Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 83.8%

Time: 5.1s

Alternatives: 20

Speedup: 10.3×

• 73.4% 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
 (*
  (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
  (+ (+ 1 (pow (/ k t) 2)) 1))))

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 / N[(N[(N[(N[(N[Power[t, 3], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[Power[N[(k / t), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.1% 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
 (*
  (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
  (+ (+ 1 (pow (/ k t) 2)) 1))))

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 / N[(N[(N[(N[(N[Power[t, 3], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[Power[N[(k / t), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin k \cdot \left|t\right|}{\ell}\\ t_2 := \left|t\right| \cdot \left|t\right|\\ t_3 := {\sin k}^{2}\\ t_4 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{4495317912455029}{12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left|t\right| \cdot \left(2 \cdot \frac{{\left(\left|t\right|\right)}^{2} \cdot t\_3}{\cos k} + \frac{{k}^{2} \cdot t\_3}{\cos k}\right)} \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq \frac{5902958103587057}{590295810358705651712}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(t\_4 \cdot \left|t\right|\right) \cdot \left(\left(\frac{k}{t\_2} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_2 \cdot \left(\left|t\right| \cdot \sin k\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k \cdot k}{t\_2} - -2\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot t\_4\right) \cdot \left(t\_1 \cdot \left|t\right|\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (* (sin k) (fabs t)) l))
       (t_2 (* (fabs t) (fabs t)))
       (t_3 (pow (sin k) 2))
       (t_4 (/ (fabs t) l)))
  (*
   (copysign 1 t)
   (if (<=
        (fabs t)
        4495317912455029/12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704)
     (*
      (/
       (* l l)
       (*
        (fabs t)
        (+
         (* 2 (/ (* (pow (fabs t) 2) t_3) (cos k)))
         (/ (* (pow k 2) t_3) (cos k)))))
      2)
     (if (<= (fabs t) 5902958103587057/590295810358705651712)
       (/
        2
        (*
         t_1
         (* (* t_4 (fabs t)) (* (- (* (/ k t_2) k) -2) (tan k)))))
       (if (<=
            (fabs t)
            102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696)
         (/
          2
          (/
           (*
            (/ (* t_2 (* (fabs t) (sin k))) l)
            (* (sin k) (- (/ (* k k) t_2) -2)))
           (* l (cos k))))
         (/ 2 (* (* (* (tan k) t_4) (* t_1 (fabs t))) 2))))))))

C

double code(double t, double l, double k) {
	double t_1 = (sin(k) * fabs(t)) / l;
	double t_2 = fabs(t) * fabs(t);
	double t_3 = pow(sin(k), 2.0);
	double t_4 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 3.6e-130) {
		tmp = ((l * l) / (fabs(t) * ((2.0 * ((pow(fabs(t), 2.0) * t_3) / cos(k))) + ((pow(k, 2.0) * t_3) / cos(k))))) * 2.0;
	} else if (fabs(t) <= 1e-5) {
		tmp = 2.0 / (t_1 * ((t_4 * fabs(t)) * ((((k / t_2) * k) - -2.0) * tan(k))));
	} else if (fabs(t) <= 1.02e+101) {
		tmp = 2.0 / ((((t_2 * (fabs(t) * sin(k))) / l) * (sin(k) * (((k * k) / t_2) - -2.0))) / (l * cos(k)));
	} else {
		tmp = 2.0 / (((tan(k) * t_4) * (t_1 * fabs(t))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (Math.sin(k) * Math.abs(t)) / l;
	double t_2 = Math.abs(t) * Math.abs(t);
	double t_3 = Math.pow(Math.sin(k), 2.0);
	double t_4 = Math.abs(t) / l;
	double tmp;
	if (Math.abs(t) <= 3.6e-130) {
		tmp = ((l * l) / (Math.abs(t) * ((2.0 * ((Math.pow(Math.abs(t), 2.0) * t_3) / Math.cos(k))) + ((Math.pow(k, 2.0) * t_3) / Math.cos(k))))) * 2.0;
	} else if (Math.abs(t) <= 1e-5) {
		tmp = 2.0 / (t_1 * ((t_4 * Math.abs(t)) * ((((k / t_2) * k) - -2.0) * Math.tan(k))));
	} else if (Math.abs(t) <= 1.02e+101) {
		tmp = 2.0 / ((((t_2 * (Math.abs(t) * Math.sin(k))) / l) * (Math.sin(k) * (((k * k) / t_2) - -2.0))) / (l * Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.tan(k) * t_4) * (t_1 * Math.abs(t))) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = (math.sin(k) * math.fabs(t)) / l
	t_2 = math.fabs(t) * math.fabs(t)
	t_3 = math.pow(math.sin(k), 2.0)
	t_4 = math.fabs(t) / l
	tmp = 0
	if math.fabs(t) <= 3.6e-130:
		tmp = ((l * l) / (math.fabs(t) * ((2.0 * ((math.pow(math.fabs(t), 2.0) * t_3) / math.cos(k))) + ((math.pow(k, 2.0) * t_3) / math.cos(k))))) * 2.0
	elif math.fabs(t) <= 1e-5:
		tmp = 2.0 / (t_1 * ((t_4 * math.fabs(t)) * ((((k / t_2) * k) - -2.0) * math.tan(k))))
	elif math.fabs(t) <= 1.02e+101:
		tmp = 2.0 / ((((t_2 * (math.fabs(t) * math.sin(k))) / l) * (math.sin(k) * (((k * k) / t_2) - -2.0))) / (l * math.cos(k)))
	else:
		tmp = 2.0 / (((math.tan(k) * t_4) * (t_1 * math.fabs(t))) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(sin(k) * abs(t)) / l)
	t_2 = Float64(abs(t) * abs(t))
	t_3 = sin(k) ^ 2.0
	t_4 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 3.6e-130)
		tmp = Float64(Float64(Float64(l * l) / Float64(abs(t) * Float64(Float64(2.0 * Float64(Float64((abs(t) ^ 2.0) * t_3) / cos(k))) + Float64(Float64((k ^ 2.0) * t_3) / cos(k))))) * 2.0);
	elseif (abs(t) <= 1e-5)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t_4 * abs(t)) * Float64(Float64(Float64(Float64(k / t_2) * k) - -2.0) * tan(k)))));
	elseif (abs(t) <= 1.02e+101)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * Float64(abs(t) * sin(k))) / l) * Float64(sin(k) * Float64(Float64(Float64(k * k) / t_2) - -2.0))) / Float64(l * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * t_4) * Float64(t_1 * abs(t))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (sin(k) * abs(t)) / l;
	t_2 = abs(t) * abs(t);
	t_3 = sin(k) ^ 2.0;
	t_4 = abs(t) / l;
	tmp = 0.0;
	if (abs(t) <= 3.6e-130)
		tmp = ((l * l) / (abs(t) * ((2.0 * (((abs(t) ^ 2.0) * t_3) / cos(k))) + (((k ^ 2.0) * t_3) / cos(k))))) * 2.0;
	elseif (abs(t) <= 1e-5)
		tmp = 2.0 / (t_1 * ((t_4 * abs(t)) * ((((k / t_2) * k) - -2.0) * tan(k))));
	elseif (abs(t) <= 1.02e+101)
		tmp = 2.0 / ((((t_2 * (abs(t) * sin(k))) / l) * (sin(k) * (((k * k) / t_2) - -2.0))) / (l * cos(k)));
	else
		tmp = 2.0 / (((tan(k) * t_4) * (t_1 * abs(t))) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4495317912455029/12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(N[(2 * N[(N[(N[Power[N[Abs[t], $MachinePrecision], 2], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[k, 2], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5902958103587057/590295810358705651712], N[(2 / N[(t$95$1 * N[(N[(t$95$4 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / t$95$2), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696], N[(2 / N[(N[(N[(N[(t$95$2 * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / t$95$2), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.6000000000000001e-130

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 58.4%

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

   9. Applied rewrites 58.4%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. lower-cos.f64 N/A

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

       10. lower-/.f64 N/A

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

   12. Applied rewrites 66.4%

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

   if 3.6000000000000001e-130 < t < 1.0000000000000001e-5

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.5%

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

   7. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

   9. Applied rewrites 68.7%

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

   if 1.0000000000000001e-5 < t < 1.02e101

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. associate-*l/ N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 50.6%

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

   if 1.02e101 < t

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. frac-2neg N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. frac-times N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

