Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 91.7%

Time: 7.6s

Alternatives: 15

Speedup: 3.7×

• 59.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

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: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right) \cdot \ell\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{\left|k\right| \cdot \left|k\right|} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{\left|k\right|} \cdot 2}{\left|k\right| \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (cos (fabs k)) l)))
  (if (<= (fabs k) 0.024)
    (*
     t_1
     (* (/ l (* (fabs k) (fabs k))) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (*
     t_1
     (/
      (* (/ l (fabs k)) 2.0)
      (*
       (fabs k)
       (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k)) * l;
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_1 * ((l / (fabs(k) * fabs(k))) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (((l / fabs(k)) * 2.0) / (fabs(k) * ((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(abs(k)) * l
    if (abs(k) <= 0.024d0) then
        tmp = t_1 * ((l / (abs(k) * abs(k))) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else
        tmp = t_1 * (((l / abs(k)) * 2.0d0) / (abs(k) * ((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k)) * l;
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_1 * ((l / (Math.abs(k) * Math.abs(k))) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (((l / Math.abs(k)) * 2.0) / (Math.abs(k) * ((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k)) * l
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_1 * ((l / (math.fabs(k) * math.fabs(k))) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	else:
		tmp = t_1 * (((l / math.fabs(k)) * 2.0) / (math.fabs(k) * ((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(abs(k)) * l)
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_1 * Float64(Float64(l / Float64(abs(k) * abs(k))) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l / abs(k)) * 2.0) / Float64(abs(k) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k)) * l;
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_1 * ((l / (abs(k) * abs(k))) * (2.0 / ((abs(k) ^ 2.0) * t)));
	else
		tmp = t_1 * (((l / abs(k)) * 2.0) / (abs(k) * ((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$1 * N[(N[(l / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

   10. Applied rewrites 82.9%

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

Alternative 2: 87.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := t\_1 \cdot \ell\\ t_3 := \frac{\ell}{\left|k\right| \cdot \left|k\right|}\\ t_4 := \cos \left(\left|k\right| + \left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{elif}\;\left|k\right| \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;t\_2 \cdot \left(t\_3 \cdot \frac{-2}{\left(t\_4 \cdot 0.5 - 0.5\right) \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\left(\left|k\right| \cdot t\right) \cdot \left|k\right|\right) \cdot \left(0.5 - 0.5 \cdot t\_4\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k)))
       (t_2 (* t_1 l))
       (t_3 (/ l (* (fabs k) (fabs k))))
       (t_4 (cos (+ (fabs k) (fabs k)))))
  (if (<= (fabs k) 0.024)
    (* t_2 (* t_3 (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (if (<= (fabs k) 8.5e+150)
      (* t_2 (* t_3 (/ -2.0 (* (- (* t_4 0.5) 0.5) t))))
      (*
       (/ t_1 (* (* (* (fabs k) t) (fabs k)) (- 0.5 (* 0.5 t_4))))
       (* (* l l) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = t_1 * l;
	double t_3 = l / (fabs(k) * fabs(k));
	double t_4 = cos((fabs(k) + fabs(k)));
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_2 * (t_3 * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else if (fabs(k) <= 8.5e+150) {
		tmp = t_2 * (t_3 * (-2.0 / (((t_4 * 0.5) - 0.5) * t)));
	} else {
		tmp = (t_1 / (((fabs(k) * t) * fabs(k)) * (0.5 - (0.5 * t_4)))) * ((l * l) * 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = t_1 * l
    t_3 = l / (abs(k) * abs(k))
    t_4 = cos((abs(k) + abs(k)))
    if (abs(k) <= 0.024d0) then
        tmp = t_2 * (t_3 * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else if (abs(k) <= 8.5d+150) then
        tmp = t_2 * (t_3 * ((-2.0d0) / (((t_4 * 0.5d0) - 0.5d0) * t)))
    else
        tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5d0 - (0.5d0 * t_4)))) * ((l * l) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = t_1 * l;
	double t_3 = l / (Math.abs(k) * Math.abs(k));
	double t_4 = Math.cos((Math.abs(k) + Math.abs(k)));
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_2 * (t_3 * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else if (Math.abs(k) <= 8.5e+150) {
		tmp = t_2 * (t_3 * (-2.0 / (((t_4 * 0.5) - 0.5) * t)));
	} else {
		tmp = (t_1 / (((Math.abs(k) * t) * Math.abs(k)) * (0.5 - (0.5 * t_4)))) * ((l * l) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = t_1 * l
	t_3 = l / (math.fabs(k) * math.fabs(k))
	t_4 = math.cos((math.fabs(k) + math.fabs(k)))
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_2 * (t_3 * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	elif math.fabs(k) <= 8.5e+150:
		tmp = t_2 * (t_3 * (-2.0 / (((t_4 * 0.5) - 0.5) * t)))
	else:
		tmp = (t_1 / (((math.fabs(k) * t) * math.fabs(k)) * (0.5 - (0.5 * t_4)))) * ((l * l) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(t_1 * l)
	t_3 = Float64(l / Float64(abs(k) * abs(k)))
	t_4 = cos(Float64(abs(k) + abs(k)))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_2 * Float64(t_3 * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	elseif (abs(k) <= 8.5e+150)
		tmp = Float64(t_2 * Float64(t_3 * Float64(-2.0 / Float64(Float64(Float64(t_4 * 0.5) - 0.5) * t))));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(abs(k) * t) * abs(k)) * Float64(0.5 - Float64(0.5 * t_4)))) * Float64(Float64(l * l) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = t_1 * l;
	t_3 = l / (abs(k) * abs(k));
	t_4 = cos((abs(k) + abs(k)));
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_2 * (t_3 * (2.0 / ((abs(k) ^ 2.0) * t)));
	elseif (abs(k) <= 8.5e+150)
		tmp = t_2 * (t_3 * (-2.0 / (((t_4 * 0.5) - 0.5) * t)));
	else
		tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5 - (0.5 * t_4)))) * ((l * l) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * l), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$2 * N[(t$95$3 * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 8.5e+150], N[(t$95$2 * N[(t$95$3 * N[(-2.0 / N[(N[(N[(t$95$4 * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k < 8.4999999999999999e150

