Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 92.3%

Time: 7.5s

Alternatives: 16

Speedup: 10.7×

• 72.7% 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: 54.5% 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: 92.3% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{\left|t\right| \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{k}{\left|t\right| \cdot \left|t\right|} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 2.4e-73)
   (*
    2.0
    (*
     (/ (* (cos k) l) k)
     (/ (/ l (- 0.5 (* (cos (+ k k)) 0.5))) (* (fabs t) k))))
   (/
    2.0
    (*
     (* (/ (* (sin k) (fabs t)) l) (fabs t))
     (*
      (/ (fabs t) l)
      (* (- (* (/ k (* (fabs t) (fabs t))) k) -2.0) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.4e-73) {
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (fabs(t) * k)));
	} else {
		tmp = 2.0 / ((((sin(k) * fabs(t)) / l) * fabs(t)) * ((fabs(t) / l) * ((((k / (fabs(t) * fabs(t))) * k) - -2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 2.4e-73) {
		tmp = 2.0 * (((Math.cos(k) * l) / k) * ((l / (0.5 - (Math.cos((k + k)) * 0.5))) / (Math.abs(t) * k)));
	} else {
		tmp = 2.0 / ((((Math.sin(k) * Math.abs(t)) / l) * Math.abs(t)) * ((Math.abs(t) / l) * ((((k / (Math.abs(t) * Math.abs(t))) * k) - -2.0) * Math.tan(k))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 2.4e-73:
		tmp = 2.0 * (((math.cos(k) * l) / k) * ((l / (0.5 - (math.cos((k + k)) * 0.5))) / (math.fabs(t) * k)))
	else:
		tmp = 2.0 / ((((math.sin(k) * math.fabs(t)) / l) * math.fabs(t)) * ((math.fabs(t) / l) * ((((k / (math.fabs(t) * math.fabs(t))) * k) - -2.0) * math.tan(k))))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.4e-73)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * l) / k) * Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) / Float64(abs(t) * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(k / Float64(abs(t) * abs(t))) * k) - -2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 2.4e-73)
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (abs(t) * k)));
	else
		tmp = 2.0 / ((((sin(k) * abs(t)) / l) * abs(t)) * ((abs(t) / l) * ((((k / (abs(t) * abs(t))) * k) - -2.0) * tan(k))));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.4e-73], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] - -2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e-73

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

   10. Applied rewrites 68.5%

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

   if 2.4000000000000001e-73 < t

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 71.7%

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

Alternative 2: 89.6% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.4 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{\left|t\right| \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{k}{\left|t\right| \cdot \left|t\right|} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 2.4e-73)
   (*
    2.0
    (*
     (/ (* (cos k) l) k)
     (/ (/ l (- 0.5 (* (cos (+ k k)) 0.5))) (* (fabs t) k))))
   (/
    2.0
    (*
     (* (* (/ (sin k) l) (fabs t)) (fabs t))
     (*
      (/ (fabs t) l)
      (* (- (* (/ k (* (fabs t) (fabs t))) k) -2.0) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.4e-73) {
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (fabs(t) * k)));
	} else {
		tmp = 2.0 / ((((sin(k) / l) * fabs(t)) * fabs(t)) * ((fabs(t) / l) * ((((k / (fabs(t) * fabs(t))) * k) - -2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 2.4e-73) {
		tmp = 2.0 * (((Math.cos(k) * l) / k) * ((l / (0.5 - (Math.cos((k + k)) * 0.5))) / (Math.abs(t) * k)));
	} else {
		tmp = 2.0 / ((((Math.sin(k) / l) * Math.abs(t)) * Math.abs(t)) * ((Math.abs(t) / l) * ((((k / (Math.abs(t) * Math.abs(t))) * k) - -2.0) * Math.tan(k))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 2.4e-73:
		tmp = 2.0 * (((math.cos(k) * l) / k) * ((l / (0.5 - (math.cos((k + k)) * 0.5))) / (math.fabs(t) * k)))
	else:
		tmp = 2.0 / ((((math.sin(k) / l) * math.fabs(t)) * math.fabs(t)) * ((math.fabs(t) / l) * ((((k / (math.fabs(t) * math.fabs(t))) * k) - -2.0) * math.tan(k))))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.4e-73)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * l) / k) * Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) / Float64(abs(t) * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * abs(t)) * abs(t)) * Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(k / Float64(abs(t) * abs(t))) * k) - -2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 2.4e-73)
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (abs(t) * k)));
	else
		tmp = 2.0 / ((((sin(k) / l) * abs(t)) * abs(t)) * ((abs(t) / l) * ((((k / (abs(t) * abs(t))) * k) - -2.0) * tan(k))));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.4e-73], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] - -2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e-73

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

   10. Applied rewrites 68.5%

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

   if 2.4000000000000001e-73 < t

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 69.1%

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

Alternative 3: 89.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\frac{\ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{\left|t\right| \cdot k}\right)\\ \mathbf{elif}\;\left|t\right| \leq 1.9 \cdot 10^{+105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot \sin k} \cdot \left(\frac{\ell}{\left(\frac{k}{t\_1} \cdot k - -2\right) \cdot \tan k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \left(\left(\sin k \cdot \left|t\right|\right) \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 7.2e-77)
     (*
      2.0
      (*
       (/ (* (cos k) l) k)
       (/ (/ l (- 0.5 (* (cos (+ k k)) 0.5))) (* (fabs t) k))))
     (if (<= (fabs t) 1.9e+105)
       (*
        (/ l (* (* t_1 (fabs t)) (sin k)))
        (* (/ l (* (- (* (/ k t_1) k) -2.0) (tan k))) 2.0))
       (/
        2.0
        (*
         (*
          (*
           (/ (fabs t) l)
           (* (fabs t) (* (* (sin k) (fabs t)) (/ 1.0 l))))
          (tan k))
         2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 7.2e-77) {
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (fabs(t) * k)));
	} else if (fabs(t) <= 1.9e+105) {
		tmp = (l / ((t_1 * fabs(t)) * sin(k))) * ((l / ((((k / t_1) * k) - -2.0) * tan(k))) * 2.0);
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * ((sin(k) * fabs(t)) * (1.0 / l)))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 7.2e-77) {
		tmp = 2.0 * (((Math.cos(k) * l) / k) * ((l / (0.5 - (Math.cos((k + k)) * 0.5))) / (Math.abs(t) * k)));
	} else if (Math.abs(t) <= 1.9e+105) {
		tmp = (l / ((t_1 * Math.abs(t)) * Math.sin(k))) * ((l / ((((k / t_1) * k) - -2.0) * Math.tan(k))) * 2.0);
	} else {
		tmp = 2.0 / ((((Math.abs(t) / l) * (Math.abs(t) * ((Math.sin(k) * Math.abs(t)) * (1.0 / l)))) * Math.tan(k)) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 7.2e-77:
		tmp = 2.0 * (((math.cos(k) * l) / k) * ((l / (0.5 - (math.cos((k + k)) * 0.5))) / (math.fabs(t) * k)))
	elif math.fabs(t) <= 1.9e+105:
		tmp = (l / ((t_1 * math.fabs(t)) * math.sin(k))) * ((l / ((((k / t_1) * k) - -2.0) * math.tan(k))) * 2.0)
	else:
		tmp = 2.0 / ((((math.fabs(t) / l) * (math.fabs(t) * ((math.sin(k) * math.fabs(t)) * (1.0 / l)))) * math.tan(k)) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 7.2e-77)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * l) / k) * Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) / Float64(abs(t) * k))));
	elseif (abs(t) <= 1.9e+105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * sin(k))) * Float64(Float64(l / Float64(Float64(Float64(Float64(k / t_1) * k) - -2.0) * tan(k))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(Float64(sin(k) * abs(t)) * Float64(1.0 / l)))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 7.2e-77)
		tmp = 2.0 * (((cos(k) * l) / k) * ((l / (0.5 - (cos((k + k)) * 0.5))) / (abs(t) * k)));
	elseif (abs(t) <= 1.9e+105)
		tmp = (l / ((t_1 * abs(t)) * sin(k))) * ((l / ((((k / t_1) * k) - -2.0) * tan(k))) * 2.0);
	else
		tmp = 2.0 / ((((abs(t) / l) * (abs(t) * ((sin(k) * abs(t)) * (1.0 / l)))) * tan(k)) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 7.2e-77], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.9e+105], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[(N[(N[(k / t$95$1), $MachinePrecision] * k), $MachinePrecision] - -2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2000000000000001e-77