   8. Applied rewrites 70.6%

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

Alternative 2: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ t_2 := \left|t\right| \cdot \left|t\right|\\ t_3 := \frac{\sin k \cdot \left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{4495317912455029}{12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot \left(\left|t\right| \cdot {\sin k}^{2}\right)}{\cos k}} \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq \frac{5902958103587057}{590295810358705651712}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\left(t\_1 \cdot \left|t\right|\right) \cdot \left(\left(\frac{k}{t\_2} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_2 \cdot \left(\left|t\right| \cdot \sin k\right)}{\ell} \cdot \left(\sin k \cdot \left(\frac{k \cdot k}{t\_2} - -2\right)\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot t\_1\right) \cdot \left(t\_3 \cdot \left|t\right|\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (fabs t) l))
       (t_2 (* (fabs t) (fabs t)))
       (t_3 (/ (* (sin k) (fabs t)) l)))
  (*
   (copysign 1 t)
   (if (<=
        (fabs t)
        4495317912455029/12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704)
     (*
      (/
       (* l l)
       (/ (* (pow k 2) (* (fabs t) (pow (sin k) 2))) (cos k)))
      2)
     (if (<= (fabs t) 5902958103587057/590295810358705651712)
       (/
        2
        (*
         t_3
         (* (* t_1 (fabs t)) (* (- (* (/ k t_2) k) -2) (tan k)))))
       (if (<=
            (fabs t)
            102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696)
         (/
          2
          (/
           (*
            (/ (* t_2 (* (fabs t) (sin k))) l)
            (* (sin k) (- (/ (* k k) t_2) -2)))
           (* l (cos k))))
         (/ 2 (* (* (* (tan k) t_1) (* t_3 (fabs t))) 2))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double t_2 = fabs(t) * fabs(t);
	double t_3 = (sin(k) * fabs(t)) / l;
	double tmp;
	if (fabs(t) <= 3.6e-130) {
		tmp = ((l * l) / ((pow(k, 2.0) * (fabs(t) * pow(sin(k), 2.0))) / cos(k))) * 2.0;
	} else if (fabs(t) <= 1e-5) {
		tmp = 2.0 / (t_3 * ((t_1 * fabs(t)) * ((((k / t_2) * k) - -2.0) * tan(k))));
	} else if (fabs(t) <= 1.02e+101) {
		tmp = 2.0 / ((((t_2 * (fabs(t) * sin(k))) / l) * (sin(k) * (((k * k) / t_2) - -2.0))) / (l * cos(k)));
	} else {
		tmp = 2.0 / (((tan(k) * t_1) * (t_3 * fabs(t))) * 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) / l;
	double t_2 = Math.abs(t) * Math.abs(t);
	double t_3 = (Math.sin(k) * Math.abs(t)) / l;
	double tmp;
	if (Math.abs(t) <= 3.6e-130) {
		tmp = ((l * l) / ((Math.pow(k, 2.0) * (Math.abs(t) * Math.pow(Math.sin(k), 2.0))) / Math.cos(k))) * 2.0;
	} else if (Math.abs(t) <= 1e-5) {
		tmp = 2.0 / (t_3 * ((t_1 * Math.abs(t)) * ((((k / t_2) * k) - -2.0) * Math.tan(k))));
	} else if (Math.abs(t) <= 1.02e+101) {
		tmp = 2.0 / ((((t_2 * (Math.abs(t) * Math.sin(k))) / l) * (Math.sin(k) * (((k * k) / t_2) - -2.0))) / (l * Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.tan(k) * t_1) * (t_3 * Math.abs(t))) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) / l
	t_2 = math.fabs(t) * math.fabs(t)
	t_3 = (math.sin(k) * math.fabs(t)) / l
	tmp = 0
	if math.fabs(t) <= 3.6e-130:
		tmp = ((l * l) / ((math.pow(k, 2.0) * (math.fabs(t) * math.pow(math.sin(k), 2.0))) / math.cos(k))) * 2.0
	elif math.fabs(t) <= 1e-5:
		tmp = 2.0 / (t_3 * ((t_1 * math.fabs(t)) * ((((k / t_2) * k) - -2.0) * math.tan(k))))
	elif math.fabs(t) <= 1.02e+101:
		tmp = 2.0 / ((((t_2 * (math.fabs(t) * math.sin(k))) / l) * (math.sin(k) * (((k * k) / t_2) - -2.0))) / (l * math.cos(k)))
	else:
		tmp = 2.0 / (((math.tan(k) * t_1) * (t_3 * math.fabs(t))) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	t_2 = Float64(abs(t) * abs(t))
	t_3 = Float64(Float64(sin(k) * abs(t)) / l)
	tmp = 0.0
	if (abs(t) <= 3.6e-130)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64((k ^ 2.0) * Float64(abs(t) * (sin(k) ^ 2.0))) / cos(k))) * 2.0);
	elseif (abs(t) <= 1e-5)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(t_1 * abs(t)) * Float64(Float64(Float64(Float64(k / t_2) * k) - -2.0) * tan(k)))));
	elseif (abs(t) <= 1.02e+101)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * Float64(abs(t) * sin(k))) / l) * Float64(sin(k) * Float64(Float64(Float64(k * k) / t_2) - -2.0))) / Float64(l * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * t_1) * Float64(t_3 * abs(t))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) / l;
	t_2 = abs(t) * abs(t);
	t_3 = (sin(k) * abs(t)) / l;
	tmp = 0.0;
	if (abs(t) <= 3.6e-130)
		tmp = ((l * l) / (((k ^ 2.0) * (abs(t) * (sin(k) ^ 2.0))) / cos(k))) * 2.0;
	elseif (abs(t) <= 1e-5)
		tmp = 2.0 / (t_3 * ((t_1 * abs(t)) * ((((k / t_2) * k) - -2.0) * tan(k))));
	elseif (abs(t) <= 1.02e+101)
		tmp = 2.0 / ((((t_2 * (abs(t) * sin(k))) / l) * (sin(k) * (((k * k) / t_2) - -2.0))) / (l * cos(k)));
	else
		tmp = 2.0 / (((tan(k) * t_1) * (t_3 * abs(t))) * 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] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4495317912455029/12486994201263968925526388919172665222994392570659884603436627838501486955279062480481224412253967884639307724485626491581791902717153141225160704], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[Power[k, 2], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5902958103587057/590295810358705651712], N[(2 / N[(t$95$3 * N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / t$95$2), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 102000000000000002010628671640657454961172628716945651504250908899455859726500368768078250896781213696], N[(2 / N[(N[(N[(N[(t$95$2 * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / t$95$2), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$3 * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.6000000000000001e-130

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 59.2%

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

   6. Applied rewrites 59.2%

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

   if 3.6000000000000001e-130 < t < 1.0000000000000001e-5

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.5%

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

   7. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

   9. Applied rewrites 68.7%

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

   if 1.0000000000000001e-5 < t < 1.02e101

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. associate-*l/ N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 50.6%

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

   if 1.02e101 < t

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. frac-2neg N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. frac-times N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

   8. Applied rewrites 70.6%

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

Alternative 3: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin \left(\left|k\right|\right) \cdot t}{\ell}\\ t_2 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq \frac{4704172149284445}{2475880078570760549798248448}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \frac{t}{\ell}\right) \cdot \left(t\_1 \cdot t\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\frac{\left|k\right|}{t \cdot t} \cdot \left|k\right| - -2\right) \cdot t\_2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (* (sin (fabs k)) t) l)) (t_2 (tan (fabs k))))
  (if (<= (fabs k) 4704172149284445/2475880078570760549798248448)
    (/ 2 (* (* (* t_2 (/ t l)) (* t_1 t)) 2))
    (/
     2
     (*
      t_1
      (*
       (* (/ t l) t)
       (* (- (* (/ (fabs k) (* t t)) (fabs k)) -2) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = (sin(fabs(k)) * t) / l;
	double t_2 = tan(fabs(k));
	double tmp;
	if (fabs(k) <= 1.9e-12) {
		tmp = 2.0 / (((t_2 * (t / l)) * (t_1 * t)) * 2.0);
	} else {
		tmp = 2.0 / (t_1 * (((t / l) * t) * ((((fabs(k) / (t * t)) * fabs(k)) - -2.0) * t_2)));
	}
	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) / l
    t_2 = tan(abs(k))
    if (abs(k) <= 1.9d-12) then
        tmp = 2.0d0 / (((t_2 * (t / l)) * (t_1 * t)) * 2.0d0)
    else
        tmp = 2.0d0 / (t_1 * (((t / l) * t) * ((((abs(k) / (t * t)) * abs(k)) - (-2.0d0)) * t_2)))
    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)) * t) / l;
	double t_2 = Math.tan(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.9e-12) {
		tmp = 2.0 / (((t_2 * (t / l)) * (t_1 * t)) * 2.0);
	} else {
		tmp = 2.0 / (t_1 * (((t / l) * t) * ((((Math.abs(k) / (t * t)) * Math.abs(k)) - -2.0) * t_2)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (math.sin(math.fabs(k)) * t) / l
	t_2 = math.tan(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.9e-12:
		tmp = 2.0 / (((t_2 * (t / l)) * (t_1 * t)) * 2.0)
	else:
		tmp = 2.0 / (t_1 * (((t / l) * t) * ((((math.fabs(k) / (t * t)) * math.fabs(k)) - -2.0) * t_2)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(sin(abs(k)) * t) / l)
	t_2 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.9e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t / l)) * Float64(t_1 * t)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(t / l) * t) * Float64(Float64(Float64(Float64(abs(k) / Float64(t * t)) * abs(k)) - -2.0) * t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (sin(abs(k)) * t) / l;
	t_2 = tan(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.9e-12)
		tmp = 2.0 / (((t_2 * (t / l)) * (t_1 * t)) * 2.0);
	else
		tmp = 2.0 / (t_1 * (((t / l) * t) * ((((abs(k) / (t * t)) * abs(k)) - -2.0) * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4704172149284445/2475880078570760549798248448], N[(2 / N[(N[(N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(2 / N[(t$95$1 * N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9e-12

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. frac-2neg N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. frac-times N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