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   if 8.4999999999999999e150 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

Alternative 3: 87.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := t\_1 \cdot \ell\\ t_3 := 0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\\ t_4 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_2 \cdot \left(\frac{\ell}{t\_4} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{elif}\;\left|k\right| \leq 7 \cdot 10^{+150}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{\ell + \ell}{t\_3 \cdot t}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\left(\left|k\right| \cdot t\right) \cdot \left|k\right|\right) \cdot t\_3} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k)))
       (t_2 (* t_1 l))
       (t_3 (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))))
       (t_4 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 0.024)
    (* t_2 (* (/ l t_4) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (if (<= (fabs k) 7e+150)
      (* t_2 (/ (/ (+ l l) (* t_3 t)) t_4))
      (*
       (/ t_1 (* (* (* (fabs k) t) (fabs k)) t_3))
       (* (* l l) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = t_1 * l;
	double t_3 = 0.5 - (0.5 * cos((fabs(k) + fabs(k))));
	double t_4 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_2 * ((l / t_4) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else if (fabs(k) <= 7e+150) {
		tmp = t_2 * (((l + l) / (t_3 * t)) / t_4);
	} else {
		tmp = (t_1 / (((fabs(k) * t) * fabs(k)) * t_3)) * ((l * l) * 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = t_1 * l
    t_3 = 0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))
    t_4 = abs(k) * abs(k)
    if (abs(k) <= 0.024d0) then
        tmp = t_2 * ((l / t_4) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else if (abs(k) <= 7d+150) then
        tmp = t_2 * (((l + l) / (t_3 * t)) / t_4)
    else
        tmp = (t_1 / (((abs(k) * t) * abs(k)) * t_3)) * ((l * l) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = t_1 * l;
	double t_3 = 0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))));
	double t_4 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_2 * ((l / t_4) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else if (Math.abs(k) <= 7e+150) {
		tmp = t_2 * (((l + l) / (t_3 * t)) / t_4);
	} else {
		tmp = (t_1 / (((Math.abs(k) * t) * Math.abs(k)) * t_3)) * ((l * l) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = t_1 * l
	t_3 = 0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))
	t_4 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_2 * ((l / t_4) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	elif math.fabs(k) <= 7e+150:
		tmp = t_2 * (((l + l) / (t_3 * t)) / t_4)
	else:
		tmp = (t_1 / (((math.fabs(k) * t) * math.fabs(k)) * t_3)) * ((l * l) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(t_1 * l)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k)))))
	t_4 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_2 * Float64(Float64(l / t_4) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	elseif (abs(k) <= 7e+150)
		tmp = Float64(t_2 * Float64(Float64(Float64(l + l) / Float64(t_3 * t)) / t_4));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(abs(k) * t) * abs(k)) * t_3)) * Float64(Float64(l * l) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = t_1 * l;
	t_3 = 0.5 - (0.5 * cos((abs(k) + abs(k))));
	t_4 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_2 * ((l / t_4) * (2.0 / ((abs(k) ^ 2.0) * t)));
	elseif (abs(k) <= 7e+150)
		tmp = t_2 * (((l + l) / (t_3 * t)) / t_4);
	else
		tmp = (t_1 / (((abs(k) * t) * abs(k)) * t_3)) * ((l * l) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * l), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$2 * N[(N[(l / t$95$4), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 7e+150], N[(t$95$2 * N[(N[(N[(l + l), $MachinePrecision] / N[(t$95$3 * t), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k < 6.9999999999999997e150