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

   10. Applied rewrites 68.5%

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

   if 7.2000000000000001e-77 < t < 1.8999999999999999e105

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

   5. Applied rewrites 54.6%

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

   if 1.8999999999999999e105 < t

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 4.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left|k\right|} \cdot \frac{\frac{\ell}{0.5 - \cos \left(\left|k\right| + \left|k\right|\right) \cdot 0.5}}{t \cdot \left|k\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 4.2e+57)
  (/
   2.0
   (*
    (*
     (* (/ t l) (* t (* (* (sin (fabs k)) t) (/ 1.0 l))))
     (tan (fabs k)))
    2.0))
  (*
   2.0
   (*
    (/ (* (cos (fabs k)) l) (fabs k))
    (/
     (/ l (- 0.5 (* (cos (+ (fabs k) (fabs k))) 0.5)))
     (* t (fabs k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 4.2e+57) {
		tmp = 2.0 / ((((t / l) * (t * ((sin(fabs(k)) * t) * (1.0 / l)))) * tan(fabs(k))) * 2.0);
	} else {
		tmp = 2.0 * (((cos(fabs(k)) * l) / fabs(k)) * ((l / (0.5 - (cos((fabs(k) + fabs(k))) * 0.5))) / (t * fabs(k))));
	}
	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(k) <= 4.2d+57) then
        tmp = 2.0d0 / ((((t / l) * (t * ((sin(abs(k)) * t) * (1.0d0 / l)))) * tan(abs(k))) * 2.0d0)
    else
        tmp = 2.0d0 * (((cos(abs(k)) * l) / abs(k)) * ((l / (0.5d0 - (cos((abs(k) + abs(k))) * 0.5d0))) / (t * abs(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 4.2e+57) {
		tmp = 2.0 / ((((t / l) * (t * ((Math.sin(Math.abs(k)) * t) * (1.0 / l)))) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(Math.abs(k)) * l) / Math.abs(k)) * ((l / (0.5 - (Math.cos((Math.abs(k) + Math.abs(k))) * 0.5))) / (t * Math.abs(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 4.2e+57:
		tmp = 2.0 / ((((t / l) * (t * ((math.sin(math.fabs(k)) * t) * (1.0 / l)))) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = 2.0 * (((math.cos(math.fabs(k)) * l) / math.fabs(k)) * ((l / (0.5 - (math.cos((math.fabs(k) + math.fabs(k))) * 0.5))) / (t * math.fabs(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 4.2e+57)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(sin(abs(k)) * t) * Float64(1.0 / l)))) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(abs(k)) * l) / abs(k)) * Float64(Float64(l / Float64(0.5 - Float64(cos(Float64(abs(k) + abs(k))) * 0.5))) / Float64(t * abs(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 4.2e+57)
		tmp = 2.0 / ((((t / l) * (t * ((sin(abs(k)) * t) * (1.0 / l)))) * tan(abs(k))) * 2.0);
	else
		tmp = 2.0 * (((cos(abs(k)) * l) / abs(k)) * ((l / (0.5 - (cos((abs(k) + abs(k))) * 0.5))) / (t * abs(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 4.2e+57], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(0.5 - N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.1999999999999998e57

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.2%

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

   if 4.1999999999999998e57 < k

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

   10. Applied rewrites 68.5%

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

Alternative 5: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left(0.5 - \cos \left(\left|k\right| + \left|k\right|\right) \cdot 0.5\right) \cdot \left|k\right|}}{\left|k\right| \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 3.4e+84)
  (/
   2.0
   (*
    (*
     (* (/ t l) (* t (* (* (sin (fabs k)) t) (/ 1.0 l))))
     (tan (fabs k)))
    2.0))
  (*
   2.0
   (/
    (*
     l
     (/
      (* (cos (fabs k)) l)
      (* (- 0.5 (* (cos (+ (fabs k) (fabs k))) 0.5)) (fabs k))))
    (* (fabs k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 3.4e+84) {
		tmp = 2.0 / ((((t / l) * (t * ((sin(fabs(k)) * t) * (1.0 / l)))) * tan(fabs(k))) * 2.0);
	} else {
		tmp = 2.0 * ((l * ((cos(fabs(k)) * l) / ((0.5 - (cos((fabs(k) + fabs(k))) * 0.5)) * 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) :: tmp
    if (abs(k) <= 3.4d+84) then
        tmp = 2.0d0 / ((((t / l) * (t * ((sin(abs(k)) * t) * (1.0d0 / l)))) * tan(abs(k))) * 2.0d0)
    else
        tmp = 2.0d0 * ((l * ((cos(abs(k)) * l) / ((0.5d0 - (cos((abs(k) + abs(k))) * 0.5d0)) * abs(k)))) / (abs(k) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 3.4e+84) {
		tmp = 2.0 / ((((t / l) * (t * ((Math.sin(Math.abs(k)) * t) * (1.0 / l)))) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = 2.0 * ((l * ((Math.cos(Math.abs(k)) * l) / ((0.5 - (Math.cos((Math.abs(k) + Math.abs(k))) * 0.5)) * Math.abs(k)))) / (Math.abs(k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 3.4e+84:
		tmp = 2.0 / ((((t / l) * (t * ((math.sin(math.fabs(k)) * t) * (1.0 / l)))) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = 2.0 * ((l * ((math.cos(math.fabs(k)) * l) / ((0.5 - (math.cos((math.fabs(k) + math.fabs(k))) * 0.5)) * math.fabs(k)))) / (math.fabs(k) * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 3.4e+84)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(sin(abs(k)) * t) * Float64(1.0 / l)))) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(Float64(cos(abs(k)) * l) / Float64(Float64(0.5 - Float64(cos(Float64(abs(k) + abs(k))) * 0.5)) * abs(k)))) / Float64(abs(k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 3.4e+84)
		tmp = 2.0 / ((((t / l) * (t * ((sin(abs(k)) * t) * (1.0 / l)))) * tan(abs(k))) * 2.0);
	else
		tmp = 2.0 * ((l * ((cos(abs(k)) * l) / ((0.5 - (cos((abs(k) + abs(k))) * 0.5)) * abs(k)))) / (abs(k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 3.4e+84], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3999999999999998e84

