
		tmp = 2.0 / (((((((0.5 - 0.5) * t) * k_m) * k_m) / l) / l) / math.cos(k_m))
	else:
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-156)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m) / l) / l) / cos(k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k_m * k_m) * t)) / Float64(k_m * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4e-156)
		tmp = 2.0 / (((((((0.5 - 0.5) * t) * k_m) * k_m) / l) / l) / cos(k_m));
	else
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4e-156], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000016e-156

   1. Initial program 50.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

   6. Applied rewrites 77.5%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 77.5%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}{\cos k}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 92.8

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

   11. Applied rewrites 92.8%

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

   if 4.00000000000000016e-156 < k

   1. Initial program 32.0%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   6. Applied rewrites 66.0%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}}{\color{blue}{k \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      9. lift-*.f64 62.7

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{k} \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t}}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\ell \cdot \ell}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 4e+282)
   (/ (/ (* (* l l) 2.0) (* (* k_m k_m) t)) (* k_m k_m))
   (/ 2.0 (/ (/ (* (* (* (- 0.5 0.5) t) k_m) k_m) (* l l)) (cos k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e+282) {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	} else {
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / cos(k_m));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 4d+282) then
        tmp = (((l * l) * 2.0d0) / ((k_m * k_m) * t)) / (k_m * k_m)
    else
        tmp = 2.0d0 / ((((((0.5d0 - 0.5d0) * t) * k_m) * k_m) / (l * l)) / cos(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e+282) {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	} else {
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / Math.cos(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 4e+282:
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m)
	else:
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / math.cos(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e+282)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k_m * k_m) * t)) / Float64(k_m * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m) / Float64(l * l)) / cos(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e+282)
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	else
		tmp = 2.0 / ((((((0.5 - 0.5) * t) * k_m) * k_m) / (l * l)) / cos(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e+282], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.00000000000000013e282

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}}{\color{blue}{k \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      9. lift-*.f64 69.9

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

   9. Applied rewrites 69.9%

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{k} \cdot k} \]

   if 4.00000000000000013e282 < (*.f64 l l)

   1. Initial program 35.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. sqr-sin-a N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot {\color{blue}{\ell}}^{2}}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}} \]

      18. lift-*.f64 64.4

          \[\leadsto \frac{2}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}} \]

   4. Applied rewrites 64.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\color{blue}{\cos k} \cdot \left(\ell \cdot \ell\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos \color{blue}{k} \cdot \left(\ell \cdot \ell\right)}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right) \cdot \left(k \cdot k\right)}{\cos \color{blue}{k} \cdot \left(\ell \cdot \ell\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{\color{blue}{\cos k} \cdot \left(\ell \cdot \ell\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{\cos k \cdot {\ell}^{\color{blue}{2}}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      17. associate-/r* N/A

          \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)\right)}{{\ell}^{2}}}{\color{blue}{\cos k}}} \]

   6. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}{\color{blue}{\cos k}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}{\cos k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\frac{\frac{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}{\cos k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t}}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 4e+282)
   (/ (/ (* (* l l) 2.0) (* (* k_m k_m) t)) (* k_m k_m))
   (/ (* 2.0 (* (cos k_m) (* l l))) (* (* (- 0.5 0.5) t) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e+282) {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	} else {
		tmp = (2.0 * (cos(k_m) * (l * l))) / (((0.5 - 0.5) * t) * (k_m * k_m));
	}
	return tmp;
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 4d+282) then
        tmp = (((l * l) * 2.0d0) / ((k_m * k_m) * t)) / (k_m * k_m)
    else
        tmp = (2.0d0 * (cos(k_m) * (l * l))) / (((0.5d0 - 0.5d0) * t) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e+282) {
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	} else {
		tmp = (2.0 * (Math.cos(k_m) * (l * l))) / (((0.5 - 0.5) * t) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 4e+282:
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m)
	else:
		tmp = (2.0 * (math.cos(k_m) * (l * l))) / (((0.5 - 0.5) * t) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e+282)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k_m * k_m) * t)) / Float64(k_m * k_m));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * Float64(l * l))) / Float64(Float64(Float64(0.5 - 0.5) * t) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e+282)
		tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
	else
		tmp = (2.0 * (cos(k_m) * (l * l))) / (((0.5 - 0.5) * t) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e+282], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.00000000000000013e282