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 76.6%

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

   if 6.9999999999999997e150 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

Alternative 4: 85.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \frac{\ell}{\left|k\right| \cdot \left|k\right|}\\ t_3 := \cos \left(\left|k\right| + \left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_1 \cdot \ell\right) \cdot \left(t\_2 \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{elif}\;\left|k\right| \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;t\_1 \cdot \frac{t\_2 \cdot \left(\ell + \ell\right)}{\left(0.5 - t\_3 \cdot 0.5\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\left(\left|k\right| \cdot t\right) \cdot \left|k\right|\right) \cdot \left(0.5 - 0.5 \cdot t\_3\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k)))
       (t_2 (/ l (* (fabs k) (fabs k))))
       (t_3 (cos (+ (fabs k) (fabs k)))))
  (if (<= (fabs k) 4.2e-7)
    (* (* t_1 l) (* t_2 (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (if (<= (fabs k) 8.5e+150)
      (* t_1 (/ (* t_2 (+ l l)) (* (- 0.5 (* t_3 0.5)) t)))
      (*
       (/ t_1 (* (* (* (fabs k) t) (fabs k)) (- 0.5 (* 0.5 t_3))))
       (* (* l l) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = l / (fabs(k) * fabs(k));
	double t_3 = cos((fabs(k) + fabs(k)));
	double tmp;
	if (fabs(k) <= 4.2e-7) {
		tmp = (t_1 * l) * (t_2 * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else if (fabs(k) <= 8.5e+150) {
		tmp = t_1 * ((t_2 * (l + l)) / ((0.5 - (t_3 * 0.5)) * t));
	} else {
		tmp = (t_1 / (((fabs(k) * t) * fabs(k)) * (0.5 - (0.5 * t_3)))) * ((l * l) * 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = l / (abs(k) * abs(k))
    t_3 = cos((abs(k) + abs(k)))
    if (abs(k) <= 4.2d-7) then
        tmp = (t_1 * l) * (t_2 * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else if (abs(k) <= 8.5d+150) then
        tmp = t_1 * ((t_2 * (l + l)) / ((0.5d0 - (t_3 * 0.5d0)) * t))
    else
        tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5d0 - (0.5d0 * t_3)))) * ((l * l) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = l / (Math.abs(k) * Math.abs(k));
	double t_3 = Math.cos((Math.abs(k) + Math.abs(k)));
	double tmp;
	if (Math.abs(k) <= 4.2e-7) {
		tmp = (t_1 * l) * (t_2 * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else if (Math.abs(k) <= 8.5e+150) {
		tmp = t_1 * ((t_2 * (l + l)) / ((0.5 - (t_3 * 0.5)) * t));
	} else {
		tmp = (t_1 / (((Math.abs(k) * t) * Math.abs(k)) * (0.5 - (0.5 * t_3)))) * ((l * l) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = l / (math.fabs(k) * math.fabs(k))
	t_3 = math.cos((math.fabs(k) + math.fabs(k)))
	tmp = 0
	if math.fabs(k) <= 4.2e-7:
		tmp = (t_1 * l) * (t_2 * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	elif math.fabs(k) <= 8.5e+150:
		tmp = t_1 * ((t_2 * (l + l)) / ((0.5 - (t_3 * 0.5)) * t))
	else:
		tmp = (t_1 / (((math.fabs(k) * t) * math.fabs(k)) * (0.5 - (0.5 * t_3)))) * ((l * l) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(l / Float64(abs(k) * abs(k)))
	t_3 = cos(Float64(abs(k) + abs(k)))
	tmp = 0.0
	if (abs(k) <= 4.2e-7)
		tmp = Float64(Float64(t_1 * l) * Float64(t_2 * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	elseif (abs(k) <= 8.5e+150)
		tmp = Float64(t_1 * Float64(Float64(t_2 * Float64(l + l)) / Float64(Float64(0.5 - Float64(t_3 * 0.5)) * t)));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(abs(k) * t) * abs(k)) * Float64(0.5 - Float64(0.5 * t_3)))) * Float64(Float64(l * l) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = l / (abs(k) * abs(k));
	t_3 = cos((abs(k) + abs(k)));
	tmp = 0.0;
	if (abs(k) <= 4.2e-7)
		tmp = (t_1 * l) * (t_2 * (2.0 / ((abs(k) ^ 2.0) * t)));
	elseif (abs(k) <= 8.5e+150)
		tmp = t_1 * ((t_2 * (l + l)) / ((0.5 - (t_3 * 0.5)) * t));
	else
		tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5 - (0.5 * t_3)))) * ((l * l) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(l / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4.2e-7], N[(N[(t$95$1 * l), $MachinePrecision] * N[(t$95$2 * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 8.5e+150], N[(t$95$1 * N[(N[(t$95$2 * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.2e-7

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 4.2e-7 < k < 8.4999999999999999e150

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 75.1%

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

   8. Applied rewrites 75.1%

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

   if 8.4999999999999999e150 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

Alternative 5: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := t\_1 \cdot \ell\\ t_3 := \left|k\right| \cdot \left|k\right|\\ t_4 := 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_2 \cdot \left(\frac{\ell}{t\_3} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{elif}\;\left|k\right| \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_2 \cdot \left(\ell \cdot \frac{-2}{\left(t\_3 \cdot t\right) \cdot \left(t\_4 - 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(\left(\left|k\right| \cdot t\right) \cdot \left|k\right|\right) \cdot \left(0.5 - t\_4\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k)))
       (t_2 (* t_1 l))
       (t_3 (* (fabs k) (fabs k)))
       (t_4 (* 0.5 (cos (+ (fabs k) (fabs k))))))
  (if (<= (fabs k) 0.024)
    (* t_2 (* (/ l t_3) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (if (<= (fabs k) 2e+150)
      (* t_2 (* l (/ -2.0 (* (* t_3 t) (- t_4 0.5)))))
      (*
       (/ t_1 (* (* (* (fabs k) t) (fabs k)) (- 0.5 t_4)))
       (* (* l l) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = t_1 * l;
	double t_3 = fabs(k) * fabs(k);
	double t_4 = 0.5 * cos((fabs(k) + fabs(k)));
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_2 * ((l / t_3) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else if (fabs(k) <= 2e+150) {
		tmp = t_2 * (l * (-2.0 / ((t_3 * t) * (t_4 - 0.5))));
	} else {
		tmp = (t_1 / (((fabs(k) * t) * fabs(k)) * (0.5 - t_4))) * ((l * l) * 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = t_1 * l
    t_3 = abs(k) * abs(k)
    t_4 = 0.5d0 * cos((abs(k) + abs(k)))
    if (abs(k) <= 0.024d0) then
        tmp = t_2 * ((l / t_3) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else if (abs(k) <= 2d+150) then
        tmp = t_2 * (l * ((-2.0d0) / ((t_3 * t) * (t_4 - 0.5d0))))
    else
        tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5d0 - t_4))) * ((l * l) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = t_1 * l;
	double t_3 = Math.abs(k) * Math.abs(k);
	double t_4 = 0.5 * Math.cos((Math.abs(k) + Math.abs(k)));
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_2 * ((l / t_3) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else if (Math.abs(k) <= 2e+150) {
		tmp = t_2 * (l * (-2.0 / ((t_3 * t) * (t_4 - 0.5))));
	} else {
		tmp = (t_1 / (((Math.abs(k) * t) * Math.abs(k)) * (0.5 - t_4))) * ((l * l) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = t_1 * l
	t_3 = math.fabs(k) * math.fabs(k)
	t_4 = 0.5 * math.cos((math.fabs(k) + math.fabs(k)))
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_2 * ((l / t_3) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	elif math.fabs(k) <= 2e+150:
		tmp = t_2 * (l * (-2.0 / ((t_3 * t) * (t_4 - 0.5))))
	else:
		tmp = (t_1 / (((math.fabs(k) * t) * math.fabs(k)) * (0.5 - t_4))) * ((l * l) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(t_1 * l)
	t_3 = Float64(abs(k) * abs(k))
	t_4 = Float64(0.5 * cos(Float64(abs(k) + abs(k))))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_2 * Float64(Float64(l / t_3) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	elseif (abs(k) <= 2e+150)
		tmp = Float64(t_2 * Float64(l * Float64(-2.0 / Float64(Float64(t_3 * t) * Float64(t_4 - 0.5)))));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(abs(k) * t) * abs(k)) * Float64(0.5 - t_4))) * Float64(Float64(l * l) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = t_1 * l;
	t_3 = abs(k) * abs(k);
	t_4 = 0.5 * cos((abs(k) + abs(k)));
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_2 * ((l / t_3) * (2.0 / ((abs(k) ^ 2.0) * t)));
	elseif (abs(k) <= 2e+150)
		tmp = t_2 * (l * (-2.0 / ((t_3 * t) * (t_4 - 0.5))));
	else
		tmp = (t_1 / (((abs(k) * t) * abs(k)) * (0.5 - t_4))) * ((l * l) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * l), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$2 * N[(N[(l / t$95$3), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2e+150], N[(t$95$2 * N[(l * N[(-2.0 / N[(N[(t$95$3 * t), $MachinePrecision] * N[(t$95$4 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k < 2e150