   8. Applied rewrites 70.6%

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

   if 1.9e-12 < k

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.5%

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

   7. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

   9. Applied rewrites 68.7%

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

Alternative 4: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right) \cdot t\\ t_2 := \tan \left(\left|k\right|\right)\\ t_3 := \left(\frac{\left|k\right|}{t \cdot t} \cdot \left|k\right| - -2\right) \cdot t\_2\\ \mathbf{if}\;\left|k\right| \leq 4000000000000000:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t\_1}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|k\right| \leq 2450000000000000003809422588439219235392891374242316420580707544617560446523970637754472568887578728787168513894952296647753728:\\ \;\;\;\;\left(\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t\_3 \cdot t\_1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\left|k\right| \cdot t}{\ell} \cdot t\right) \cdot t\_3\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (sin (fabs k)) t))
       (t_2 (tan (fabs k)))
       (t_3 (* (- (* (/ (fabs k) (* t t)) (fabs k)) -2) t_2)))
  (if (<= (fabs k) 4000000000000000)
    (/ 2 (* (* (* t_2 (/ t l)) (* (/ t_1 l) t)) 2))
    (if (<=
         (fabs k)
         2450000000000000003809422588439219235392891374242316420580707544617560446523970637754472568887578728787168513894952296647753728)
      (* (* (/ l (* t t)) (/ l (* t_3 t_1))) 2)
      (/ 2 (* (/ t l) (* (* (/ (* (fabs k) t) l) t) t_3)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k)) * t;
	double t_2 = tan(fabs(k));
	double t_3 = (((fabs(k) / (t * t)) * fabs(k)) - -2.0) * t_2;
	double tmp;
	if (fabs(k) <= 4e+15) {
		tmp = 2.0 / (((t_2 * (t / l)) * ((t_1 / l) * t)) * 2.0);
	} else if (fabs(k) <= 2.45e+126) {
		tmp = ((l / (t * t)) * (l / (t_3 * t_1))) * 2.0;
	} else {
		tmp = 2.0 / ((t / l) * ((((fabs(k) * t) / l) * t) * t_3));
	}
	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 = sin(abs(k)) * t
    t_2 = tan(abs(k))
    t_3 = (((abs(k) / (t * t)) * abs(k)) - (-2.0d0)) * t_2
    if (abs(k) <= 4d+15) then
        tmp = 2.0d0 / (((t_2 * (t / l)) * ((t_1 / l) * t)) * 2.0d0)
    else if (abs(k) <= 2.45d+126) then
        tmp = ((l / (t * t)) * (l / (t_3 * t_1))) * 2.0d0
    else
        tmp = 2.0d0 / ((t / l) * ((((abs(k) * t) / l) * t) * t_3))
    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)) * t;
	double t_2 = Math.tan(Math.abs(k));
	double t_3 = (((Math.abs(k) / (t * t)) * Math.abs(k)) - -2.0) * t_2;
	double tmp;
	if (Math.abs(k) <= 4e+15) {
		tmp = 2.0 / (((t_2 * (t / l)) * ((t_1 / l) * t)) * 2.0);
	} else if (Math.abs(k) <= 2.45e+126) {
		tmp = ((l / (t * t)) * (l / (t_3 * t_1))) * 2.0;
	} else {
		tmp = 2.0 / ((t / l) * ((((Math.abs(k) * t) / l) * t) * t_3));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k)) * t
	t_2 = math.tan(math.fabs(k))
	t_3 = (((math.fabs(k) / (t * t)) * math.fabs(k)) - -2.0) * t_2
	tmp = 0
	if math.fabs(k) <= 4e+15:
		tmp = 2.0 / (((t_2 * (t / l)) * ((t_1 / l) * t)) * 2.0)
	elif math.fabs(k) <= 2.45e+126:
		tmp = ((l / (t * t)) * (l / (t_3 * t_1))) * 2.0
	else:
		tmp = 2.0 / ((t / l) * ((((math.fabs(k) * t) / l) * t) * t_3))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(abs(k)) * t)
	t_2 = tan(abs(k))
	t_3 = Float64(Float64(Float64(Float64(abs(k) / Float64(t * t)) * abs(k)) - -2.0) * t_2)
	tmp = 0.0
	if (abs(k) <= 4e+15)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t / l)) * Float64(Float64(t_1 / l) * t)) * 2.0));
	elseif (abs(k) <= 2.45e+126)
		tmp = Float64(Float64(Float64(l / Float64(t * t)) * Float64(l / Float64(t_3 * t_1))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(Float64(abs(k) * t) / l) * t) * t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k)) * t;
	t_2 = tan(abs(k));
	t_3 = (((abs(k) / (t * t)) * abs(k)) - -2.0) * t_2;
	tmp = 0.0;
	if (abs(k) <= 4e+15)
		tmp = 2.0 / (((t_2 * (t / l)) * ((t_1 / l) * t)) * 2.0);
	elseif (abs(k) <= 2.45e+126)
		tmp = ((l / (t * t)) * (l / (t_3 * t_1))) * 2.0;
	else
		tmp = 2.0 / ((t / l) * ((((abs(k) * t) / l) * t) * t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4000000000000000], N[(2 / N[(N[(N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2450000000000000003809422588439219235392891374242316420580707544617560446523970637754472568887578728787168513894952296647753728], N[(N[(N[(l / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(2 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4e15

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. frac-2neg N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. frac-times N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

   8. Applied rewrites 70.6%

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

   if 4e15 < k < 2.45e126

   1. Initial program 55.1%

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

      1. 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 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.6%

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

   5. Applied rewrites 57.7%

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

   if 2.45e126 < k

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 64.1%

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

Alternative 5: 76.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := \frac{\left|t\right|}{\ell}\\ t_3 := k \cdot \left|t\right|\\ t_4 := \frac{t\_3}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{2022342995529785}{297403381695556612559612499629980112026252040331878891811154371863188131432080874709033662899231270117959744758038594610090917049108981141558166116220478925156594168089491974788537281966859547374047839156470287441213549741375576017631419788069731616602409021090828782564753069762936832}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{2}{k + k}\\ \mathbf{elif}\;\left|t\right| \leq 4999999999999999727876154935214080:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_2 \cdot \left|t\right|\right) \cdot t\_4\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\frac{k}{\left|t\right|} \cdot k}{\left|t\right|}\right) + 1\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224:\\ \;\;\;\;\frac{\ell}{t\_3 \cdot t\_1} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(t\_4 \cdot \left|t\right|\right) \cdot \left(\left(\frac{k}{t\_1} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t)))
       (t_2 (/ (fabs t) l))
       (t_3 (* k (fabs t)))
       (t_4 (/ t_3 l)))
  (*
   (copysign 1 t)
   (if (<=
        (fabs t)
        2022342995529785/297403381695556612559612499629980112026252040331878891811154371863188131432080874709033662899231270117959744758038594610090917049108981141558166116220478925156594168089491974788537281966859547374047839156470287441213549741375576017631419788069731616602409021090828782564753069762936832)
     (*
      (/
       (* l l)
       (*
        (* (* (* k (+ 1 (* -1/6 (pow k 2)))) (fabs t)) (fabs t))
        (fabs t)))
      (/ 2 (+ k k)))
     (if (<= (fabs t) 4999999999999999727876154935214080)
       (/
        2
        (*
         (* (* (* t_2 (fabs t)) t_4) (tan k))
         (+ (+ 1 (/ (* (/ k (fabs t)) k) (fabs t))) 1)))
       (if (<=
            (fabs t)
            3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224)
         (* (/ l (* t_3 t_1)) (* (/ l (+ k k)) 2))
         (/
          2
          (*
           t_2
           (*
            (* t_4 (fabs t))
            (* (- (* (/ k t_1) k) -2) (tan k)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = fabs(t) / l;
	double t_3 = k * fabs(t);
	double t_4 = t_3 / l;
	double tmp;
	if (fabs(t) <= 6.8e-270) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * fabs(t)) * fabs(t)) * fabs(t))) * (2.0 / (k + k));
	} else if (fabs(t) <= 5e+33) {
		tmp = 2.0 / ((((t_2 * fabs(t)) * t_4) * tan(k)) * ((1.0 + (((k / fabs(t)) * k) / fabs(t))) + 1.0));
	} else if (fabs(t) <= 4e+99) {
		tmp = (l / (t_3 * t_1)) * ((l / (k + k)) * 2.0);
	} else {
		tmp = 2.0 / (t_2 * ((t_4 * fabs(t)) * ((((k / t_1) * k) - -2.0) * tan(k))));
	}
	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 t_2 = Math.abs(t) / l;
	double t_3 = k * Math.abs(t);
	double t_4 = t_3 / l;
	double tmp;
	if (Math.abs(t) <= 6.8e-270) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * Math.pow(k, 2.0)))) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * (2.0 / (k + k));
	} else if (Math.abs(t) <= 5e+33) {
		tmp = 2.0 / ((((t_2 * Math.abs(t)) * t_4) * Math.tan(k)) * ((1.0 + (((k / Math.abs(t)) * k) / Math.abs(t))) + 1.0));
	} else if (Math.abs(t) <= 4e+99) {
		tmp = (l / (t_3 * t_1)) * ((l / (k + k)) * 2.0);
	} else {
		tmp = 2.0 / (t_2 * ((t_4 * Math.abs(t)) * ((((k / t_1) * k) - -2.0) * Math.tan(k))));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * math.fabs(t)
	t_2 = math.fabs(t) / l
	t_3 = k * math.fabs(t)
	t_4 = t_3 / l
	tmp = 0
	if math.fabs(t) <= 6.8e-270:
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * math.pow(k, 2.0)))) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * (2.0 / (k + k))
	elif math.fabs(t) <= 5e+33:
		tmp = 2.0 / ((((t_2 * math.fabs(t)) * t_4) * math.tan(k)) * ((1.0 + (((k / math.fabs(t)) * k) / math.fabs(t))) + 1.0))
	elif math.fabs(t) <= 4e+99:
		tmp = (l / (t_3 * t_1)) * ((l / (k + k)) * 2.0)
	else:
		tmp = 2.0 / (t_2 * ((t_4 * math.fabs(t)) * ((((k / t_1) * k) - -2.0) * math.tan(k))))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(abs(t) / l)
	t_3 = Float64(k * abs(t))
	t_4 = Float64(t_3 / l)
	tmp = 0.0
	if (abs(t) <= 6.8e-270)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * Float64(2.0 / Float64(k + k)));
	elseif (abs(t) <= 5e+33)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * abs(t)) * t_4) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(Float64(k / abs(t)) * k) / abs(t))) + 1.0)));
	elseif (abs(t) <= 4e+99)
		tmp = Float64(Float64(l / Float64(t_3 * t_1)) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_4 * abs(t)) * Float64(Float64(Float64(Float64(k / t_1) * k) - -2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * abs(t);
	t_2 = abs(t) / l;
	t_3 = k * abs(t);
	t_4 = t_3 / l;
	tmp = 0.0;
	if (abs(t) <= 6.8e-270)
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * (2.0 / (k + k));
	elseif (abs(t) <= 5e+33)
		tmp = 2.0 / ((((t_2 * abs(t)) * t_4) * tan(k)) * ((1.0 + (((k / abs(t)) * k) / abs(t))) + 1.0));
	elseif (abs(t) <= 4e+99)
		tmp = (l / (t_3 * t_1)) * ((l / (k + k)) * 2.0);
	else
		tmp = 2.0 / (t_2 * ((t_4 * abs(t)) * ((((k / t_1) * k) - -2.0) * tan(k))));
	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]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2022342995529785/297403381695556612559612499629980112026252040331878891811154371863188131432080874709033662899231270117959744758038594610090917049108981141558166116220478925156594168089491974788537281966859547374047839156470287441213549741375576017631419788069731616602409021090828782564753069762936832], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(k * N[(1 + N[(-1/6 * N[Power[k, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2 / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4999999999999999727876154935214080], N[(2 / N[(N[(N[(N[(t$95$2 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[(N[(N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224], N[(N[(l / N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(2 / N[(t$95$2 * N[(N[(t$95$4 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / t$95$1), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 6.8000000000000001e-270

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {k}^{2}\right)}\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       4. lower-pow.f64 62.2%

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   if 6.8000000000000001e-270 < t < 4.9999999999999997e33