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.2%

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

   if 3.3999999999999998e84 < k

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 66.1%

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

   10. Applied rewrites 66.1%

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

Alternative 6: 83.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ t_2 := \cos t\_1\\ t_3 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t\_1\right)} \cdot 2\\ \mathbf{elif}\;\left|k\right| \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\ell}{\left(\left|k\right| \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{t\_1} \cdot 2\right)\\ \mathbf{elif}\;\left|k\right| \leq 15:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{t}\\ \mathbf{elif}\;\left|k\right| \leq 3.9 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{t\_3}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot t\_2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{t\_3 \cdot \ell}{\left(0.5 - t\_2 \cdot 0.5\right) \cdot \left|k\right|}}{\left|k\right| \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k)))
       (t_2 (cos t_1))
       (t_3 (cos (fabs k))))
  (if (<= (fabs k) 3.3e-208)
    (* (/ (* l l) (* t (* (* (* (sin (fabs k)) t) t) t_1))) 2.0)
    (if (<= (fabs k) 6.5e-126)
      (* (/ l (* (* (fabs k) (* t t)) t)) (* (/ l t_1) 2.0))
      (if (<= (fabs k) 15.0)
        (/ (* (/ l t) (/ l (* (pow (fabs k) 2.0) t))) t)
        (if (<= (fabs k) 3.9e+102)
          (*
           2.0
           (*
            l
            (*
             l
             (/
              t_3
              (* (* (* (fabs k) (fabs k)) t) (- 0.5 (* 0.5 t_2)))))))
          (*
           2.0
           (/
            (* l (/ (* t_3 l) (* (- 0.5 (* t_2 0.5)) (fabs k))))
            (* (fabs k) t)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double t_2 = cos(t_1);
	double t_3 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((sin(fabs(k)) * t) * t) * t_1))) * 2.0;
	} else if (fabs(k) <= 6.5e-126) {
		tmp = (l / ((fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (fabs(k) <= 15.0) {
		tmp = ((l / t) * (l / (pow(fabs(k), 2.0) * t))) / t;
	} else if (fabs(k) <= 3.9e+102) {
		tmp = 2.0 * (l * (l * (t_3 / (((fabs(k) * fabs(k)) * t) * (0.5 - (0.5 * t_2))))));
	} else {
		tmp = 2.0 * ((l * ((t_3 * l) / ((0.5 - (t_2 * 0.5)) * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = abs(k) + abs(k)
    t_2 = cos(t_1)
    t_3 = cos(abs(k))
    if (abs(k) <= 3.3d-208) then
        tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0d0
    else if (abs(k) <= 6.5d-126) then
        tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0d0)
    else if (abs(k) <= 15.0d0) then
        tmp = ((l / t) * (l / ((abs(k) ** 2.0d0) * t))) / t
    else if (abs(k) <= 3.9d+102) then
        tmp = 2.0d0 * (l * (l * (t_3 / (((abs(k) * abs(k)) * t) * (0.5d0 - (0.5d0 * t_2))))))
    else
        tmp = 2.0d0 * ((l * ((t_3 * l) / ((0.5d0 - (t_2 * 0.5d0)) * 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.abs(k) + Math.abs(k);
	double t_2 = Math.cos(t_1);
	double t_3 = Math.cos(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((Math.sin(Math.abs(k)) * t) * t) * t_1))) * 2.0;
	} else if (Math.abs(k) <= 6.5e-126) {
		tmp = (l / ((Math.abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (Math.abs(k) <= 15.0) {
		tmp = ((l / t) * (l / (Math.pow(Math.abs(k), 2.0) * t))) / t;
	} else if (Math.abs(k) <= 3.9e+102) {
		tmp = 2.0 * (l * (l * (t_3 / (((Math.abs(k) * Math.abs(k)) * t) * (0.5 - (0.5 * t_2))))));
	} else {
		tmp = 2.0 * ((l * ((t_3 * l) / ((0.5 - (t_2 * 0.5)) * Math.abs(k)))) / (Math.abs(k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	t_2 = math.cos(t_1)
	t_3 = math.cos(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 3.3e-208:
		tmp = ((l * l) / (t * (((math.sin(math.fabs(k)) * t) * t) * t_1))) * 2.0
	elif math.fabs(k) <= 6.5e-126:
		tmp = (l / ((math.fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0)
	elif math.fabs(k) <= 15.0:
		tmp = ((l / t) * (l / (math.pow(math.fabs(k), 2.0) * t))) / t
	elif math.fabs(k) <= 3.9e+102:
		tmp = 2.0 * (l * (l * (t_3 / (((math.fabs(k) * math.fabs(k)) * t) * (0.5 - (0.5 * t_2))))))
	else:
		tmp = 2.0 * ((l * ((t_3 * l) / ((0.5 - (t_2 * 0.5)) * math.fabs(k)))) / (math.fabs(k) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	t_2 = cos(t_1)
	t_3 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.3e-208)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(sin(abs(k)) * t) * t) * t_1))) * 2.0);
	elseif (abs(k) <= 6.5e-126)
		tmp = Float64(Float64(l / Float64(Float64(abs(k) * Float64(t * t)) * t)) * Float64(Float64(l / t_1) * 2.0));
	elseif (abs(k) <= 15.0)
		tmp = Float64(Float64(Float64(l / t) * Float64(l / Float64((abs(k) ^ 2.0) * t))) / t);
	elseif (abs(k) <= 3.9e+102)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(t_3 / Float64(Float64(Float64(abs(k) * abs(k)) * t) * Float64(0.5 - Float64(0.5 * t_2)))))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(Float64(t_3 * l) / Float64(Float64(0.5 - Float64(t_2 * 0.5)) * abs(k)))) / Float64(abs(k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) + abs(k);
	t_2 = cos(t_1);
	t_3 = cos(abs(k));
	tmp = 0.0;
	if (abs(k) <= 3.3e-208)
		tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0;
	elseif (abs(k) <= 6.5e-126)
		tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	elseif (abs(k) <= 15.0)
		tmp = ((l / t) * (l / ((abs(k) ^ 2.0) * t))) / t;
	elseif (abs(k) <= 3.9e+102)
		tmp = 2.0 * (l * (l * (t_3 / (((abs(k) * abs(k)) * t) * (0.5 - (0.5 * t_2))))));
	else
		tmp = 2.0 * ((l * ((t_3 * l) / ((0.5 - (t_2 * 0.5)) * abs(k)))) / (abs(k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e-208], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 6.5e-126], N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15.0], N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 3.9e+102], N[(2.0 * N[(l * N[(l * N[(t$95$3 / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(N[(t$95$3 * l), $MachinePrecision] / N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 3.3000000000000001e-208

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.8%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 58.8%

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

   8. Applied rewrites 58.8%

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

   if 3.3000000000000001e-208 < k < 6.5000000000000001e-126

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 54.3%

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

   9. Applied rewrites 54.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 63.6%

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

   if 6.5000000000000001e-126 < k < 15

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.6%

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

   10. Applied rewrites 64.6%

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

   if 15 < k < 3.8999999999999998e102

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.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. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 64.4%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 64.4%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