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 39.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.5%

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

   10. Applied rewrites 39.5%

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

   if 2e150 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

Alternative 6: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right| + \left|k\right|\right)\\ t_2 := \cos \left(\left|k\right|\right)\\ t_3 := t\_2 \cdot \ell\\ t_4 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_3 \cdot \left(\frac{\ell}{t\_4} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{elif}\;\left|k\right| \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_3 \cdot \left(\ell \cdot \frac{-2}{\left(t\_4 \cdot t\right) \cdot \left(0.5 \cdot t\_1 - 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left|k\right| \cdot \left(\left|k\right| \cdot \left(\left(0.5 - t\_1 \cdot 0.5\right) \cdot t\right)\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (+ (fabs k) (fabs k))))
       (t_2 (cos (fabs k)))
       (t_3 (* t_2 l))
       (t_4 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 0.024)
    (* t_3 (* (/ l t_4) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (if (<= (fabs k) 2e+150)
      (* t_3 (* l (/ -2.0 (* (* t_4 t) (- (* 0.5 t_1) 0.5)))))
      (*
       (/ t_2 (* (fabs k) (* (fabs k) (* (- 0.5 (* t_1 0.5)) t))))
       (* (* l l) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cos((fabs(k) + fabs(k)));
	double t_2 = cos(fabs(k));
	double t_3 = t_2 * l;
	double t_4 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_3 * ((l / t_4) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else if (fabs(k) <= 2e+150) {
		tmp = t_3 * (l * (-2.0 / ((t_4 * t) * ((0.5 * t_1) - 0.5))));
	} else {
		tmp = (t_2 / (fabs(k) * (fabs(k) * ((0.5 - (t_1 * 0.5)) * t)))) * ((l * l) * 2.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos((abs(k) + abs(k)))
    t_2 = cos(abs(k))
    t_3 = t_2 * l
    t_4 = abs(k) * abs(k)
    if (abs(k) <= 0.024d0) then
        tmp = t_3 * ((l / t_4) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else if (abs(k) <= 2d+150) then
        tmp = t_3 * (l * ((-2.0d0) / ((t_4 * t) * ((0.5d0 * t_1) - 0.5d0))))
    else
        tmp = (t_2 / (abs(k) * (abs(k) * ((0.5d0 - (t_1 * 0.5d0)) * t)))) * ((l * l) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos((Math.abs(k) + Math.abs(k)));
	double t_2 = Math.cos(Math.abs(k));
	double t_3 = t_2 * l;
	double t_4 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_3 * ((l / t_4) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else if (Math.abs(k) <= 2e+150) {
		tmp = t_3 * (l * (-2.0 / ((t_4 * t) * ((0.5 * t_1) - 0.5))));
	} else {
		tmp = (t_2 / (Math.abs(k) * (Math.abs(k) * ((0.5 - (t_1 * 0.5)) * t)))) * ((l * l) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos((math.fabs(k) + math.fabs(k)))
	t_2 = math.cos(math.fabs(k))
	t_3 = t_2 * l
	t_4 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_3 * ((l / t_4) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	elif math.fabs(k) <= 2e+150:
		tmp = t_3 * (l * (-2.0 / ((t_4 * t) * ((0.5 * t_1) - 0.5))))
	else:
		tmp = (t_2 / (math.fabs(k) * (math.fabs(k) * ((0.5 - (t_1 * 0.5)) * t)))) * ((l * l) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(Float64(abs(k) + abs(k)))
	t_2 = cos(abs(k))
	t_3 = Float64(t_2 * l)
	t_4 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_3 * Float64(Float64(l / t_4) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	elseif (abs(k) <= 2e+150)
		tmp = Float64(t_3 * Float64(l * Float64(-2.0 / Float64(Float64(t_4 * t) * Float64(Float64(0.5 * t_1) - 0.5)))));
	else
		tmp = Float64(Float64(t_2 / Float64(abs(k) * Float64(abs(k) * Float64(Float64(0.5 - Float64(t_1 * 0.5)) * t)))) * Float64(Float64(l * l) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos((abs(k) + abs(k)));
	t_2 = cos(abs(k));
	t_3 = t_2 * l;
	t_4 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_3 * ((l / t_4) * (2.0 / ((abs(k) ^ 2.0) * t)));
	elseif (abs(k) <= 2e+150)
		tmp = t_3 * (l * (-2.0 / ((t_4 * t) * ((0.5 * t_1) - 0.5))));
	else
		tmp = (t_2 / (abs(k) * (abs(k) * ((0.5 - (t_1 * 0.5)) * t)))) * ((l * l) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * l), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$3 * N[(N[(l / t$95$4), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2e+150], N[(t$95$3 * N[(l * N[(-2.0 / N[(N[(t$95$4 * t), $MachinePrecision] * N[(N[(0.5 * t$95$1), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k < 2e150