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 68.6%

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

   10. Applied rewrites 68.6%

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

   if 4.9999999999999997e33 < t < 3.9999999999999999e99

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 58.4%

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

   9. Applied rewrites 58.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 64.2%

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

   if 3.9999999999999999e99 < t

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 64.1%

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

Alternative 6: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 19600000000000000374956579330606504855576818647416687785348925165027140076453871189112209616714959517606064769747386368:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin \left(\left|k\right|\right) \cdot t}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\left|k\right| \cdot t}{\ell} \cdot t\right) \cdot \left(\left(\frac{\left|k\right|}{t \cdot t} \cdot \left|k\right| - -2\right) \cdot t\_1\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (tan (fabs k))))
  (if (<=
       (fabs k)
       19600000000000000374956579330606504855576818647416687785348925165027140076453871189112209616714959517606064769747386368)
    (/ 2 (* (* (* t_1 (/ t l)) (* (/ (* (sin (fabs k)) t) l) t)) 2))
    (/
     2
     (*
      (/ t l)
      (*
       (* (/ (* (fabs k) t) l) t)
       (* (- (* (/ (fabs k) (* t t)) (fabs k)) -2) t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k));
	double tmp;
	if (fabs(k) <= 1.96e+118) {
		tmp = 2.0 / (((t_1 * (t / l)) * (((sin(fabs(k)) * t) / l) * t)) * 2.0);
	} else {
		tmp = 2.0 / ((t / l) * ((((fabs(k) * t) / l) * t) * ((((fabs(k) / (t * t)) * fabs(k)) - -2.0) * 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 = tan(abs(k))
    if (abs(k) <= 1.96d+118) then
        tmp = 2.0d0 / (((t_1 * (t / l)) * (((sin(abs(k)) * t) / l) * t)) * 2.0d0)
    else
        tmp = 2.0d0 / ((t / l) * ((((abs(k) * t) / l) * t) * ((((abs(k) / (t * t)) * abs(k)) - (-2.0d0)) * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.96e+118) {
		tmp = 2.0 / (((t_1 * (t / l)) * (((Math.sin(Math.abs(k)) * t) / l) * t)) * 2.0);
	} else {
		tmp = 2.0 / ((t / l) * ((((Math.abs(k) * t) / l) * t) * ((((Math.abs(k) / (t * t)) * Math.abs(k)) - -2.0) * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.96e+118:
		tmp = 2.0 / (((t_1 * (t / l)) * (((math.sin(math.fabs(k)) * t) / l) * t)) * 2.0)
	else:
		tmp = 2.0 / ((t / l) * ((((math.fabs(k) * t) / l) * t) * ((((math.fabs(k) / (t * t)) * math.fabs(k)) - -2.0) * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.96e+118)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t / l)) * Float64(Float64(Float64(sin(abs(k)) * t) / l) * t)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(Float64(abs(k) * t) / l) * t) * Float64(Float64(Float64(Float64(abs(k) / Float64(t * t)) * abs(k)) - -2.0) * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.96e+118)
		tmp = 2.0 / (((t_1 * (t / l)) * (((sin(abs(k)) * t) / l) * t)) * 2.0);
	else
		tmp = 2.0 / ((t / l) * ((((abs(k) * t) / l) * t) * ((((abs(k) / (t * t)) * abs(k)) - -2.0) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 19600000000000000374956579330606504855576818647416687785348925165027140076453871189112209616714959517606064769747386368], N[(2 / N[(N[(N[(t$95$1 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.96e118

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. frac-2neg N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. frac-times N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. lift-*.f64 N/A

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

   8. Applied rewrites 70.6%

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

   if 1.96e118 < k

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 64.1%

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

Alternative 7: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := \frac{t\_1}{\ell}\\ t_3 := \frac{\left|t\right|}{\ell}\\ t_4 := \left|t\right| \cdot \left|t\right|\\ t_5 := \left(\frac{k}{t\_4} \cdot k - -2\right) \cdot \tan k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{3455165794209175}{7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{2}{k + k}\\ \mathbf{elif}\;\left|t\right| \leq 4999999999999999727876154935214080:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(t\_3 \cdot \left|t\right|\right) \cdot t\_5\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224:\\ \;\;\;\;\frac{\ell}{t\_1 \cdot t\_4} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\left(t\_2 \cdot \left|t\right|\right) \cdot t\_5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t)))
       (t_2 (/ t_1 l))
       (t_3 (/ (fabs t) l))
       (t_4 (* (fabs t) (fabs t)))
       (t_5 (* (- (* (/ k t_4) k) -2) (tan k))))
  (*
   (copysign 1 t)
   (if (<=
        (fabs t)
        3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552)
     (*
      (/
       (* l l)
       (*
        (* (* (* k (+ 1 (* -1/6 (pow k 2)))) (fabs t)) (fabs t))
        (fabs t)))
      (/ 2 (+ k k)))
     (if (<= (fabs t) 4999999999999999727876154935214080)
       (/ 2 (* t_2 (* (* t_3 (fabs t)) t_5)))
       (if (<=
            (fabs t)
            3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224)
         (* (/ l (* t_1 t_4)) (* (/ l (+ k k)) 2))
         (/ 2 (* t_3 (* (* t_2 (fabs t)) t_5)))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = t_1 / l;
	double t_3 = fabs(t) / l;
	double t_4 = fabs(t) * fabs(t);
	double t_5 = (((k / t_4) * k) - -2.0) * tan(k);
	double tmp;
	if (fabs(t) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * fabs(t)) * fabs(t)) * fabs(t))) * (2.0 / (k + k));
	} else if (fabs(t) <= 5e+33) {
		tmp = 2.0 / (t_2 * ((t_3 * fabs(t)) * t_5));
	} else if (fabs(t) <= 4e+99) {
		tmp = (l / (t_1 * t_4)) * ((l / (k + k)) * 2.0);
	} else {
		tmp = 2.0 / (t_3 * ((t_2 * fabs(t)) * t_5));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double t_2 = t_1 / l;
	double t_3 = Math.abs(t) / l;
	double t_4 = Math.abs(t) * Math.abs(t);
	double t_5 = (((k / t_4) * k) - -2.0) * Math.tan(k);
	double tmp;
	if (Math.abs(t) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * Math.pow(k, 2.0)))) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * (2.0 / (k + k));
	} else if (Math.abs(t) <= 5e+33) {
		tmp = 2.0 / (t_2 * ((t_3 * Math.abs(t)) * t_5));
	} else if (Math.abs(t) <= 4e+99) {
		tmp = (l / (t_1 * t_4)) * ((l / (k + k)) * 2.0);
	} else {
		tmp = 2.0 / (t_3 * ((t_2 * Math.abs(t)) * t_5));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	t_2 = t_1 / l
	t_3 = math.fabs(t) / l
	t_4 = math.fabs(t) * math.fabs(t)
	t_5 = (((k / t_4) * k) - -2.0) * math.tan(k)
	tmp = 0
	if math.fabs(t) <= 4.8e-148:
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * math.pow(k, 2.0)))) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * (2.0 / (k + k))
	elif math.fabs(t) <= 5e+33:
		tmp = 2.0 / (t_2 * ((t_3 * math.fabs(t)) * t_5))
	elif math.fabs(t) <= 4e+99:
		tmp = (l / (t_1 * t_4)) * ((l / (k + k)) * 2.0)
	else:
		tmp = 2.0 / (t_3 * ((t_2 * math.fabs(t)) * t_5))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(t_1 / l)
	t_3 = Float64(abs(t) / l)
	t_4 = Float64(abs(t) * abs(t))
	t_5 = Float64(Float64(Float64(Float64(k / t_4) * k) - -2.0) * tan(k))
	tmp = 0.0
	if (abs(t) <= 4.8e-148)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * Float64(2.0 / Float64(k + k)));
	elseif (abs(t) <= 5e+33)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_3 * abs(t)) * t_5)));
	elseif (abs(t) <= 4e+99)
		tmp = Float64(Float64(l / Float64(t_1 * t_4)) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(t_2 * abs(t)) * t_5)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	t_2 = t_1 / l;
	t_3 = abs(t) / l;
	t_4 = abs(t) * abs(t);
	t_5 = (((k / t_4) * k) - -2.0) * tan(k);
	tmp = 0.0;
	if (abs(t) <= 4.8e-148)
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * (2.0 / (k + k));
	elseif (abs(t) <= 5e+33)
		tmp = 2.0 / (t_2 * ((t_3 * abs(t)) * t_5));
	elseif (abs(t) <= 4e+99)
		tmp = (l / (t_1 * t_4)) * ((l / (k + k)) * 2.0);
	else
		tmp = 2.0 / (t_3 * ((t_2 * abs(t)) * t_5));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(k / t$95$4), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(k * N[(1 + N[(-1/6 * N[Power[k, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2 / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4999999999999999727876154935214080], N[(2 / N[(t$95$2 * N[(N[(t$95$3 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3999999999999999869344675216466765095398132743226221891671847117885183383686911450434959473821876224], N[(N[(l / N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(2 / N[(t$95$3 * N[(N[(t$95$2 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.8000000000000002e-148

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {k}^{2}\right)}\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       4. lower-pow.f64 62.2%

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   if 4.8000000000000002e-148 < t < 4.9999999999999997e33

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 69.6%

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

   10. Applied rewrites 61.3%

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

   if 4.9999999999999997e33 < t < 3.9999999999999999e99

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 58.4%

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

   9. Applied rewrites 58.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 64.2%

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

   if 3.9999999999999999e99 < t

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 64.1%

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

Alternative 8: 73.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 2349999999999999944735847609781977088:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_1 \cdot t\right) \cdot \frac{k \cdot t}{\left|\ell\right|}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\frac{k}{t} \cdot k}{t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(t\_1 \cdot \frac{\sin k \cdot t}{\left|\ell\right|}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ t (fabs l))))
  (if (<= (fabs l) 2349999999999999944735847609781977088)
    (/
     2
     (*
      (* (* (* t_1 t) (/ (* k t) (fabs l))) (tan k))
      (+ (+ 1 (/ (* (/ k t) k) t)) 1)))
    (/ 2 (* (* (* t (* t_1 (/ (* (sin k) t) (fabs l)))) (tan k)) 2)))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 2.35e+36) {
		tmp = 2.0 / ((((t_1 * t) * ((k * t) / fabs(l))) * tan(k)) * ((1.0 + (((k / t) * k) / t)) + 1.0));
	} else {
		tmp = 2.0 / (((t * (t_1 * ((sin(k) * t) / fabs(l)))) * tan(k)) * 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 = t / abs(l)
    if (abs(l) <= 2.35d+36) then
        tmp = 2.0d0 / ((((t_1 * t) * ((k * t) / abs(l))) * tan(k)) * ((1.0d0 + (((k / t) * k) / t)) + 1.0d0))
    else
        tmp = 2.0d0 / (((t * (t_1 * ((sin(k) * t) / abs(l)))) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 2.35e+36) {
		tmp = 2.0 / ((((t_1 * t) * ((k * t) / Math.abs(l))) * Math.tan(k)) * ((1.0 + (((k / t) * k) / t)) + 1.0));
	} else {
		tmp = 2.0 / (((t * (t_1 * ((Math.sin(k) * t) / Math.abs(l)))) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 2.35e+36:
		tmp = 2.0 / ((((t_1 * t) * ((k * t) / math.fabs(l))) * math.tan(k)) * ((1.0 + (((k / t) * k) / t)) + 1.0))
	else:
		tmp = 2.0 / (((t * (t_1 * ((math.sin(k) * t) / math.fabs(l)))) * math.tan(k)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 2.35e+36)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_1 * t) * Float64(Float64(k * t) / abs(l))) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(Float64(k / t) * k) / t)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(t_1 * Float64(Float64(sin(k) * t) / abs(l)))) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 2.35e+36)
		tmp = 2.0 / ((((t_1 * t) * ((k * t) / abs(l))) * tan(k)) * ((1.0 + (((k / t) * k) / t)) + 1.0));
	else
		tmp = 2.0 / (((t * (t_1 * ((sin(k) * t) / abs(l)))) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2349999999999999944735847609781977088], N[(2 / N[(N[(N[(N[(t$95$1 * t), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1 + N[(N[(N[(k / t), $MachinePrecision] * k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(t * N[(t$95$1 * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3499999999999999e36