   6. Applied rewrites 61.5%

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

   if 3.8999999999999998e102 < k

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 66.1%

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

   10. Applied rewrites 66.1%

       \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\cos k \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot k}}{\color{blue}{k} \cdot t} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 3.4 \cdot 10^{-197}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(t\_1 \cdot t\right) \cdot t\right) \cdot t\_2\right)} \cdot 2\\ \mathbf{elif}\;\left|k\right| \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot t\_1}{\ell}\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left(0.5 - \cos t\_2 \cdot 0.5\right) \cdot \left|k\right|}}{\left|k\right| \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (sin (fabs k))) (t_2 (+ (fabs k) (fabs k))))
  (if (<= (fabs k) 3.4e-197)
    (* (/ (* l l) (* t (* (* (* t_1 t) t) t_2))) 2.0)
    (if (<= (fabs k) 3.4e+84)
      (/
       2.0
       (* (* (* (/ t l) (/ (* (* t t) t_1) l)) (tan (fabs k))) 2.0))
      (*
       2.0
       (/
        (*
         l
         (/
          (* (cos (fabs k)) l)
          (* (- 0.5 (* (cos t_2) 0.5)) (fabs k))))
        (* (fabs k) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double t_2 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 3.4e-197) {
		tmp = ((l * l) / (t * (((t_1 * t) * t) * t_2))) * 2.0;
	} else if (fabs(k) <= 3.4e+84) {
		tmp = 2.0 / ((((t / l) * (((t * t) * t_1) / l)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = 2.0 * ((l * ((cos(fabs(k)) * l) / ((0.5 - (cos(t_2) * 0.5)) * 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) :: t_2
    real(8) :: tmp
    t_1 = sin(abs(k))
    t_2 = abs(k) + abs(k)
    if (abs(k) <= 3.4d-197) then
        tmp = ((l * l) / (t * (((t_1 * t) * t) * t_2))) * 2.0d0
    else if (abs(k) <= 3.4d+84) then
        tmp = 2.0d0 / ((((t / l) * (((t * t) * t_1) / l)) * tan(abs(k))) * 2.0d0)
    else
        tmp = 2.0d0 * ((l * ((cos(abs(k)) * l) / ((0.5d0 - (cos(t_2) * 0.5d0)) * 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.sin(Math.abs(k));
	double t_2 = Math.abs(k) + Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 3.4e-197) {
		tmp = ((l * l) / (t * (((t_1 * t) * t) * t_2))) * 2.0;
	} else if (Math.abs(k) <= 3.4e+84) {
		tmp = 2.0 / ((((t / l) * (((t * t) * t_1) / l)) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = 2.0 * ((l * ((Math.cos(Math.abs(k)) * l) / ((0.5 - (Math.cos(t_2) * 0.5)) * Math.abs(k)))) / (Math.abs(k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	t_2 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 3.4e-197:
		tmp = ((l * l) / (t * (((t_1 * t) * t) * t_2))) * 2.0
	elif math.fabs(k) <= 3.4e+84:
		tmp = 2.0 / ((((t / l) * (((t * t) * t_1) / l)) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = 2.0 * ((l * ((math.cos(math.fabs(k)) * l) / ((0.5 - (math.cos(t_2) * 0.5)) * math.fabs(k)))) / (math.fabs(k) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 3.4e-197)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(t_1 * t) * t) * t_2))) * 2.0);
	elseif (abs(k) <= 3.4e+84)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(Float64(Float64(t * t) * t_1) / l)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(Float64(cos(abs(k)) * l) / Float64(Float64(0.5 - Float64(cos(t_2) * 0.5)) * abs(k)))) / Float64(abs(k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	t_2 = abs(k) + abs(k);
	tmp = 0.0;
	if (abs(k) <= 3.4e-197)
		tmp = ((l * l) / (t * (((t_1 * t) * t) * t_2))) * 2.0;
	elseif (abs(k) <= 3.4e+84)
		tmp = 2.0 / ((((t / l) * (((t * t) * t_1) / l)) * tan(abs(k))) * 2.0);
	else
		tmp = 2.0 * ((l * ((cos(abs(k)) * l) / ((0.5 - (cos(t_2) * 0.5)) * abs(k)))) / (abs(k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[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], 3.4e-197], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(t$95$1 * t), $MachinePrecision] * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 3.4e+84], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[t$95$2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.3999999999999998e-197

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.8%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 58.8%

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

   8. Applied rewrites 58.8%

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

   if 3.3999999999999998e-197 < k < 3.3999999999999998e84

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 62.5%

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

   if 3.3999999999999998e84 < k

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 60.9%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 66.1%

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

   10. Applied rewrites 66.1%

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

Alternative 8: 78.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ t_2 := \cos t\_1\\ t_3 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t\_1\right)} \cdot 2\\ \mathbf{elif}\;\left|k\right| \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\ell}{\left(\left|k\right| \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{t\_1} \cdot 2\right)\\ \mathbf{elif}\;\left|k\right| \leq 15:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{t}\\ \mathbf{elif}\;\left|k\right| \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{t\_3}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot t\_2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(t\_3 \cdot \ell\right) \cdot \ell}{\left(\left(0.5 - t\_2 \cdot 0.5\right) \cdot \left|k\right|\right) \cdot \left(t \cdot \left|k\right|\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k)))
       (t_2 (cos t_1))
       (t_3 (cos (fabs k))))
  (if (<= (fabs k) 3.3e-208)
    (* (/ (* l l) (* t (* (* (* (sin (fabs k)) t) t) t_1))) 2.0)
    (if (<= (fabs k) 6.5e-126)
      (* (/ l (* (* (fabs k) (* t t)) t)) (* (/ l t_1) 2.0))
      (if (<= (fabs k) 15.0)
        (/ (* (/ l t) (/ l (* (pow (fabs k) 2.0) t))) t)
        (if (<= (fabs k) 1.05e+154)
          (*
           2.0
           (*
            l
            (*
             l
             (/
              t_3
              (* (* (* (fabs k) (fabs k)) t) (- 0.5 (* 0.5 t_2)))))))
          (*
           2.0
           (/
            (* (* t_3 l) l)
            (* (* (- 0.5 (* t_2 0.5)) (fabs k)) (* t (fabs k)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double t_2 = cos(t_1);
	double t_3 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((sin(fabs(k)) * t) * t) * t_1))) * 2.0;
	} else if (fabs(k) <= 6.5e-126) {
		tmp = (l / ((fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (fabs(k) <= 15.0) {
		tmp = ((l / t) * (l / (pow(fabs(k), 2.0) * t))) / t;
	} else if (fabs(k) <= 1.05e+154) {
		tmp = 2.0 * (l * (l * (t_3 / (((fabs(k) * fabs(k)) * t) * (0.5 - (0.5 * t_2))))));
	} else {
		tmp = 2.0 * (((t_3 * l) * l) / (((0.5 - (t_2 * 0.5)) * fabs(k)) * (t * fabs(k))));
	}
	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 = abs(k) + abs(k)
    t_2 = cos(t_1)
    t_3 = cos(abs(k))
    if (abs(k) <= 3.3d-208) then
        tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0d0
    else if (abs(k) <= 6.5d-126) then
        tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0d0)
    else if (abs(k) <= 15.0d0) then
        tmp = ((l / t) * (l / ((abs(k) ** 2.0d0) * t))) / t
    else if (abs(k) <= 1.05d+154) then
        tmp = 2.0d0 * (l * (l * (t_3 / (((abs(k) * abs(k)) * t) * (0.5d0 - (0.5d0 * t_2))))))
    else
        tmp = 2.0d0 * (((t_3 * l) * l) / (((0.5d0 - (t_2 * 0.5d0)) * abs(k)) * (t * abs(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) + Math.abs(k);
	double t_2 = Math.cos(t_1);
	double t_3 = Math.cos(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((Math.sin(Math.abs(k)) * t) * t) * t_1))) * 2.0;
	} else if (Math.abs(k) <= 6.5e-126) {
		tmp = (l / ((Math.abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (Math.abs(k) <= 15.0) {
		tmp = ((l / t) * (l / (Math.pow(Math.abs(k), 2.0) * t))) / t;
	} else if (Math.abs(k) <= 1.05e+154) {
		tmp = 2.0 * (l * (l * (t_3 / (((Math.abs(k) * Math.abs(k)) * t) * (0.5 - (0.5 * t_2))))));
	} else {
		tmp = 2.0 * (((t_3 * l) * l) / (((0.5 - (t_2 * 0.5)) * Math.abs(k)) * (t * Math.abs(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	t_2 = math.cos(t_1)
	t_3 = math.cos(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 3.3e-208:
		tmp = ((l * l) / (t * (((math.sin(math.fabs(k)) * t) * t) * t_1))) * 2.0
	elif math.fabs(k) <= 6.5e-126:
		tmp = (l / ((math.fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0)
	elif math.fabs(k) <= 15.0:
		tmp = ((l / t) * (l / (math.pow(math.fabs(k), 2.0) * t))) / t
	elif math.fabs(k) <= 1.05e+154:
		tmp = 2.0 * (l * (l * (t_3 / (((math.fabs(k) * math.fabs(k)) * t) * (0.5 - (0.5 * t_2))))))
	else:
		tmp = 2.0 * (((t_3 * l) * l) / (((0.5 - (t_2 * 0.5)) * math.fabs(k)) * (t * math.fabs(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	t_2 = cos(t_1)
	t_3 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.3e-208)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(sin(abs(k)) * t) * t) * t_1))) * 2.0);
	elseif (abs(k) <= 6.5e-126)
		tmp = Float64(Float64(l / Float64(Float64(abs(k) * Float64(t * t)) * t)) * Float64(Float64(l / t_1) * 2.0));
	elseif (abs(k) <= 15.0)
		tmp = Float64(Float64(Float64(l / t) * Float64(l / Float64((abs(k) ^ 2.0) * t))) / t);
	elseif (abs(k) <= 1.05e+154)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(t_3 / Float64(Float64(Float64(abs(k) * abs(k)) * t) * Float64(0.5 - Float64(0.5 * t_2)))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(t_3 * l) * l) / Float64(Float64(Float64(0.5 - Float64(t_2 * 0.5)) * abs(k)) * Float64(t * abs(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) + abs(k);
	t_2 = cos(t_1);
	t_3 = cos(abs(k));
	tmp = 0.0;
	if (abs(k) <= 3.3e-208)
		tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0;
	elseif (abs(k) <= 6.5e-126)
		tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	elseif (abs(k) <= 15.0)
		tmp = ((l / t) * (l / ((abs(k) ^ 2.0) * t))) / t;
	elseif (abs(k) <= 1.05e+154)
		tmp = 2.0 * (l * (l * (t_3 / (((abs(k) * abs(k)) * t) * (0.5 - (0.5 * t_2))))));
	else
		tmp = 2.0 * (((t_3 * l) * l) / (((0.5 - (t_2 * 0.5)) * abs(k)) * (t * abs(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e-208], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 6.5e-126], N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15.0], N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.05e+154], N[(2.0 * N[(l * N[(l * N[(t$95$3 / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(t$95$3 * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 3.3000000000000001e-208