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 39.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.5%

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

   10. Applied rewrites 39.5%

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

   if 2e150 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 70.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

Alternative 7: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right) \cdot \ell\\ t_2 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;t\_1 \cdot \left(\frac{\ell}{t\_2} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{-2}{\left(t\_2 \cdot t\right) \cdot \left(0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right) - 0.5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (cos (fabs k)) l)) (t_2 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 0.024)
    (* t_1 (* (/ l t_2) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (*
     t_1
     (*
      l
      (/
       -2.0
       (* (* t_2 t) (- (* 0.5 (cos (+ (fabs k) (fabs k)))) 0.5))))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k)) * l;
	double t_2 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = t_1 * ((l / t_2) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (l * (-2.0 / ((t_2 * t) * ((0.5 * cos((fabs(k) + fabs(k)))) - 0.5))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(abs(k)) * l
    t_2 = abs(k) * abs(k)
    if (abs(k) <= 0.024d0) then
        tmp = t_1 * ((l / t_2) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else
        tmp = t_1 * (l * ((-2.0d0) / ((t_2 * t) * ((0.5d0 * cos((abs(k) + abs(k)))) - 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k)) * l;
	double t_2 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = t_1 * ((l / t_2) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (l * (-2.0 / ((t_2 * t) * ((0.5 * Math.cos((Math.abs(k) + Math.abs(k)))) - 0.5))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k)) * l
	t_2 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = t_1 * ((l / t_2) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	else:
		tmp = t_1 * (l * (-2.0 / ((t_2 * t) * ((0.5 * math.cos((math.fabs(k) + math.fabs(k)))) - 0.5))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(cos(abs(k)) * l)
	t_2 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(t_1 * Float64(Float64(l / t_2) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	else
		tmp = Float64(t_1 * Float64(l * Float64(-2.0 / Float64(Float64(t_2 * t) * Float64(Float64(0.5 * cos(Float64(abs(k) + abs(k)))) - 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k)) * l;
	t_2 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = t_1 * ((l / t_2) * (2.0 / ((abs(k) ^ 2.0) * t)));
	else
		tmp = t_1 * (l * (-2.0 / ((t_2 * t) * ((0.5 * cos((abs(k) + abs(k)))) - 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(t$95$1 * N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l * N[(-2.0 / N[(N[(t$95$2 * t), $MachinePrecision] * N[(N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 39.5%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 39.5%

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

   10. Applied rewrites 39.5%

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

Alternative 8: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;\left(t\_1 \cdot \ell\right) \cdot \left(\frac{\ell}{t\_2} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell + \ell}{\left(\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t\right) \cdot t\_2} \cdot \ell\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k))) (t_2 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 0.024)
    (* (* t_1 l) (* (/ l t_2) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (*
     (*
      (/
       (+ l l)
       (* (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t) t_2))
      l)
     t_1))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else {
		tmp = (((l + l) / (((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t) * t_2)) * l) * t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = abs(k) * abs(k)
    if (abs(k) <= 0.024d0) then
        tmp = (t_1 * l) * ((l / t_2) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else
        tmp = (((l + l) / (((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t) * t_2)) * l) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else {
		tmp = (((l + l) / (((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t) * t_2)) * l) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	else:
		tmp = (((l + l) / (((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t) * t_2)) * l) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(Float64(t_1 * l) * Float64(Float64(l / t_2) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t) * t_2)) * l) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / ((abs(k) ^ 2.0) * t)));
	else
		tmp = (((l + l) / (((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t) * t_2)) * l) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(N[(t$95$1 * l), $MachinePrecision] * N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.024

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.5%

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

   4. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.8%

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

   11. Applied rewrites 72.8%

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

   if 0.024 < k

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   6. Applied rewrites 75.1%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      5. count-2 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t} \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\sin k \cdot \sin k\right) \cdot t} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\sin k \cdot \sin k\right) \cdot t} \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(\sin k \cdot \sin k\right) \cdot t} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{{\sin k}^{2} \cdot t} \]

      10. lower-pow.f64 83.6%

          \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{{\sin k}^{2} \cdot t} \]

   8. Applied rewrites 83.6%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{{\sin k}^{2} \cdot t} \]

   10. Applied rewrites 75.2%

       \[\leadsto \left(\frac{\ell + \ell}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell\right) \cdot \color{blue}{\cos k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 0.024:\\ \;\;\;\;\left(t\_1 \cdot \ell\right) \cdot \left(\frac{\ell}{t\_2} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_1}{\left(t\_2 \cdot t\right) \cdot \left(0.5 - \cos \left(\left|k\right| + \left|k\right|\right) \cdot 0.5\right)} \cdot \ell\right) \cdot \left(\ell + \ell\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k))) (t_2 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 0.024)
    (* (* t_1 l) (* (/ l t_2) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (*
     (*
      (/
       t_1
       (* (* t_2 t) (- 0.5 (* (cos (+ (fabs k) (fabs k))) 0.5))))
      l)
     (+ l l)))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 0.024) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else {
		tmp = ((t_1 / ((t_2 * t) * (0.5 - (cos((fabs(k) + fabs(k))) * 0.5)))) * l) * (l + l);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = abs(k) * abs(k)
    if (abs(k) <= 0.024d0) then
        tmp = (t_1 * l) * ((l / t_2) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else
        tmp = ((t_1 / ((t_2 * t) * (0.5d0 - (cos((abs(k) + abs(k))) * 0.5d0)))) * l) * (l + l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 0.024) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else {
		tmp = ((t_1 / ((t_2 * t) * (0.5 - (Math.cos((Math.abs(k) + Math.abs(k))) * 0.5)))) * l) * (l + l);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 0.024:
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	else:
		tmp = ((t_1 / ((t_2 * t) * (0.5 - (math.cos((math.fabs(k) + math.fabs(k))) * 0.5)))) * l) * (l + l)
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 0.024)
		tmp = Float64(Float64(t_1 * l) * Float64(Float64(l / t_2) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	else
		tmp = Float64(Float64(Float64(t_1 / Float64(Float64(t_2 * t) * Float64(0.5 - Float64(cos(Float64(abs(k) + abs(k))) * 0.5)))) * l) * Float64(l + l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 0.024)
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / ((abs(k) ^ 2.0) * t)));
	else
		tmp = ((t_1 / ((t_2 * t) * (0.5 - (cos((abs(k) + abs(k))) * 0.5)))) * l) * (l + l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 0.024], N[(N[(t$95$1 * l), $MachinePrecision] * N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / N[(N[(t$95$2 * t), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.024