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 68.6%

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

   10. Applied rewrites 68.6%

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

   if 2.3499999999999999e36 < l

   1. Initial program 55.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 68.4%

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

   6. Applied rewrites 68.4%

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

Alternative 9: 72.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell} \cdot \left|t\right|\\ t_2 := \frac{k \cdot \left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{3455165794209175}{7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{2}{k + k}\\ \mathbf{elif}\;\left|t\right| \leq \frac{6965490562232727}{1180591620717411303424}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\left|t\right| \cdot \left|t\right|} \cdot k - -2\right) \cdot t\_1\right) \cdot \left(t\_2 \cdot \tan k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 9999999999999999338604948347429745623719502164303315186116928223077006466996036476256924325958459471709145545996985214755393808134448127932794585054037286174943850004480:\\ \;\;\;\;\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{\ell \cdot 2}{k + k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot t\_2\right) \cdot \tan k\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (/ (fabs t) l) (fabs t))) (t_2 (/ (* k (fabs t)) l)))
  (*
   (copysign 1 t)
   (if (<=
        (fabs t)
        3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552)
     (*
      (/
       (* l l)
       (*
        (* (* (* k (+ 1 (* -1/6 (pow k 2)))) (fabs t)) (fabs t))
        (fabs t)))
      (/ 2 (+ k k)))
     (if (<= (fabs t) 6965490562232727/1180591620717411303424)
       (/
        2
        (*
         (* (- (* (/ k (* (fabs t) (fabs t))) k) -2) t_1)
         (* t_2 (tan k))))
       (if (<=
            (fabs t)
            9999999999999999338604948347429745623719502164303315186116928223077006466996036476256924325958459471709145545996985214755393808134448127932794585054037286174943850004480)
         (*
          (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t)))
          (/ (* l 2) (+ k k)))
         (/ 2 (* (* (* t_1 t_2) (tan k)) (+ 1 1)))))))))

C

double code(double t, double l, double k) {
	double t_1 = (fabs(t) / l) * fabs(t);
	double t_2 = (k * fabs(t)) / l;
	double tmp;
	if (fabs(t) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * fabs(t)) * fabs(t)) * fabs(t))) * (2.0 / (k + k));
	} else if (fabs(t) <= 5.9e-6) {
		tmp = 2.0 / (((((k / (fabs(t) * fabs(t))) * k) - -2.0) * t_1) * (t_2 * tan(k)));
	} else if (fabs(t) <= 1e+169) {
		tmp = (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * ((l * 2.0) / (k + k));
	} else {
		tmp = 2.0 / (((t_1 * t_2) * tan(k)) * (1.0 + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = (Math.abs(t) / l) * Math.abs(t);
	double t_2 = (k * Math.abs(t)) / l;
	double tmp;
	if (Math.abs(t) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * Math.pow(k, 2.0)))) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * (2.0 / (k + k));
	} else if (Math.abs(t) <= 5.9e-6) {
		tmp = 2.0 / (((((k / (Math.abs(t) * Math.abs(t))) * k) - -2.0) * t_1) * (t_2 * Math.tan(k)));
	} else if (Math.abs(t) <= 1e+169) {
		tmp = (l / (((Math.sin(k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * ((l * 2.0) / (k + k));
	} else {
		tmp = 2.0 / (((t_1 * t_2) * Math.tan(k)) * (1.0 + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = (math.fabs(t) / l) * math.fabs(t)
	t_2 = (k * math.fabs(t)) / l
	tmp = 0
	if math.fabs(t) <= 4.8e-148:
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * math.pow(k, 2.0)))) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * (2.0 / (k + k))
	elif math.fabs(t) <= 5.9e-6:
		tmp = 2.0 / (((((k / (math.fabs(t) * math.fabs(t))) * k) - -2.0) * t_1) * (t_2 * math.tan(k)))
	elif math.fabs(t) <= 1e+169:
		tmp = (l / (((math.sin(k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * ((l * 2.0) / (k + k))
	else:
		tmp = 2.0 / (((t_1 * t_2) * math.tan(k)) * (1.0 + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(abs(t) / l) * abs(t))
	t_2 = Float64(Float64(k * abs(t)) / l)
	tmp = 0.0
	if (abs(t) <= 4.8e-148)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * Float64(2.0 / Float64(k + k)));
	elseif (abs(t) <= 5.9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / Float64(abs(t) * abs(t))) * k) - -2.0) * t_1) * Float64(t_2 * tan(k))));
	elseif (abs(t) <= 1e+169)
		tmp = Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * Float64(Float64(l * 2.0) / Float64(k + k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * t_2) * tan(k)) * Float64(1.0 + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (abs(t) / l) * abs(t);
	t_2 = (k * abs(t)) / l;
	tmp = 0.0;
	if (abs(t) <= 4.8e-148)
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * (2.0 / (k + k));
	elseif (abs(t) <= 5.9e-6)
		tmp = 2.0 / (((((k / (abs(t) * abs(t))) * k) - -2.0) * t_1) * (t_2 * tan(k)));
	elseif (abs(t) <= 1e+169)
		tmp = (l / (((sin(k) * abs(t)) * abs(t)) * abs(t))) * ((l * 2.0) / (k + k));
	else
		tmp = 2.0 / (((t_1 * t_2) * tan(k)) * (1.0 + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(k * N[(1 + N[(-1/6 * N[Power[k, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2 / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6965490562232727/1180591620717411303424], N[(2 / N[(N[(N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 9999999999999999338604948347429745623719502164303315186116928223077006466996036476256924325958459471709145545996985214755393808134448127932794585054037286174943850004480], N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2), $MachinePrecision] / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1 + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.8000000000000002e-148

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {k}^{2}\right)}\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       4. lower-pow.f64 62.2%

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   if 4.8000000000000002e-148 < t < 5.9000000000000003e-6

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 59.1%

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

   if 5.9000000000000003e-6 < t < 9.9999999999999993e168

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 62.1%

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

   if 9.9999999999999993e168 < t

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 65.1%

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

Alternative 10: 70.5% accurate, 1.5× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{3455165794209175}{7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{2}{k + k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\left(\frac{k}{\left|t\right| \cdot \left|t\right|} \cdot k - -2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1 t)
 (if (<=
      (fabs t)
      3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552)
   (*
    (/
     (* l l)
     (*
      (* (* (* k (+ 1 (* -1/6 (pow k 2)))) (fabs t)) (fabs t))
      (fabs t)))
    (/ 2 (+ k k)))
   (/
    2
    (*
     (/ (fabs t) l)
     (*
      (* (/ (* k (fabs t)) l) (fabs t))
      (* (- (* (/ k (* (fabs t) (fabs t))) k) -2) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * fabs(t)) * fabs(t)) * fabs(t))) * (2.0 / (k + k));
	} else {
		tmp = 2.0 / ((fabs(t) / l) * ((((k * fabs(t)) / l) * fabs(t)) * ((((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) <= 4.8e-148) {
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * Math.pow(k, 2.0)))) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * (2.0 / (k + k));
	} else {
		tmp = 2.0 / ((Math.abs(t) / l) * ((((k * Math.abs(t)) / l) * Math.abs(t)) * ((((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) <= 4.8e-148:
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * math.pow(k, 2.0)))) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * (2.0 / (k + k))
	else:
		tmp = 2.0 / ((math.fabs(t) / l) * ((((k * math.fabs(t)) / l) * math.fabs(t)) * ((((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) <= 4.8e-148)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * Float64(2.0 / Float64(k + k)));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(k * abs(t)) / l) * abs(t)) * 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) <= 4.8e-148)
		tmp = ((l * l) / ((((k * (1.0 + (-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * (2.0 / (k + k));
	else
		tmp = 2.0 / ((abs(t) / l) * ((((k * abs(t)) / l) * abs(t)) * ((((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], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3455165794209175/7198262071269114212496861612297570974191515389283066612961208916178940129074380592510465097766225371439873457013633432197133225688790879502413624289384262168215552], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(k * N[(1 + N[(-1/6 * N[Power[k, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2 / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] - -2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000002e-148

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {k}^{2}\right)}\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       4. lower-pow.f64 62.2%

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   if 4.8000000000000002e-148 < t