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.8%

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 58.8%

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

   8. Applied rewrites 58.8%

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

   if 3.3000000000000001e-208 < k < 6.5000000000000001e-126

   1. Initial program 54.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.7%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 54.3%

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

   9. Applied rewrites 54.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 63.6%

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

   if 6.5000000000000001e-126 < k < 15

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 67.1%

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

   3. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 76.1%

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 64.6%

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

   10. Applied rewrites 64.6%

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

   if 15 < k < 1.05e154

   1. Initial program 54.5%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.8%

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.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. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 64.4%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 64.4%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}}\right)\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{\color{blue}{2}}}\right)\right) \]

      21. unpow2 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\sin k \cdot \color{blue}{\sin k}\right)}\right)\right) \]

   6. Applied rewrites 61.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\right)}\right) \]

   if 1.05e154 < k

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. 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 59.8%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      12. associate-*r* N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      13. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 60.9%

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

   8. Applied rewrites 59.8%

      \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{\color{blue}{k \cdot t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}}}{k}}{\color{blue}{k \cdot t}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}}}{k}}{\color{blue}{k} \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}}}{k}}{k \cdot t} \]

      4. associate-/l/ N/A

         \[\leadsto 2 \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k}}{\color{blue}{k} \cdot t} \]

      5. associate-/l/ N/A

         \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      8. lower-*.f64 58.6%

         \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot k\right) \cdot \left(\color{blue}{k} \cdot t\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k\right) \cdot \left(k \cdot \color{blue}{t}\right)} \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(\frac{1}{2} - \cos \left(k + k\right) \cdot \frac{1}{2}\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      11. lower-*.f64 58.6%

          \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   10. Applied rewrites 58.6%

       \[\leadsto 2 \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t\_1\right)} \cdot 2\\ \mathbf{elif}\;\left|k\right| \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\ell}{\left(\left|k\right| \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{t\_1} \cdot 2\right)\\ \mathbf{elif}\;\left|k\right| \leq 15:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos \left(\left|k\right|\right)}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos t\_1\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k))))
  (if (<= (fabs k) 3.3e-208)
    (* (/ (* l l) (* t (* (* (* (sin (fabs k)) t) t) t_1))) 2.0)
    (if (<= (fabs k) 6.5e-126)
      (* (/ l (* (* (fabs k) (* t t)) t)) (* (/ l t_1) 2.0))
      (if (<= (fabs k) 15.0)
        (/ (* (/ l t) (/ l (* (pow (fabs k) 2.0) t))) t)
        (*
         2.0
         (*
          l
          (*
           l
           (/
            (cos (fabs k))
            (*
             (* (* (fabs k) (fabs k)) t)
             (- 0.5 (* 0.5 (cos t_1)))))))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((sin(fabs(k)) * t) * t) * t_1))) * 2.0;
	} else if (fabs(k) <= 6.5e-126) {
		tmp = (l / ((fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (fabs(k) <= 15.0) {
		tmp = ((l / t) * (l / (pow(fabs(k), 2.0) * t))) / t;
	} else {
		tmp = 2.0 * (l * (l * (cos(fabs(k)) / (((fabs(k) * fabs(k)) * t) * (0.5 - (0.5 * cos(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) :: tmp
    t_1 = abs(k) + abs(k)
    if (abs(k) <= 3.3d-208) then
        tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0d0
    else if (abs(k) <= 6.5d-126) then
        tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0d0)
    else if (abs(k) <= 15.0d0) then
        tmp = ((l / t) * (l / ((abs(k) ** 2.0d0) * t))) / t
    else
        tmp = 2.0d0 * (l * (l * (cos(abs(k)) / (((abs(k) * abs(k)) * t) * (0.5d0 - (0.5d0 * cos(t_1)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) + Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((Math.sin(Math.abs(k)) * t) * t) * t_1))) * 2.0;
	} else if (Math.abs(k) <= 6.5e-126) {
		tmp = (l / ((Math.abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (Math.abs(k) <= 15.0) {
		tmp = ((l / t) * (l / (Math.pow(Math.abs(k), 2.0) * t))) / t;
	} else {
		tmp = 2.0 * (l * (l * (Math.cos(Math.abs(k)) / (((Math.abs(k) * Math.abs(k)) * t) * (0.5 - (0.5 * Math.cos(t_1)))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 3.3e-208:
		tmp = ((l * l) / (t * (((math.sin(math.fabs(k)) * t) * t) * t_1))) * 2.0
	elif math.fabs(k) <= 6.5e-126:
		tmp = (l / ((math.fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0)
	elif math.fabs(k) <= 15.0:
		tmp = ((l / t) * (l / (math.pow(math.fabs(k), 2.0) * t))) / t
	else:
		tmp = 2.0 * (l * (l * (math.cos(math.fabs(k)) / (((math.fabs(k) * math.fabs(k)) * t) * (0.5 - (0.5 * math.cos(t_1)))))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 3.3e-208)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(sin(abs(k)) * t) * t) * t_1))) * 2.0);
	elseif (abs(k) <= 6.5e-126)
		tmp = Float64(Float64(l / Float64(Float64(abs(k) * Float64(t * t)) * t)) * Float64(Float64(l / t_1) * 2.0));
	elseif (abs(k) <= 15.0)
		tmp = Float64(Float64(Float64(l / t) * Float64(l / Float64((abs(k) ^ 2.0) * t))) / t);
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(abs(k)) / Float64(Float64(Float64(abs(k) * abs(k)) * t) * Float64(0.5 - Float64(0.5 * cos(t_1))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) + abs(k);
	tmp = 0.0;
	if (abs(k) <= 3.3e-208)
		tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0;
	elseif (abs(k) <= 6.5e-126)
		tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	elseif (abs(k) <= 15.0)
		tmp = ((l / t) * (l / ((abs(k) ^ 2.0) * t))) / t;
	else
		tmp = 2.0 * (l * (l * (cos(abs(k)) / (((abs(k) * abs(k)) * t) * (0.5 - (0.5 * cos(t_1)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e-208], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 6.5e-126], N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15.0], N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.3000000000000001e-208