   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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   6. Applied rewrites 75.1%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot t} \]

      3. associate-/l* N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}}\right) \]

      11. frac-2neg N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)}\right) \]

   8. Applied rewrites 41.2%

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\left(\cos \left(k + k\right) \cdot 0.5 - 0.5\right) \cdot t}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot \color{blue}{t}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

       3. lower-pow.f64 72.8%

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

   11. Applied rewrites 72.8%

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]

   if 0.024 < k

   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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2}} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + {\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left({\ell}^{2} + {\ell}^{2}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(2 \cdot \color{blue}{{\ell}^{2}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot 2\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \ell\right) \cdot \left(\ell \cdot 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right)} \cdot \ell\right) \cdot \left(\ell \cdot 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right)} \cdot \ell\right) \cdot \left(\ell \cdot 2\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right)} \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\ell}\right) \]

      12. count-2-rev N/A

          \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right)} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

      13. lower-+.f64 75.1%

          \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

   8. Applied rewrites 75.1%

      \[\leadsto \left(\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)} \cdot \ell\right) \cdot \color{blue}{\left(\ell + \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_1 \cdot \ell\right) \cdot \left(\frac{\ell}{t\_2} \cdot \frac{2}{{\left(\left|k\right|\right)}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(t\_2 \cdot t\right) \cdot \left(0.5 - \cos \left(\left|k\right| + \left|k\right|\right) \cdot 0.5\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (cos (fabs k))) (t_2 (* (fabs k) (fabs k))))
  (if (<= (fabs k) 5.8e-7)
    (* (* t_1 l) (* (/ l t_2) (/ 2.0 (* (pow (fabs k) 2.0) t))))
    (*
     t_1
     (/
      (* (+ l l) l)
      (* (* t_2 t) (- 0.5 (* (cos (+ (fabs k) (fabs k))) 0.5))))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 5.8e-7) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (pow(fabs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (((l + l) * l) / ((t_2 * t) * (0.5 - (cos((fabs(k) + fabs(k))) * 0.5))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(abs(k))
    t_2 = abs(k) * abs(k)
    if (abs(k) <= 5.8d-7) then
        tmp = (t_1 * l) * ((l / t_2) * (2.0d0 / ((abs(k) ** 2.0d0) * t)))
    else
        tmp = t_1 * (((l + l) * l) / ((t_2 * t) * (0.5d0 - (cos((abs(k) + abs(k))) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cos(Math.abs(k));
	double t_2 = Math.abs(k) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 5.8e-7) {
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (Math.pow(Math.abs(k), 2.0) * t)));
	} else {
		tmp = t_1 * (((l + l) * l) / ((t_2 * t) * (0.5 - (Math.cos((Math.abs(k) + Math.abs(k))) * 0.5))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.cos(math.fabs(k))
	t_2 = math.fabs(k) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 5.8e-7:
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / (math.pow(math.fabs(k), 2.0) * t)))
	else:
		tmp = t_1 * (((l + l) * l) / ((t_2 * t) * (0.5 - (math.cos((math.fabs(k) + math.fabs(k))) * 0.5))))
	return tmp

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 5.8e-7)
		tmp = Float64(Float64(t_1 * l) * Float64(Float64(l / t_2) * Float64(2.0 / Float64((abs(k) ^ 2.0) * t))));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l + l) * l) / Float64(Float64(t_2 * t) * Float64(0.5 - Float64(cos(Float64(abs(k) + abs(k))) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = cos(abs(k));
	t_2 = abs(k) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 5.8e-7)
		tmp = (t_1 * l) * ((l / t_2) * (2.0 / ((abs(k) ^ 2.0) * t)));
	else
		tmp = t_1 * (((l + l) * l) / ((t_2 * t) * (0.5 - (cos((abs(k) + abs(k))) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5.8e-7], N[(N[(t$95$1 * l), $MachinePrecision] * N[(N[(l / t$95$2), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$2 * t), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.7999999999999995e-7

   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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   6. Applied rewrites 75.1%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot t} \]

      3. associate-/l* N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}}\right) \]

      11. frac-2neg N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)}\right) \]

   8. Applied rewrites 41.2%

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\left(\cos \left(k + k\right) \cdot 0.5 - 0.5\right) \cdot t}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot \color{blue}{t}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

       3. lower-pow.f64 72.8%

          \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

   11. Applied rewrites 72.8%

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]

   if 5.7999999999999995e-7 < k

   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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2}} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + {\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left({\ell}^{2} + {\ell}^{2}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(2 \cdot \color{blue}{{\ell}^{2}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot 2\right) \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\cos k \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)}} \]

      4. associate-/l* N/A

         \[\leadsto \cos k \cdot \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)}} \]

      6. lower-/.f64 67.6%

         \[\leadsto \cos k \cdot \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \cos k \cdot \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \cos k \cdot \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \cos k \cdot \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \cos k \cdot \frac{\left(\ell \cdot 2\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \cos k \cdot \frac{\left(\ell \cdot 2\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \cos k \cdot \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      13. count-2-rev N/A

          \[\leadsto \cos k \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \]