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 68.6%

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 64.1%

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

Alternative 11: 69.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq \frac{3334007216439927}{1667003608219963568519962947680314449286189580578977040099064452941009309454408017880358050217888572685732477648358310111472200413529841270090839013082707511523789394878503639615769571477953506182241254033971650495422687009369115322790969344}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t}\right) \cdot 2}{t \cdot t\_1}\\ \mathbf{elif}\;\left|k\right| \leq 28000000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\left|k\right| \cdot t}{\ell}\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot \left(1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(\left|k\right| \cdot \left(1 + \frac{-1}{6} \cdot {\left(\left|k\right|\right)}^{2}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{t\_1}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k))))
  (if (<=
       (fabs k)
       3334007216439927/1667003608219963568519962947680314449286189580578977040099064452941009309454408017880358050217888572685732477648358310111472200413529841270090839013082707511523789394878503639615769571477953506182241254033971650495422687009369115322790969344)
    (/ (* (* l (/ l (* (* (sin (fabs k)) t) t))) 2) (* t t_1))
    (if (<= (fabs k) 28000000000000000)
      (/
       2
       (*
        (* (* (* (/ t l) t) (/ (* (fabs k) t) l)) (tan (fabs k)))
        (+ 1 1)))
      (*
       (/
        (* l l)
        (*
         (* (* (* (fabs k) (+ 1 (* -1/6 (pow (fabs k) 2)))) t) t)
         t))
       (/ 2 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 2e-225) {
		tmp = ((l * (l / ((sin(fabs(k)) * t) * t))) * 2.0) / (t * t_1);
	} else if (fabs(k) <= 2.8e+16) {
		tmp = 2.0 / (((((t / l) * t) * ((fabs(k) * t) / l)) * tan(fabs(k))) * (1.0 + 1.0));
	} else {
		tmp = ((l * l) / ((((fabs(k) * (1.0 + (-0.16666666666666666 * pow(fabs(k), 2.0)))) * t) * t) * t)) * (2.0 / 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) <= 2d-225) then
        tmp = ((l * (l / ((sin(abs(k)) * t) * t))) * 2.0d0) / (t * t_1)
    else if (abs(k) <= 2.8d+16) then
        tmp = 2.0d0 / (((((t / l) * t) * ((abs(k) * t) / l)) * tan(abs(k))) * (1.0d0 + 1.0d0))
    else
        tmp = ((l * l) / ((((abs(k) * (1.0d0 + ((-0.16666666666666666d0) * (abs(k) ** 2.0d0)))) * t) * t) * t)) * (2.0d0 / 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) <= 2e-225) {
		tmp = ((l * (l / ((Math.sin(Math.abs(k)) * t) * t))) * 2.0) / (t * t_1);
	} else if (Math.abs(k) <= 2.8e+16) {
		tmp = 2.0 / (((((t / l) * t) * ((Math.abs(k) * t) / l)) * Math.tan(Math.abs(k))) * (1.0 + 1.0));
	} else {
		tmp = ((l * l) / ((((Math.abs(k) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(k), 2.0)))) * t) * t) * t)) * (2.0 / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 2e-225:
		tmp = ((l * (l / ((math.sin(math.fabs(k)) * t) * t))) * 2.0) / (t * t_1)
	elif math.fabs(k) <= 2.8e+16:
		tmp = 2.0 / (((((t / l) * t) * ((math.fabs(k) * t) / l)) * math.tan(math.fabs(k))) * (1.0 + 1.0))
	else:
		tmp = ((l * l) / ((((math.fabs(k) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(k), 2.0)))) * t) * t) * t)) * (2.0 / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 2e-225)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(Float64(sin(abs(k)) * t) * t))) * 2.0) / Float64(t * t_1));
	elseif (abs(k) <= 2.8e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * t) * Float64(Float64(abs(k) * t) / l)) * tan(abs(k))) * Float64(1.0 + 1.0)));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Float64(abs(k) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(k) ^ 2.0)))) * t) * t) * t)) * Float64(2.0 / 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) <= 2e-225)
		tmp = ((l * (l / ((sin(abs(k)) * t) * t))) * 2.0) / (t * t_1);
	elseif (abs(k) <= 2.8e+16)
		tmp = 2.0 / (((((t / l) * t) * ((abs(k) * t) / l)) * tan(abs(k))) * (1.0 + 1.0));
	else
		tmp = ((l * l) / ((((abs(k) * (1.0 + (-0.16666666666666666 * (abs(k) ^ 2.0)))) * t) * t) * t)) * (2.0 / 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], 3334007216439927/1667003608219963568519962947680314449286189580578977040099064452941009309454408017880358050217888572685732477648358310111472200413529841270090839013082707511523789394878503639615769571477953506182241254033971650495422687009369115322790969344], N[(N[(N[(l * N[(l / N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 28000000000000000], N[(2 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1 + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * N[(1 + N[(-1/6 * N[Power[N[Abs[k], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(2 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999999e-225

   1. Initial program 55.1%

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

      1. 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 46.1%

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

   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 55.8%

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

   6. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 65.7%

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

   if 1.9999999999999999e-225 < k < 2.8e16

   1. Initial program 55.1%

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

      1. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.0%

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 75.0%

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot t\right) \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 68.6%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot \color{blue}{t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.6%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.1%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]

   if 2.8e16 < k

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

   8. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {k}^{2}\right)}\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{k}^{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

       4. lower-pow.f64 62.2%

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq \frac{5159126280621731}{515912628062173092140956821207535748553561841832149923953086629908861232965551620580485601452790222553392963860602664725471042538576841344971229471155214430574596371092778402508526872730885196340843977449424988675925808879640197010584371971452184059071815020407947264}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 999999999999999965084388885482519417592855130626093842171043595190833186399051537317196816706799625297221478016185520727674168639944850288849622355474122345476546392575499689981548348018063279122228410984187505225498624:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot t\right) \cdot t}\right) \cdot 2}{t \cdot \left(k + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(1 + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (* l l)
     5159126280621731/515912628062173092140956821207535748553561841832149923953086629908861232965551620580485601452790222553392963860602664725471042538576841344971229471155214430574596371092778402508526872730885196340843977449424988675925808879640197010584371971452184059071815020407947264)
  (* (/ l (* (* k t) (* t t))) (* (/ l (+ k k)) 2))
  (if (<=
       (* l l)
       999999999999999965084388885482519417592855130626093842171043595190833186399051537317196816706799625297221478016185520727674168639944850288849622355474122345476546392575499689981548348018063279122228410984187505225498624)
    (/ (* (* l (/ l (* (* (sin k) t) t))) 2) (* t (+ k k)))
    (/ 2 (* (* (* (* (/ t l) t) (/ (* k t) l)) (tan k)) (+ 1 1))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-251) {
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	} else if ((l * l) <= 1e+219) {
		tmp = ((l * (l / ((sin(k) * t) * t))) * 2.0) / (t * (k + k));
	} else {
		tmp = 2.0 / (((((t / l) * t) * ((k * t) / l)) * tan(k)) * (1.0 + 1.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-251) then
        tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0d0)
    else if ((l * l) <= 1d+219) then
        tmp = ((l * (l / ((sin(k) * t) * t))) * 2.0d0) / (t * (k + k))
    else
        tmp = 2.0d0 / (((((t / l) * t) * ((k * t) / l)) * tan(k)) * (1.0d0 + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-251) {
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	} else if ((l * l) <= 1e+219) {
		tmp = ((l * (l / ((Math.sin(k) * t) * t))) * 2.0) / (t * (k + k));
	} else {
		tmp = 2.0 / (((((t / l) * t) * ((k * t) / l)) * Math.tan(k)) * (1.0 + 1.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-251:
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0)
	elif (l * l) <= 1e+219:
		tmp = ((l * (l / ((math.sin(k) * t) * t))) * 2.0) / (t * (k + k))
	else:
		tmp = 2.0 / (((((t / l) * t) * ((k * t) / l)) * math.tan(k)) * (1.0 + 1.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-251)
		tmp = Float64(Float64(l / Float64(Float64(k * t) * Float64(t * t))) * Float64(Float64(l / Float64(k + k)) * 2.0));
	elseif (Float64(l * l) <= 1e+219)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(Float64(sin(k) * t) * t))) * 2.0) / Float64(t * Float64(k + k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * t) * Float64(Float64(k * t) / l)) * tan(k)) * Float64(1.0 + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-251)
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	elseif ((l * l) <= 1e+219)
		tmp = ((l * (l / ((sin(k) * t) * t))) * 2.0) / (t * (k + k));
	else
		tmp = 2.0 / (((((t / l) * t) * ((k * t) / l)) * tan(k)) * (1.0 + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5159126280621731/515912628062173092140956821207535748553561841832149923953086629908861232965551620580485601452790222553392963860602664725471042538576841344971229471155214430574596371092778402508526872730885196340843977449424988675925808879640197010584371971452184059071815020407947264], N[(N[(l / N[(N[(k * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 999999999999999965084388885482519417592855130626093842171043595190833186399051537317196816706799625297221478016185520727674168639944850288849622355474122345476546392575499689981548348018063279122228410984187505225498624], N[(N[(N[(l * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision] / N[(t * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1 + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1e-251

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 64.2%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 1e-251 < (*.f64 l l) < 9.9999999999999997e218

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

   8. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t}} \cdot \frac{2}{k + k} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t} \cdot \color{blue}{\frac{2}{k + k}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

   10. Applied rewrites 65.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot t\right) \cdot t}\right) \cdot 2}{t \cdot \left(k + k\right)}} \]

   if 9.9999999999999997e218 < (*.f64 l l)

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.0%

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 75.0%

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot t\right) \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 68.6%