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      12. lower-*.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(2 \cdot \color{blue}{k}\right)\right)} \cdot 2 \]

      14. count-2-rev N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

      15. lift-+.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + k\right)\right)}} \cdot 2 \]

   if 3.3000000000000001e-208 < k < 6.5000000000000001e-126

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      2. lower-pow.f64 54.3%

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 6.5000000000000001e-126 < k < 15

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. mult-flip N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(t \cdot t\right) \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 76.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)}}{t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}}}{t} \]

      3. lower-pow.f64 64.6%

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot t}}{t} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]

   if 15 < k

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. 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 59.8%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.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. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right)\right) \]

      11. lower-/.f64 64.4%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}\right)\right) \]

      13. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)}\right)\right) \]

      14. pow2 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)}\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)}\right)\right) \]

      16. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)}\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}}\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}}\right)\right) \]

      19. lower-*.f64 64.4%

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}}\right)\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{\color{blue}{2}}}\right)\right) \]

      21. unpow2 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\sin k \cdot \color{blue}{\sin k}\right)}\right)\right) \]

   6. Applied rewrites 61.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\right)}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t\_1\right)} \cdot 2\\ \mathbf{elif}\;\left|k\right| \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\ell}{\left(\left|k\right| \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{t\_1} \cdot 2\right)\\ \mathbf{elif}\;\left|k\right| \leq 1.35 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos t\_1 \cdot 0.5}}{\left|k\right|}}{\left|k\right| \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k))))
  (if (<= (fabs k) 3.3e-208)
    (* (/ (* l l) (* t (* (* (* (sin (fabs k)) t) t) t_1))) 2.0)
    (if (<= (fabs k) 6.5e-126)
      (* (/ l (* (* (fabs k) (* t t)) t)) (* (/ l t_1) 2.0))
      (if (<= (fabs k) 1.35e+85)
        (/ (* (/ l t) (/ l (* (pow (fabs k) 2.0) t))) t)
        (*
         2.0
         (/
          (/ (/ (* (* 1.0 l) l) (- 0.5 (* (cos t_1) 0.5))) (fabs k))
          (* (fabs k) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((sin(fabs(k)) * t) * t) * t_1))) * 2.0;
	} else if (fabs(k) <= 6.5e-126) {
		tmp = (l / ((fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (fabs(k) <= 1.35e+85) {
		tmp = ((l / t) * (l / (pow(fabs(k), 2.0) * t))) / t;
	} else {
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (cos(t_1) * 0.5))) / 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 = abs(k) + abs(k)
    if (abs(k) <= 3.3d-208) then
        tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0d0
    else if (abs(k) <= 6.5d-126) then
        tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0d0)
    else if (abs(k) <= 1.35d+85) then
        tmp = ((l / t) * (l / ((abs(k) ** 2.0d0) * t))) / t
    else
        tmp = 2.0d0 * (((((1.0d0 * l) * l) / (0.5d0 - (cos(t_1) * 0.5d0))) / 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.abs(k) + Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((Math.sin(Math.abs(k)) * t) * t) * t_1))) * 2.0;
	} else if (Math.abs(k) <= 6.5e-126) {
		tmp = (l / ((Math.abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	} else if (Math.abs(k) <= 1.35e+85) {
		tmp = ((l / t) * (l / (Math.pow(Math.abs(k), 2.0) * t))) / t;
	} else {
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (Math.cos(t_1) * 0.5))) / Math.abs(k)) / (Math.abs(k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 3.3e-208:
		tmp = ((l * l) / (t * (((math.sin(math.fabs(k)) * t) * t) * t_1))) * 2.0
	elif math.fabs(k) <= 6.5e-126:
		tmp = (l / ((math.fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0)
	elif math.fabs(k) <= 1.35e+85:
		tmp = ((l / t) * (l / (math.pow(math.fabs(k), 2.0) * t))) / t
	else:
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (math.cos(t_1) * 0.5))) / math.fabs(k)) / (math.fabs(k) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 3.3e-208)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(sin(abs(k)) * t) * t) * t_1))) * 2.0);
	elseif (abs(k) <= 6.5e-126)
		tmp = Float64(Float64(l / Float64(Float64(abs(k) * Float64(t * t)) * t)) * Float64(Float64(l / t_1) * 2.0));
	elseif (abs(k) <= 1.35e+85)
		tmp = Float64(Float64(Float64(l / t) * Float64(l / Float64((abs(k) ^ 2.0) * t))) / t);
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(Float64(1.0 * l) * l) / Float64(0.5 - Float64(cos(t_1) * 0.5))) / abs(k)) / Float64(abs(k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) + abs(k);
	tmp = 0.0;
	if (abs(k) <= 3.3e-208)
		tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0;
	elseif (abs(k) <= 6.5e-126)
		tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	elseif (abs(k) <= 1.35e+85)
		tmp = ((l / t) * (l / ((abs(k) ^ 2.0) * t))) / t;
	else
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (cos(t_1) * 0.5))) / abs(k)) / (abs(k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e-208], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 6.5e-126], N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.35e+85], N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[(1.0 * l), $MachinePrecision] * l), $MachinePrecision] / N[(0.5 - N[(N[Cos[t$95$1], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.3000000000000001e-208

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      12. lower-*.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(2 \cdot \color{blue}{k}\right)\right)} \cdot 2 \]

      14. count-2-rev N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

      15. lift-+.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + k\right)\right)}} \cdot 2 \]

   if 3.3000000000000001e-208 < k < 6.5000000000000001e-126

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      2. lower-pow.f64 54.3%

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 6.5000000000000001e-126 < k < 1.3499999999999999e85

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. mult-flip N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(t \cdot t\right) \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 76.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)}}{t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}}}{t} \]

      3. lower-pow.f64 64.6%

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot t}}{t} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]

   if 1.3499999999999999e85 < k

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. 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 59.8%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      12. associate-*r* N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      13. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 60.9%