      14. lower-+.f64 67.6%

          \[\leadsto \cos k \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \cos k \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \cos k \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \cos \left(k + k\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      17. lower-*.f64 67.6%

          \[\leadsto \cos k \cdot \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot \color{blue}{0.5}\right)} \]

   8. Applied rewrites 67.6%

      \[\leadsto \cos k \cdot \color{blue}{\frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.8% accurate, 1.9× speedup

Math

\[\left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (* (cos k) l) (* (/ l (* k k)) (/ 2.0 (* (pow k 2.0) t)))))

C

double code(double t, double l, double k) {
	return (cos(k) * l) * ((l / (k * k)) * (2.0 / (pow(k, 2.0) * t)));
}

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 = (cos(k) * l) * ((l / (k * k)) * (2.0d0 / ((k ** 2.0d0) * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (Math.cos(k) * l) * ((l / (k * k)) * (2.0 / (Math.pow(k, 2.0) * t)));
}

Python

def code(t, l, k):
	return (math.cos(k) * l) * ((l / (k * k)) * (2.0 / (math.pow(k, 2.0) * t)))

Julia

function code(t, l, k)
	return Float64(Float64(cos(k) * l) * Float64(Float64(l / Float64(k * k)) * Float64(2.0 / Float64((k ^ 2.0) * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (cos(k) * l) * ((l / (k * k)) * (2.0 / ((k ^ 2.0) * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. 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. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   7. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

   10. lower-sin.f64 73.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

4. Applied rewrites 73.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

6. Applied rewrites 75.1%

   \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot t} \]

   3. associate-/l* N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   7. associate-*l* N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}}\right) \]

   11. frac-2neg N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)}\right) \]

8. Applied rewrites 41.2%

   \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\left(\cos \left(k + k\right) \cdot 0.5 - 0.5\right) \cdot t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot \color{blue}{t}}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

    3. lower-pow.f64 72.8%

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{{k}^{2} \cdot t}\right) \]

11. Applied rewrites 72.8%

    \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}}\right) \]
12. Add Preprocessing

Alternative 12: 69.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 6.2 \cdot 10^{+184}:\\ \;\;\;\;\left(\left|\ell\right| + \left|\ell\right|\right) \cdot \frac{\left|\ell\right|}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k \cdot \left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{k \cdot k}\right)\right) \cdot 2}{\left(0.5 - 0.5\right) \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs l) 6.2e+184)
  (* (+ (fabs l) (fabs l)) (/ (fabs l) (* (pow k 4.0) t)))
  (/
   (* (* (cos k) (* (fabs l) (/ (fabs l) (* k k)))) 2.0)
   (* (- 0.5 0.5) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 6.2e+184) {
		tmp = (fabs(l) + fabs(l)) * (fabs(l) / (pow(k, 4.0) * t));
	} else {
		tmp = ((cos(k) * (fabs(l) * (fabs(l) / (k * k)))) * 2.0) / ((0.5 - 0.5) * t);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(l) <= 6.2d+184) then
        tmp = (abs(l) + abs(l)) * (abs(l) / ((k ** 4.0d0) * t))
    else
        tmp = ((cos(k) * (abs(l) * (abs(l) / (k * k)))) * 2.0d0) / ((0.5d0 - 0.5d0) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 6.2e+184) {
		tmp = (Math.abs(l) + Math.abs(l)) * (Math.abs(l) / (Math.pow(k, 4.0) * t));
	} else {
		tmp = ((Math.cos(k) * (Math.abs(l) * (Math.abs(l) / (k * k)))) * 2.0) / ((0.5 - 0.5) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 6.2e+184:
		tmp = (math.fabs(l) + math.fabs(l)) * (math.fabs(l) / (math.pow(k, 4.0) * t))
	else:
		tmp = ((math.cos(k) * (math.fabs(l) * (math.fabs(l) / (k * k)))) * 2.0) / ((0.5 - 0.5) * t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 6.2e+184)
		tmp = Float64(Float64(abs(l) + abs(l)) * Float64(abs(l) / Float64((k ^ 4.0) * t)));
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(abs(l) * Float64(abs(l) / Float64(k * k)))) * 2.0) / Float64(Float64(0.5 - 0.5) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 6.2e+184)
		tmp = (abs(l) + abs(l)) * (abs(l) / ((k ^ 4.0) * t));
	else
		tmp = ((cos(k) * (abs(l) * (abs(l) / (k * k)))) * 2.0) / ((0.5 - 0.5) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 6.2e+184], N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.1999999999999997e184

   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 k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 61.8%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      14. metadata-eval 68.0%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

   6. Applied rewrites 68.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\ell \cdot 2\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      7. count-2-rev N/A

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      8. lower-+.f64 68.0%

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

      11. lower-*.f64 68.0%

          \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot \color{blue}{t}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

       3. lower-pow.f64 68.1%

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

   11. Applied rewrites 68.1%

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]

   if 6.1999999999999997e184 < 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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

      5. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   6. Applied rewrites 75.1%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(0.5 - \frac{1}{2}\right) \cdot t} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.5%