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   6. Applied rewrites 70.3%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}}{\color{blue}{k \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      5. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

      9. lift-*.f64 69.9

         \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

   9. Applied rewrites 69.9%

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{k} \cdot k} \]

   if 4.00000000000000013e282 < (*.f64 l l)

   1. Initial program 35.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.2%

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k\_m \cdot k\_m\right) \cdot t}}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (/ (* (* l l) 2.0) (* (* k_m k_m) t)) (* k_m k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((l * l) * 2.0d0) / ((k_m * k_m) * t)) / (k_m * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k_m * k_m) * t)) / Float64(k_m * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l * l) * 2.0) / ((k_m * k_m) * t)) / (k_m * k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. 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. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

4. Applied rewrites 68.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

6. Applied rewrites 68.7%

   \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}}{\color{blue}{k \cdot k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot t}}{k \cdot k} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

   9. lift-*.f64 66.3

      \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

9. Applied rewrites 66.3%

   \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{k} \cdot k} \]
10. Add Preprocessing

Alternative 12: 66.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* k_m k_m)) (/ (* l l) (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / (k_m * k_m)) * ((l * l) / ((k_m * k_m) * t));
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / (k_m * k_m)) * ((l * l) / ((k_m * k_m) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / (k_m * k_m)) * ((l * l) / ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / (k_m * k_m)) * ((l * l) / ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(k_m * k_m)) * Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / (k_m * k_m)) * ((l * l) / ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 63.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 65.2

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 65.2%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{2}} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{2}} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{{k}^{2}} \cdot \frac{\color{blue}{{\ell}^{2}}}{\left(k \cdot k\right) \cdot t} \]

   11. pow2 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{\color{blue}{2}}}{\left(k \cdot k\right) \cdot t} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{\color{blue}{2}}}{\left(k \cdot k\right) \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   15. pow2 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   17. pow2 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot t} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot t} \]

   19. pow2 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

   21. lift-*.f64 66.2

       \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

8. Applied rewrites 66.2%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}} \]
9. Add Preprocessing

Alternative 13: 65.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* 2.0 (* l l)) (* (* k_m k_m) (* k_m (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(k_m * k_m) * Float64(k_m * Float64(k_m * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * (l * l)) / ((k_m * k_m) * (k_m * (k_m * t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 63.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 65.2

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 65.2%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   5. lower-*.f64 65.2

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 65.2%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]
9. Add Preprocessing

Alternative 14: 65.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \left(\ell \cdot \ell\right)}{k\_m \cdot \left(k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* 2.0 (* l l)) (* k_m (* k_m (* (* k_m k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * (l * l)) / (k_m * (k_m * ((k_m * k_m) * t)));
}

Fortran

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, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 * (l * l)) / (k_m * (k_m * ((k_m * k_m) * t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * (l * l)) / (k_m * (k_m * ((k_m * k_m) * t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * (l * l)) / (k_m * (k_m * ((k_m * k_m) * t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * Float64(l * l)) / Float64(k_m * Float64(k_m * Float64(Float64(k_m * k_m) * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * (l * l)) / (k_m * (k_m * ((k_m * k_m) * t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

   8. pow-prod-up N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. lower-*.f64 63.5

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

4. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   3. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(k \cdot k\right)}^{2} \cdot t} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   12. lift-*.f64 65.2

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

6. Applied rewrites 65.2%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left({k}^{2} \cdot t\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   11. lift-*.f64 65.2

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 65.2%

   \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025116 
(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))))