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

   8. Applied rewrites 59.8%

      \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{\color{blue}{k \cdot t}} \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{k \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.8%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{k \cdot t} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.55 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{k \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{1}{\frac{\left(\left(\frac{k}{t\_1} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left|t\right|\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \frac{\ell}{\left(k + k\right) \cdot \left(\left(k \cdot t\_1\right) \cdot \left|t\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.55e-125)
     (*
      2.0
      (/
       (/ (/ (* (* 1.0 l) l) (- 0.5 (* (cos (+ k k)) 0.5))) k)
       (* k (fabs t))))
     (if (<= (fabs t) 1.8e+43)
       (*
        (/ 2.0 k)
        (/
         1.0
         (/
          (*
           (* (- (* (/ k t_1) k) -2.0) (tan k))
           (* (* (/ (fabs t) l) (fabs t)) (fabs t)))
          l)))
       (* (+ l l) (/ l (* (+ k k) (* (* k t_1) (fabs t))))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.55e-125) {
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (cos((k + k)) * 0.5))) / k) / (k * fabs(t)));
	} else if (fabs(t) <= 1.8e+43) {
		tmp = (2.0 / k) * (1.0 / ((((((k / t_1) * k) - -2.0) * tan(k)) * (((fabs(t) / l) * fabs(t)) * fabs(t))) / l));
	} else {
		tmp = (l + l) * (l / ((k + k) * ((k * t_1) * fabs(t))));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.55e-125) {
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (Math.cos((k + k)) * 0.5))) / k) / (k * Math.abs(t)));
	} else if (Math.abs(t) <= 1.8e+43) {
		tmp = (2.0 / k) * (1.0 / ((((((k / t_1) * k) - -2.0) * Math.tan(k)) * (((Math.abs(t) / l) * Math.abs(t)) * Math.abs(t))) / l));
	} else {
		tmp = (l + l) * (l / ((k + k) * ((k * t_1) * Math.abs(t))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.55e-125:
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (math.cos((k + k)) * 0.5))) / k) / (k * math.fabs(t)))
	elif math.fabs(t) <= 1.8e+43:
		tmp = (2.0 / k) * (1.0 / ((((((k / t_1) * k) - -2.0) * math.tan(k)) * (((math.fabs(t) / l) * math.fabs(t)) * math.fabs(t))) / l))
	else:
		tmp = (l + l) * (l / ((k + k) * ((k * t_1) * math.fabs(t))))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.55e-125)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(Float64(1.0 * l) * l) / Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))) / k) / Float64(k * abs(t))));
	elseif (abs(t) <= 1.8e+43)
		tmp = Float64(Float64(2.0 / k) * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(Float64(k / t_1) * k) - -2.0) * tan(k)) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * abs(t))) / l)));
	else
		tmp = Float64(Float64(l + l) * Float64(l / Float64(Float64(k + k) * Float64(Float64(k * t_1) * abs(t)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.55e-125)
		tmp = 2.0 * (((((1.0 * l) * l) / (0.5 - (cos((k + k)) * 0.5))) / k) / (k * abs(t)));
	elseif (abs(t) <= 1.8e+43)
		tmp = (2.0 / k) * (1.0 / ((((((k / t_1) * k) - -2.0) * tan(k)) * (((abs(t) / l) * abs(t)) * abs(t))) / l));
	else
		tmp = (l + l) * (l / ((k + k) * ((k * t_1) * abs(t))));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.55e-125], N[(2.0 * N[(N[(N[(N[(N[(1.0 * l), $MachinePrecision] * l), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.8e+43], N[(N[(2.0 / k), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[(N[(N[(k / t$95$1), $MachinePrecision] * k), $MachinePrecision] - -2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[(k + k), $MachinePrecision] * N[(N[(k * t$95$1), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5500000000000001e-125

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. 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 59.8%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\color{blue}{k} \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      12. associate-*r* N/A

          \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      13. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 60.9%

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.3%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}\right) \]

      3. associate-*l/ N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{\left(k \cdot k\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}}{k}}{\color{blue}{k \cdot t}} \]

   8. Applied rewrites 59.8%

      \[\leadsto 2 \cdot \frac{\frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{\color{blue}{k \cdot t}} \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{k \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.8%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\left(1 \cdot \ell\right) \cdot \ell}{0.5 - \cos \left(k + k\right) \cdot 0.5}}{k}}{k \cdot t} \]

   if 1.5500000000000001e-125 < t < 1.8e43

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k}} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{\frac{t}{\ell}}{\ell}}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{\frac{t}{\ell}}}{\ell}\right)} \]

      8. associate-/l* N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \frac{t}{\ell}}{\ell}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \frac{t}{\ell}}}{\ell}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\frac{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}{\ell}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\frac{\left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}{\ell}}} \]

   5. Applied rewrites 60.2%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\frac{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot t\right)}{\ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2}{k}} \cdot \frac{1}{\frac{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot t\right)}{\ell}} \]
   7. Step-by-step derivation

      1. lower-/.f64 54.6%

         \[\leadsto \frac{2}{\color{blue}{k}} \cdot \frac{1}{\frac{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot t\right)}{\ell}} \]

   8. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{2}{k}} \cdot \frac{1}{\frac{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot t\right)}{\ell}} \]

   if 1.8e43 < t

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      2. lower-pow.f64 54.3%

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

       3. lift-/.f64 N/A

          \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

       4. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

       5. associate-/l* N/A

          \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}\right)} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

   11. Applied rewrites 62.4%

       \[\leadsto \color{blue}{\left(\ell + \ell\right) \cdot \frac{\ell}{\left(k + k\right) \cdot \left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{\left(\left|t\right|\right)}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left|t\right| \cdot k\right) \cdot t\_1} \cdot \ell}{\left(\frac{k}{t\_1} \cdot k - -2\right) \cdot \tan k} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot \frac{\ell}{{k}^{2} \cdot \left|t\right|}}{\left|t\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow (fabs t) 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))
        2e+269)
     (*
      (/
       (* (/ l (* (* (fabs t) k) t_1)) l)
       (* (- (* (/ k t_1) k) -2.0) (tan k)))
      2.0)
     (/ (* (/ l (fabs t)) (/ l (* (pow k 2.0) (fabs t)))) (fabs t))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if ((2.0 / ((((pow(fabs(t), 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0))) <= 2e+269) {
		tmp = (((l / ((fabs(t) * k) * t_1)) * l) / ((((k / t_1) * k) - -2.0) * tan(k))) * 2.0;
	} else {
		tmp = ((l / fabs(t)) * (l / (pow(k, 2.0) * fabs(t)))) / fabs(t);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * Math.abs(t);
	double tmp;
	if ((2.0 / ((((Math.pow(Math.abs(t), 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0))) <= 2e+269) {
		tmp = (((l / ((Math.abs(t) * k) * t_1)) * l) / ((((k / t_1) * k) - -2.0) * Math.tan(k))) * 2.0;
	} else {
		tmp = ((l / Math.abs(t)) * (l / (Math.pow(k, 2.0) * Math.abs(t)))) / Math.abs(t);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * math.fabs(t)
	tmp = 0
	if (2.0 / ((((math.pow(math.fabs(t), 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))) <= 2e+269:
		tmp = (((l / ((math.fabs(t) * k) * t_1)) * l) / ((((k / t_1) * k) - -2.0) * math.tan(k))) * 2.0
	else:
		tmp = ((l / math.fabs(t)) * (l / (math.pow(k, 2.0) * math.fabs(t)))) / math.fabs(t)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((abs(t) ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0))) <= 2e+269)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(abs(t) * k) * t_1)) * l) / Float64(Float64(Float64(Float64(k / t_1) * k) - -2.0) * tan(k))) * 2.0);
	else
		tmp = Float64(Float64(Float64(l / abs(t)) * Float64(l / Float64((k ^ 2.0) * abs(t)))) / abs(t));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * abs(t);
	tmp = 0.0;
	if ((2.0 / (((((abs(t) ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0))) <= 2e+269)
		tmp = (((l / ((abs(t) * k) * t_1)) * l) / ((((k / t_1) * k) - -2.0) * tan(k))) * 2.0;
	else
		tmp = ((l / abs(t)) * (l / ((k ^ 2.0) * abs(t)))) / abs(t);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[N[Abs[t], $MachinePrecision], 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 / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+269], N[(N[(N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(N[(k / t$95$1), $MachinePrecision] * k), $MachinePrecision] - -2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e269

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 44.2%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2 \]

   6. Applied rewrites 44.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)}} \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)}} \cdot 2 \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k}\right)} \cdot 2 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\color{blue}{\frac{k \cdot k}{t \cdot t}} - -2\right) \cdot \tan k}\right) \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\frac{\color{blue}{k \cdot k}}{t \cdot t} - -2\right) \cdot \tan k}\right) \cdot 2 \]

      7. associate-*l/ N/A

         \[\leadsto \left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\color{blue}{\frac{k}{t \cdot t} \cdot k} - -2\right) \cdot \tan k}\right) \cdot 2 \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\color{blue}{\frac{k}{t \cdot t}} \cdot k - -2\right) \cdot \tan k}\right) \cdot 2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\left(\color{blue}{\frac{k}{t \cdot t} \cdot k} - -2\right) \cdot \tan k}\right) \cdot 2 \]