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\left(0.5 - 0.5\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 6.2 \cdot 10^{+184}:\\ \;\;\;\;\left(\left|\ell\right| + \left|\ell\right|\right) \cdot \frac{\left|\ell\right|}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5\right)} \cdot \left(\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs l) 6.2e+184)
  (* (+ (fabs l) (fabs l)) (/ (fabs l) (* (pow k 4.0) t)))
  (*
   (/ (cos k) (* (* (* k k) t) (- 0.5 0.5)))
   (* (* (fabs l) (fabs l)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 6.2e+184) {
		tmp = (fabs(l) + fabs(l)) * (fabs(l) / (pow(k, 4.0) * t));
	} else {
		tmp = (cos(k) / (((k * k) * t) * (0.5 - 0.5))) * ((fabs(l) * fabs(l)) * 2.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(l) <= 6.2d+184) then
        tmp = (abs(l) + abs(l)) * (abs(l) / ((k ** 4.0d0) * t))
    else
        tmp = (cos(k) / (((k * k) * t) * (0.5d0 - 0.5d0))) * ((abs(l) * abs(l)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 6.2e+184) {
		tmp = (Math.abs(l) + Math.abs(l)) * (Math.abs(l) / (Math.pow(k, 4.0) * t));
	} else {
		tmp = (Math.cos(k) / (((k * k) * t) * (0.5 - 0.5))) * ((Math.abs(l) * Math.abs(l)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 6.2e+184:
		tmp = (math.fabs(l) + math.fabs(l)) * (math.fabs(l) / (math.pow(k, 4.0) * t))
	else:
		tmp = (math.cos(k) / (((k * k) * t) * (0.5 - 0.5))) * ((math.fabs(l) * math.fabs(l)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 6.2e+184)
		tmp = Float64(Float64(abs(l) + abs(l)) * Float64(abs(l) / Float64((k ^ 4.0) * t)));
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64(k * k) * t) * Float64(0.5 - 0.5))) * Float64(Float64(abs(l) * abs(l)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 6.2e+184)
		tmp = (abs(l) + abs(l)) * (abs(l) / ((k ^ 4.0) * t));
	else
		tmp = (cos(k) / (((k * k) * t) * (0.5 - 0.5))) * ((abs(l) * abs(l)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 6.2e+184], N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.1999999999999997e184

   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 k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 61.8%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. mult-flip N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

      12. pow-flip N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

      14. metadata-eval 68.0%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

   6. Applied rewrites 68.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\ell \cdot 2\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      7. count-2-rev N/A

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      8. lower-+.f64 68.0%

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\ell + \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

      11. lower-*.f64 68.0%

          \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot \color{blue}{t}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

       3. lower-pow.f64 68.1%

          \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

   11. Applied rewrites 68.1%

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]

   if 6.1999999999999997e184 < 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

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 73.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2}} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. associate-/l* N/A

         \[\leadsto {\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} + {\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left({\ell}^{2} + {\ell}^{2}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(2 \cdot \color{blue}{{\ell}^{2}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - \frac{1}{2}\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 35.7%

      \[\leadsto \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.1% accurate, 2.0× speedup

Math

\[\left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (* (cos k) l) (* 2.0 (/ l (* (pow k 4.0) t)))))

C

double code(double t, double l, double k) {
	return (cos(k) * l) * (2.0 * (l / (pow(k, 4.0) * t)));
}

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 = (cos(k) * l) * (2.0d0 * (l / ((k ** 4.0d0) * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (Math.cos(k) * l) * (2.0 * (l / (Math.pow(k, 4.0) * t)));
}

Python

def code(t, l, k):
	return (math.cos(k) * l) * (2.0 * (l / (math.pow(k, 4.0) * t)))

Julia

function code(t, l, k)
	return Float64(Float64(cos(k) * l) * Float64(2.0 * Float64(l / Float64((k ^ 4.0) * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (cos(k) * l) * (2.0 * (l / ((k ^ 4.0) * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(2.0 * N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. 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. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   7. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

   10. lower-sin.f64 73.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

4. Applied rewrites 73.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \cdot 2 \]

   6. associate-*l/ N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}} \cdot 2}{\color{blue}{t \cdot {\sin k}^{2}}} \]

6. Applied rewrites 75.1%

   \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot 2}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right)} \cdot t} \]

   3. associate-/l* N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\right) \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{2}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \]

   7. associate-*l* N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}}\right) \]

   11. frac-2neg N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\mathsf{neg}\left(2\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t}\right)}\right) \]

8. Applied rewrites 41.2%

   \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{-2}{\left(\cos \left(k + k\right) \cdot 0.5 - 0.5\right) \cdot t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}}\right) \]

    2. lower-/.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot \color{blue}{t}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \]

    4. lower-pow.f64 69.1%

       \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \]

11. Applied rewrites 69.1%

    \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \]
12. Add Preprocessing

Alternative 15: 68.1% accurate, 3.7× speedup

Math

\[\left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (+ l l) (/ l (* (pow k 4.0) t))))

C

double code(double t, double l, double k) {
	return (l + l) * (l / (pow(k, 4.0) * t));
}

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 = (l + l) * (l / ((k ** 4.0d0) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * (l / (Math.pow(k, 4.0) * t));
}

Python

def code(t, l, k):
	return (l + l) * (l / (math.pow(k, 4.0) * t))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(l / Float64((k ^ 4.0) * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (l / ((k ^ 4.0) * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 61.8%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 61.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. mult-flip N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{1}}{{k}^{4} \cdot t}\right) \]

   5. associate-*l* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4} \cdot t}\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{\color{blue}{t}}\right)\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\frac{1}{{k}^{4}}}{t}\right)\right) \]

   12. pow-flip N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t}\right)\right) \]

   14. metadata-eval 68.0%

       \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right) \]

6. Applied rewrites 68.0%

   \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{{k}^{-4}}{t}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\ell \cdot 2\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{-4}}{t}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

   7. count-2-rev N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

   8. lower-+.f64 68.0%

      \[\leadsto \left(\ell + \ell\right) \cdot \left(\color{blue}{\ell} \cdot \frac{{k}^{-4}}{t}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{-4}}{t}}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

   11. lower-*.f64 68.0%

       \[\leadsto \left(\ell + \ell\right) \cdot \left(\frac{{k}^{-4}}{t} \cdot \color{blue}{\ell}\right) \]

8. Applied rewrites 68.0%

   \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right)} \]

9. Taylor expanded in t around 0

   \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot \color{blue}{t}} \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

    3. lower-pow.f64 68.1%

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot t} \]

11. Applied rewrites 68.1%

    \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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))))