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot \color{blue}{t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.6%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.1%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.3% accurate, 1.6× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq \frac{8742010015015781}{971334446112864535459730953411759453321203419526069760625906204869452142602604249088}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{\left|t\right|}{\ell \cdot \ell} \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \sin k} \cdot \frac{2}{k + k}\\ \mathbf{elif}\;\left|t\right| \leq 19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \left|t\right|\right) \cdot \left(\left|t\right| \cdot \left|t\right|\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|}\right) \cdot 2}{\left|t\right| \cdot \left(k + k\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1 t)
 (if (<=
      (fabs t)
      8742010015015781/971334446112864535459730953411759453321203419526069760625906204869452142602604249088)
   (*
    (/ 1 (* (* (* (/ (fabs t) (* l l)) (fabs t)) (fabs t)) (sin k)))
    (/ 2 (+ k k)))
   (if (<=
        (fabs t)
        19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816)
     (*
      (/ l (* (* k (fabs t)) (* (fabs t) (fabs t))))
      (* (/ l (+ k k)) 2))
     (/
      (* (* l (/ l (* (* (sin k) (fabs t)) (fabs t)))) 2)
      (* (fabs t) (+ k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 9e-69) {
		tmp = (1.0 / ((((fabs(t) / (l * l)) * fabs(t)) * fabs(t)) * sin(k))) * (2.0 / (k + k));
	} else if (fabs(t) <= 1.95e+106) {
		tmp = (l / ((k * fabs(t)) * (fabs(t) * fabs(t)))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l * (l / ((sin(k) * fabs(t)) * fabs(t)))) * 2.0) / (fabs(t) * (k + k));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 9e-69) {
		tmp = (1.0 / ((((Math.abs(t) / (l * l)) * Math.abs(t)) * Math.abs(t)) * Math.sin(k))) * (2.0 / (k + k));
	} else if (Math.abs(t) <= 1.95e+106) {
		tmp = (l / ((k * Math.abs(t)) * (Math.abs(t) * Math.abs(t)))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l * (l / ((Math.sin(k) * Math.abs(t)) * Math.abs(t)))) * 2.0) / (Math.abs(t) * (k + k));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 9e-69:
		tmp = (1.0 / ((((math.fabs(t) / (l * l)) * math.fabs(t)) * math.fabs(t)) * math.sin(k))) * (2.0 / (k + k))
	elif math.fabs(t) <= 1.95e+106:
		tmp = (l / ((k * math.fabs(t)) * (math.fabs(t) * math.fabs(t)))) * ((l / (k + k)) * 2.0)
	else:
		tmp = ((l * (l / ((math.sin(k) * math.fabs(t)) * math.fabs(t)))) * 2.0) / (math.fabs(t) * (k + k))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 9e-69)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(Float64(abs(t) / Float64(l * l)) * abs(t)) * abs(t)) * sin(k))) * Float64(2.0 / Float64(k + k)));
	elseif (abs(t) <= 1.95e+106)
		tmp = Float64(Float64(l / Float64(Float64(k * abs(t)) * Float64(abs(t) * abs(t)))) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(Float64(sin(k) * abs(t)) * abs(t)))) * 2.0) / Float64(abs(t) * Float64(k + k)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 9e-69)
		tmp = (1.0 / ((((abs(t) / (l * l)) * abs(t)) * abs(t)) * sin(k))) * (2.0 / (k + k));
	elseif (abs(t) <= 1.95e+106)
		tmp = (l / ((k * abs(t)) * (abs(t) * abs(t)))) * ((l / (k + k)) * 2.0);
	else
		tmp = ((l * (l / ((sin(k) * abs(t)) * abs(t)))) * 2.0) / (abs(t) * (k + k));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 8742010015015781/971334446112864535459730953411759453321203419526069760625906204869452142602604249088], N[(N[(1 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2 / N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816], N[(N[(l / N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.0000000000000002e-69

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

   8. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}{\ell \cdot \ell}}} \cdot \frac{2}{k + k} \]

      3. lower-unsound-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}{\ell \cdot \ell}}} \cdot \frac{2}{k + k} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}{\ell \cdot \ell}}} \cdot \frac{2}{k + k} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot t}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sin k \cdot t\right) \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sin k \cdot t\right)} \cdot \left(t \cdot t\right)}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \sin k\right)} \cdot \left(t \cdot t\right)}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \sin k\right)} \cdot \left(t \cdot t\right)}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(t \cdot \sin k\right) \cdot \color{blue}{\left(t \cdot t\right)}}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{1}{\color{blue}{\left(t \cdot \sin k\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}}} \cdot \frac{2}{k + k} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(t \cdot \sin k\right) \cdot \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}} \cdot \frac{2}{k + k} \]

      14. associate-*r/ N/A

          \[\leadsto \frac{1}{\left(t \cdot \sin k\right) \cdot \color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)}} \cdot \frac{2}{k + k} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(t \cdot \sin k\right) \cdot \left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \cdot \frac{2}{k + k} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(t \cdot \sin k\right)} \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \frac{2}{k + k} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\sin k \cdot t\right)} \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \frac{2}{k + k} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\sin k \cdot t\right) \cdot \color{blue}{\left(t \cdot \frac{t}{\ell \cdot \ell}\right)}} \cdot \frac{2}{k + k} \]

      19. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\sin k \cdot \left(t \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \cdot \frac{2}{k + k} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sin k \cdot \color{blue}{\left(t \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \cdot \frac{2}{k + k} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\sin k \cdot \left(t \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \cdot \frac{2}{k + k} \]

   10. Applied rewrites 59.2%

       \[\leadsto \color{blue}{\frac{1}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot t\right) \cdot \sin k}} \cdot \frac{2}{k + k} \]

   if 9.0000000000000002e-69 < t < 1.9499999999999998e106

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 64.2%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 1.9499999999999998e106 < t

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

   8. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t}} \cdot \frac{2}{k + k} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t} \cdot \color{blue}{\frac{2}{k + k}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

   10. Applied rewrites 65.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot t\right) \cdot t}\right) \cdot 2}{t \cdot \left(k + k\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.2% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \left|t\right|\right) \cdot \left(\left|t\right| \cdot \left|t\right|\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|}\right) \cdot 2}{\left|t\right| \cdot \left(k + k\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1 t)
 (if (<=
      (fabs t)
      19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816)
   (*
    (/ l (* (* k (fabs t)) (* (fabs t) (fabs t))))
    (* (/ l (+ k k)) 2))
   (/
    (* (* l (/ l (* (* (sin k) (fabs t)) (fabs t)))) 2)
    (* (fabs t) (+ k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.95e+106) {
		tmp = (l / ((k * fabs(t)) * (fabs(t) * fabs(t)))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l * (l / ((sin(k) * fabs(t)) * fabs(t)))) * 2.0) / (fabs(t) * (k + k));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.95e+106) {
		tmp = (l / ((k * Math.abs(t)) * (Math.abs(t) * Math.abs(t)))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = ((l * (l / ((Math.sin(k) * Math.abs(t)) * Math.abs(t)))) * 2.0) / (Math.abs(t) * (k + k));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.95e+106:
		tmp = (l / ((k * math.fabs(t)) * (math.fabs(t) * math.fabs(t)))) * ((l / (k + k)) * 2.0)
	else:
		tmp = ((l * (l / ((math.sin(k) * math.fabs(t)) * math.fabs(t)))) * 2.0) / (math.fabs(t) * (k + k))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.95e+106)
		tmp = Float64(Float64(l / Float64(Float64(k * abs(t)) * Float64(abs(t) * abs(t)))) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(Float64(sin(k) * abs(t)) * abs(t)))) * 2.0) / Float64(abs(t) * Float64(k + k)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.95e+106)
		tmp = (l / ((k * abs(t)) * (abs(t) * abs(t)))) * ((l / (k + k)) * 2.0);
	else
		tmp = ((l * (l / ((sin(k) * abs(t)) * abs(t)))) * 2.0) / (abs(t) * (k + k));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 19499999999999998414092620530282440361043467030448022129590792332144947096303360161971506580700578568994816], N[(N[(l / N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9499999999999998e106

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 64.2%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 1.9499999999999998e106 < t

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{2 \cdot k}} \]

   8. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{2}{k + k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t}} \cdot \frac{2}{k + k} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t}} \cdot \frac{2}{k + k} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t}}{t} \cdot \color{blue}{\frac{2}{k + k}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{\left(\sin k \cdot t\right) \cdot t} \cdot 2}{t \cdot \left(k + k\right)}} \]

   10. Applied rewrites 65.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \frac{\ell}{\left(\sin k \cdot t\right) \cdot t}\right) \cdot 2}{t \cdot \left(k + k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 1499999999999999989831423011542441964692562662916096:\\ \;\;\;\;\frac{\ell}{\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{\ell \cdot 2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t\_1 \cdot \left(\left|k\right| \cdot t\right)}\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (+ (fabs k) (fabs k))))
  (if (<=
       (fabs k)
       1499999999999999989831423011542441964692562662916096)
    (* (/ l (* (* (* (sin (fabs k)) t) t) t)) (/ (* l 2) t_1))
    (* (* (/ l (* t t)) (/ l (* t_1 (* (fabs k) t)))) 2))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 1.5e+51) {
		tmp = (l / (((sin(fabs(k)) * t) * t) * t)) * ((l * 2.0) / t_1);
	} else {
		tmp = ((l / (t * t)) * (l / (t_1 * (fabs(k) * t)))) * 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) <= 1.5d+51) then
        tmp = (l / (((sin(abs(k)) * t) * t) * t)) * ((l * 2.0d0) / t_1)
    else
        tmp = ((l / (t * t)) * (l / (t_1 * (abs(k) * t)))) * 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) <= 1.5e+51) {
		tmp = (l / (((Math.sin(Math.abs(k)) * t) * t) * t)) * ((l * 2.0) / t_1);
	} else {
		tmp = ((l / (t * t)) * (l / (t_1 * (Math.abs(k) * t)))) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 1.5e+51:
		tmp = (l / (((math.sin(math.fabs(k)) * t) * t) * t)) * ((l * 2.0) / t_1)
	else:
		tmp = ((l / (t * t)) * (l / (t_1 * (math.fabs(k) * t)))) * 2.0
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 1.5e+51)
		tmp = Float64(Float64(l / Float64(Float64(Float64(sin(abs(k)) * t) * t) * t)) * Float64(Float64(l * 2.0) / t_1));
	else
		tmp = Float64(Float64(Float64(l / Float64(t * t)) * Float64(l / Float64(t_1 * Float64(abs(k) * t)))) * 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) <= 1.5e+51)
		tmp = (l / (((sin(abs(k)) * t) * t) * t)) * ((l * 2.0) / t_1);
	else
		tmp = ((l / (t * t)) * (l / (t_1 * (abs(k) * t)))) * 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], 1499999999999999989831423011542441964692562662916096], N[(N[(l / N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e51

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{\ell \cdot 2}{2 \cdot k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)} \cdot \frac{\ell \cdot 2}{2 \cdot k}} \]

   8. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{\ell \cdot 2}{k + k}} \]

   if 1.5e51 < k

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \left(2 \cdot k\right)} \cdot 2 \]

       5. associate-*l* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

       6. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{t \cdot t} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(2 \cdot k\right)}\right)} \cdot 2 \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{t \cdot t} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(2 \cdot k\right)}\right)} \cdot 2 \]

       8. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\ell}{t \cdot t}} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(2 \cdot k\right)}\right) \cdot 2 \]

       9. lower-/.f64 N/A

          \[\leadsto \left(\frac{\ell}{t \cdot t} \cdot \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(2 \cdot k\right)}}\right) \cdot 2 \]