      10. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell}{\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k}} \cdot 2 \]

   8. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell}{\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k}} \cdot 2 \]

   if 2.0000000000000001e269 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. mult-flip N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(t \cdot t\right) \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 76.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)}}{t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}}}{t} \]

      3. lower-pow.f64 64.6%

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot t}}{t} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.4% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{\left(\left|t\right|\right)}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \left(\left|t\right| \cdot \left|t\right|\right)\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot \frac{\ell}{{k}^{2} \cdot \left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow (fabs t) 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))
      2e+269)
   (*
    (/ l (* (* k (* (fabs t) (fabs t))) (fabs t)))
    (* (/ l (+ k k)) 2.0))
   (/ (* (/ l (fabs t)) (/ l (* (pow k 2.0) (fabs t)))) (fabs t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(fabs(t), 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0))) <= 2e+269) {
		tmp = (l / ((k * (fabs(t) * fabs(t))) * fabs(t))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l / fabs(t)) * (l / (pow(k, 2.0) * fabs(t)))) / fabs(t);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(Math.abs(t), 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0))) <= 2e+269) {
		tmp = (l / ((k * (Math.abs(t) * Math.abs(t))) * Math.abs(t))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l / Math.abs(t)) * (l / (Math.pow(k, 2.0) * Math.abs(t)))) / Math.abs(t);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(math.fabs(t), 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))) <= 2e+269:
		tmp = (l / ((k * (math.fabs(t) * math.fabs(t))) * math.fabs(t))) * ((l / (k + k)) * 2.0)
	else:
		tmp = ((l / math.fabs(t)) * (l / (math.pow(k, 2.0) * math.fabs(t)))) / math.fabs(t)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((abs(t) ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0))) <= 2e+269)
		tmp = Float64(Float64(l / Float64(Float64(k * Float64(abs(t) * abs(t))) * abs(t))) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(l / abs(t)) * Float64(l / Float64((k ^ 2.0) * abs(t)))) / abs(t));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((abs(t) ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0))) <= 2e+269)
		tmp = (l / ((k * (abs(t) * abs(t))) * abs(t))) * ((l / (k + k)) * 2.0);
	else
		tmp = ((l / abs(t)) * (l / ((k ^ 2.0) * abs(t)))) / abs(t);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[N[Abs[t], $MachinePrecision], 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 / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+269], N[(N[(l / N[(N[(k * N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e269

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      2. lower-pow.f64 54.3%

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 2.0000000000000001e269 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. mult-flip N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(t \cdot t\right) \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{1}{\ell}\right)}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot \sin k\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 76.1%

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \color{blue}{\frac{1}{\ell}}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{1}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell \cdot 2}{\left(\left(\frac{k}{t \cdot t} \cdot k - -2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)}}{t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}}}{t} \]

      3. lower-pow.f64 64.6%

         \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{2} \cdot t}}{t} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{-208}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t\_1\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left|k\right| \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{t\_1} \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k))))
  (if (<= (fabs k) 3.3e-208)
    (* (/ (* l l) (* t (* (* (* (sin (fabs k)) t) t) t_1))) 2.0)
    (* (/ l (* (* (fabs k) (* t t)) t)) (* (/ l t_1) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((sin(fabs(k)) * t) * t) * t_1))) * 2.0;
	} else {
		tmp = (l / ((fabs(k) * (t * t)) * t)) * ((l / t_1) * 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) :: tmp
    t_1 = abs(k) + abs(k)
    if (abs(k) <= 3.3d-208) then
        tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0d0
    else
        tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) + Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 3.3e-208) {
		tmp = ((l * l) / (t * (((Math.sin(Math.abs(k)) * t) * t) * t_1))) * 2.0;
	} else {
		tmp = (l / ((Math.abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 3.3e-208:
		tmp = ((l * l) / (t * (((math.sin(math.fabs(k)) * t) * t) * t_1))) * 2.0
	else:
		tmp = (l / ((math.fabs(k) * (t * t)) * t)) * ((l / t_1) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 3.3e-208)
		tmp = Float64(Float64(Float64(l * l) / Float64(t * Float64(Float64(Float64(sin(abs(k)) * t) * t) * t_1))) * 2.0);
	else
		tmp = Float64(Float64(l / Float64(Float64(abs(k) * Float64(t * t)) * t)) * Float64(Float64(l / t_1) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) + abs(k);
	tmp = 0.0;
	if (abs(k) <= 3.3e-208)
		tmp = ((l * l) / (t * (((sin(abs(k)) * t) * t) * t_1))) * 2.0;
	else
		tmp = (l / ((abs(k) * (t * t)) * t)) * ((l / t_1) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e-208], N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3000000000000001e-208

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot t\right)}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(\sin k \cdot t\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      12. lower-*.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(2 \cdot \color{blue}{k}\right)\right)} \cdot 2 \]

      14. count-2-rev N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

      15. lift-+.f64 58.8%

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + \color{blue}{k}\right)\right)} \cdot 2 \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(k + k\right)\right)}} \cdot 2 \]

   if 3.3000000000000001e-208 < k

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   3. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 55.5%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

      2. lower-pow.f64 54.3%

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 63.6%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.7% accurate, 9.1× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (/ l (* (* k (* t t)) t)) (* (/ l (+ k k)) 2.0)))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * t)) * ((l / (k + k)) * 2.0);
}

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 = (l / ((k * (t * t)) * t)) * ((l / (k + k)) * 2.0d0)
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * t)) * ((l / (k + k)) * 2.0);
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * t)) * ((l / (k + k)) * 2.0)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * t)) * Float64(Float64(l / Float64(k + k)) * 2.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * t)) * ((l / (k + k)) * 2.0);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

3. Applied rewrites 45.7%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

4. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
5. Step-by-step derivation

   1. lower-*.f64 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

6. Applied rewrites 55.5%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   2. lower-pow.f64 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

9. Applied rewrites 54.3%

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

    2. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    5. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

    6. associate-*l* N/A

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{3}} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

11. Applied rewrites 63.6%

    \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]
12. Add Preprocessing

Alternative 16: 62.4% accurate, 10.7× speedup

Math

\[\left(\ell + \ell\right) \cdot \frac{\ell}{\left(k + k\right) \cdot \left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (+ l l) (/ l (* (+ k k) (* (* k (* t t)) t)))))

C

double code(double t, double l, double k) {
	return (l + l) * (l / ((k + k) * ((k * (t * t)) * 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 + k) * ((k * (t * t)) * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * (l / ((k + k) * ((k * (t * t)) * t)));
}

Python

def code(t, l, k):
	return (l + l) * (l / ((k + k) * ((k * (t * t)) * t)))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(l / Float64(Float64(k + k) * Float64(Float64(k * Float64(t * t)) * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (l / ((k + k) * ((k * (t * t)) * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[(k + k), $MachinePrecision] * N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot 2} \]

3. Applied rewrites 45.7%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

4. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
5. Step-by-step derivation

   1. lower-*.f64 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

6. Applied rewrites 55.5%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{{t}^{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   2. lower-pow.f64 54.3%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot {t}^{\color{blue}{3}}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

9. Applied rewrites 54.3%

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

    3. lift-/.f64 N/A

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

    4. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

    5. associate-/l* N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}\right)} \]

    6. associate-*r* N/A

       \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)}} \]

    7. *-commutative N/A

       \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

    8. lift-*.f64 N/A

       \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot \frac{\ell}{\left(k \cdot {t}^{3}\right) \cdot \left(2 \cdot k\right)} \]

11. Applied rewrites 62.4%

    \[\leadsto \color{blue}{\left(\ell + \ell\right) \cdot \frac{\ell}{\left(k + k\right) \cdot \left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)}} \]
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))))