   11. Applied rewrites 62.1%

       \[\leadsto \color{blue}{\left(\frac{\ell}{t \cdot t} \cdot \frac{\ell}{\left(k + k\right) \cdot \left(k \cdot t\right)}\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 64.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq \frac{4856672230564323}{485667223056432267729865476705879726660601709763034880312953102434726071301302124544}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\ell\right) \cdot \frac{\ell}{\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(-t\right)\right) \cdot \left(k + k\right)}\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (* l l)
     4856672230564323/485667223056432267729865476705879726660601709763034880312953102434726071301302124544)
  (* (/ l (* (* k t) (* t t))) (* (/ l (+ k k)) 2))
  (* (* (- l) (/ l (* (* (* (* (sin k) t) t) (- t)) (+ k k)))) 2)))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-68) {
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = (-l * (l / ((((sin(k) * t) * t) * -t) * (k + k)))) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-68) then
        tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0d0)
    else
        tmp = (-l * (l / ((((sin(k) * t) * t) * -t) * (k + k)))) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-68) {
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	} else {
		tmp = (-l * (l / ((((Math.sin(k) * t) * t) * -t) * (k + k)))) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-68:
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0)
	else:
		tmp = (-l * (l / ((((math.sin(k) * t) * t) * -t) * (k + k)))) * 2.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-68)
		tmp = Float64(Float64(l / Float64(Float64(k * t) * Float64(t * t))) * Float64(Float64(l / Float64(k + k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(-l) * Float64(l / Float64(Float64(Float64(Float64(sin(k) * t) * t) * Float64(-t)) * Float64(k + k)))) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-68)
		tmp = (l / ((k * t) * (t * t))) * ((l / (k + k)) * 2.0);
	else
		tmp = (-l * (l / ((((sin(k) * t) * t) * -t) * (k + k)))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 4856672230564323/485667223056432267729865476705879726660601709763034880312953102434726071301302124544], N[(N[(l / N[(N[(k * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision], N[(N[((-l) * N[(l / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * (-t)), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e-68

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

       2. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       5. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

   11. Applied rewrites 64.2%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]

   if 1.0000000000000001e-68 < (*.f64 l l)

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\ell \cdot \ell\right)}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)}} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\ell \cdot \ell}\right)}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot \ell}}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-\ell\right)} \cdot \ell}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)} \cdot 2 \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(-\ell\right) \cdot \frac{\ell}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)}\right)} \cdot 2 \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-\ell\right) \cdot \frac{\ell}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)}\right)} \cdot 2 \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-\ell\right) \cdot \color{blue}{\frac{\ell}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)\right)}}\right) \cdot 2 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-\ell\right) \cdot \frac{\ell}{\mathsf{neg}\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot k\right)}\right)}\right) \cdot 2 \]

   8. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\left(\left(-\ell\right) \cdot \frac{\ell}{\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(-t\right)\right) \cdot \left(k + k\right)}\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 64.1% accurate, 9.1× speedup

Math

\[\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right) \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (/ l (* (* k t) (* t t))) (* (/ l (+ k k)) 2)))

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 * t) * Float64(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 * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k + k), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. 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 46.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

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 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

6. Applied rewrites 55.8%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
8. Step-by-step derivation

   1. lower-*.f64 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

9. Applied rewrites 58.4%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

    2. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    5. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{2 \cdot k}\right)} \cdot 2 \]

    6. associate-*l* N/A

       \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{2 \cdot k} \cdot 2\right)} \]

11. Applied rewrites 64.2%

    \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{k + k} \cdot 2\right)} \]
12. Add Preprocessing

Alternative 18: 63.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ t_2 := \left|k\right| + \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq \frac{4904977144712527}{20437404769635530871361256581497226916530700906859085224986083762557049772738192033637969566644589579154866655684531151298277765001150399085969119214436673744076858091019117327539586267590276988750370373064129781691707499060437712782221877948907972172872918086407741866417750991158722661661540352}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t\_2 \cdot t\right) \cdot \left(t\_1 \cdot t\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(t\_2 \cdot \left(t \cdot t\right)\right) \cdot t\_1} \cdot \ell\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs k) t)) (t_2 (+ (fabs k) (fabs k))))
  (if (<=
       (fabs k)
       4904977144712527/20437404769635530871361256581497226916530700906859085224986083762557049772738192033637969566644589579154866655684531151298277765001150399085969119214436673744076858091019117327539586267590276988750370373064129781691707499060437712782221877948907972172872918086407741866417750991158722661661540352)
    (* (/ (* l l) (* (* t_2 t) (* t_1 t))) 2)
    (* (* (/ l (* (* t_2 (* t t)) t_1)) l) 2))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double t_2 = fabs(k) + fabs(k);
	double tmp;
	if (fabs(k) <= 2.4e-280) {
		tmp = ((l * l) / ((t_2 * t) * (t_1 * t))) * 2.0;
	} else {
		tmp = ((l / ((t_2 * (t * t)) * t_1)) * l) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(k) * t
    t_2 = abs(k) + abs(k)
    if (abs(k) <= 2.4d-280) then
        tmp = ((l * l) / ((t_2 * t) * (t_1 * t))) * 2.0d0
    else
        tmp = ((l / ((t_2 * (t * t)) * t_1)) * l) * 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) * t;
	double t_2 = Math.abs(k) + Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 2.4e-280) {
		tmp = ((l * l) / ((t_2 * t) * (t_1 * t))) * 2.0;
	} else {
		tmp = ((l / ((t_2 * (t * t)) * t_1)) * l) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	t_2 = math.fabs(k) + math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 2.4e-280:
		tmp = ((l * l) / ((t_2 * t) * (t_1 * t))) * 2.0
	else:
		tmp = ((l / ((t_2 * (t * t)) * t_1)) * l) * 2.0
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	t_2 = Float64(abs(k) + abs(k))
	tmp = 0.0
	if (abs(k) <= 2.4e-280)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(t_2 * t) * Float64(t_1 * t))) * 2.0);
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(t_2 * Float64(t * t)) * t_1)) * l) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	t_2 = abs(k) + abs(k);
	tmp = 0.0;
	if (abs(k) <= 2.4e-280)
		tmp = ((l * l) / ((t_2 * t) * (t_1 * t))) * 2.0;
	else
		tmp = ((l / ((t_2 * (t * t)) * t_1)) * l) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4904977144712527/20437404769635530871361256581497226916530700906859085224986083762557049772738192033637969566644589579154866655684531151298277765001150399085969119214436673744076858091019117327539586267590276988750370373064129781691707499060437712782221877948907972172872918086407741866417750991158722661661540352], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$2 * t), $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision], N[(N[(N[(l / N[(N[(t$95$2 * N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.3999999999999998e-280

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(2 \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(2 \cdot \color{blue}{k}\right) \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot 2 \]

       4. count-2-rev N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k + \color{blue}{k}\right) \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot 2 \]

       5. lift-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k + \color{blue}{k}\right) \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot 2 \]

       6. lift-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k + \color{blue}{k}\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)\right)} \cdot 2 \]

       7. lift-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k + \color{blue}{k}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)\right)} \cdot 2 \]

       8. lift-+.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k + \color{blue}{k}\right) \cdot \mathsf{Rewrite=>}\left(associate-*l*, \left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right)\right)} \cdot 2 \]

       9. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k + k\right) \cdot t\right) \cdot \left(t \cdot \left(k \cdot t\right)\right)}} \cdot 2 \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k + k\right) \cdot t\right) \cdot \left(t \cdot \left(k \cdot t\right)\right)}} \cdot 2 \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k + k\right) \cdot t\right)} \cdot \left(t \cdot \left(k \cdot t\right)\right)} \cdot 2 \]

       12. *-commutative N/A

           \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k + k\right) \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot 2 \]

       13. lower-*.f64 62.2%

           \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k + k\right) \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot 2 \]

   11. Applied rewrites 62.2%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k + k\right) \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot 2 \]

   if 2.3999999999999998e-280 < k

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. 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 46.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

   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 55.8%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

   6. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lower-*.f64 58.4%

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}\right)} \cdot 2 \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot \ell\right)} \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot \ell\right)} \cdot 2 \]

   11. Applied rewrites 63.2%

       \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(k + k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.2% accurate, 10.3× speedup

Math

\[\left(\frac{\ell}{\left(\left(k + k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell\right) \cdot 2 \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (* (/ l (* (* (+ k k) (* t t)) (* k t))) l) 2))

C

double code(double t, double l, double k) {
	return ((l / (((k + k) * (t * t)) * (k * t))) * l) * 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 + k) * (t * t)) * (k * t))) * l) * 2.0d0
end function

Java

public static double code(double t, double l, double k) {
	return ((l / (((k + k) * (t * t)) * (k * t))) * l) * 2.0;
}

Python

def code(t, l, k):
	return ((l / (((k + k) * (t * t)) * (k * t))) * l) * 2.0

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / Float64(Float64(Float64(k + k) * Float64(t * t)) * Float64(k * t))) * l) * 2.0)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / (((k + k) * (t * t)) * (k * t))) * l) * 2.0;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[(N[(N[(k + k), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. 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 46.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

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 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

6. Applied rewrites 55.8%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
8. Step-by-step derivation

   1. lower-*.f64 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

9. Applied rewrites 58.4%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

    3. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}\right)} \cdot 2 \]

    4. *-commutative N/A

       \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot \ell\right)} \cdot 2 \]

    5. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot \ell\right)} \cdot 2 \]

11. Applied rewrites 63.2%

    \[\leadsto \color{blue}{\left(\frac{\ell}{\left(\left(k + k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell\right)} \cdot 2 \]
12. Add Preprocessing

Alternative 20: 58.7% accurate, 10.3× speedup

Math

\[\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\left(k + k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (* l l) (/ 2 (* (* (+ k k) (* t t)) (* k t)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (((k + k) * (t * t)) * (k * 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) * (2.0d0 / (((k + k) * (t * t)) * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (((k + k) * (t * t)) * (k * t)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (((k + k) * (t * t)) * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(Float64(k + k) * Float64(t * t)) * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (((k + k) * (t * t)) * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2 / N[(N[(N[(k + k), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. 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 46.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k \cdot k}{t \cdot t} - -2\right) \cdot \tan k\right)} \cdot 2} \]

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 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \cdot 2 \]

6. Applied rewrites 55.8%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \cdot 2 \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
8. Step-by-step derivation

   1. lower-*.f64 58.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]

9. Applied rewrites 58.4%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \cdot 2 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \cdot 2} \]

    2. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \cdot 2 \]

    3. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \]

    4. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \]

    5. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \]

    6. lower-/.f64 58.3%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \left(2 \cdot k\right)}} \]

11. Applied rewrites 58.7%

    \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\left(k + k\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))