Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 87.8%

Time: 8.6s

Alternatives: 23

Speedup: 6.6×

• 73.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\left(\frac{\ell}{\left({k}^{2} \cdot \left(\left|t\right| \cdot \sin k\right)\right) \cdot \tan k} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot \frac{\left|t\right|}{\ell}\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.4e-41)
      (* (* (/ l (* (* (pow k 2.0) (* (fabs t) (sin k))) (tan k))) l) 2.0)
      (/
       2.0
       (*
        (* (* (tan k) (* (/ (* (sin k) (fabs t)) l) (fabs t))) (/ (fabs t) l))
        (fma t_1 t_1 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double tmp;
	if (fabs(t) <= 3.4e-41) {
		tmp = ((l / ((pow(k, 2.0) * (fabs(t) * sin(k))) * tan(k))) * l) * 2.0;
	} else {
		tmp = 2.0 / (((tan(k) * (((sin(k) * fabs(t)) / l) * fabs(t))) * (fabs(t) / l)) * fma(t_1, t_1, 2.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(t))
	tmp = 0.0
	if (abs(t) <= 3.4e-41)
		tmp = Float64(Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * sin(k))) * tan(k))) * l) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))) * Float64(abs(t) / l)) * fma(t_1, t_1, 2.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.4e-41], N[(N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.3999999999999998e-41

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.64 63.9%

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

   8. Applied rewrites 63.9%

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

   if 3.3999999999999998e-41 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. add-flip N/A

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

      5. lift-pow.64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate--l+ N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   7. Applied rewrites 77.5%

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

Alternative 2: 87.6% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{-42}:\\ \;\;\;\;\left(\frac{\ell}{\left({k}^{2} \cdot \left(\left|t\right| \cdot \sin k\right)\right) \cdot \tan k} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1e-42)
    (* (* (/ l (* (* (pow k 2.0) (* (fabs t) (sin k))) (tan k))) l) 2.0)
    (/
     2.0
     (*
      (* (/ (* (sin k) (fabs t)) l) (fabs t))
      (*
       (/ (fabs t) l)
       (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1e-42) {
		tmp = ((l / ((pow(k, 2.0) * (fabs(t) * sin(k))) * tan(k))) * l) * 2.0;
	} else {
		tmp = 2.0 / ((((sin(k) * fabs(t)) / l) * fabs(t)) * ((fabs(t) / l) * (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1e-42)
		tmp = Float64(Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * sin(k))) * tan(k))) * l) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(Float64(abs(t) / l) * Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e-42], N[(N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000004e-42

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.64 63.9%

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

   8. Applied rewrites 63.9%

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

   if 1.00000000000000004e-42 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 71.2%

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

Alternative 3: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\left(\frac{\ell}{\left({k}^{2} \cdot \left(\left|t\right| \cdot \sin k\right)\right) \cdot \tan k} \cdot \ell\right) \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq 1.05 \cdot 10^{+89}:\\ \;\;\;\;\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\tan k \cdot \left|t\right|\right) \cdot \left(\sin k \cdot t\_1\right)}}{2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.4e-41)
      (* (* (/ l (* (* (pow k 2.0) (* (fabs t) (sin k))) (tan k))) l) 2.0)
      (if (<= (fabs t) 1.05e+89)
        (*
         (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t)))
         (* (/ l (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k))) 2.0))
        (/ (/ 2.0 (* (* (tan k) (fabs t)) (* (sin k) t_1))) (* 2.0 t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 3.4e-41) {
		tmp = ((l / ((pow(k, 2.0) * (fabs(t) * sin(k))) * tan(k))) * l) * 2.0;
	} else if (fabs(t) <= 1.05e+89) {
		tmp = (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * ((l / (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))) * 2.0);
	} else {
		tmp = (2.0 / ((tan(k) * fabs(t)) * (sin(k) * t_1))) / (2.0 * t_1);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 3.4e-41)
		tmp = Float64(Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * sin(k))) * tan(k))) * l) * 2.0);
	elseif (abs(t) <= 1.05e+89)
		tmp = Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * Float64(Float64(l / Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k))) * 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * abs(t)) * Float64(sin(k) * t_1))) / Float64(2.0 * t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.4e-41], N[(N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.05e+89], 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 / N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.3999999999999998e-41

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.64 63.9%

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

   8. Applied rewrites 63.9%

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

   if 3.3999999999999998e-41 < t < 1.04999999999999993e89

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 56.9%

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

   if 1.04999999999999993e89 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 69.9%

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

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{-42}:\\ \;\;\;\;\left(\frac{\ell}{\left({k}^{2} \cdot \left(\left|t\right| \cdot \sin k\right)\right) \cdot \tan k} \cdot \ell\right) \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq 1.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_1}, k, 2\right)\right) \cdot t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\tan k \cdot \left|t\right|\right) \cdot \left(\sin k \cdot t\_2\right)}}{2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) (fabs t))) (t_2 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e-42)
      (* (* (/ l (* (* (pow k 2.0) (* (fabs t) (sin k))) (tan k))) l) 2.0)
      (if (<= (fabs t) 1.5e+102)
        (*
         (/
          2.0
          (*
           (* (* (/ (* (sin k) (fabs t)) l) (tan k)) (fma (/ k t_1) k 2.0))
           t_1))
         l)
        (/ (/ 2.0 (* (* (tan k) (fabs t)) (* (sin k) t_2))) (* 2.0 t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1e-42) {
		tmp = ((l / ((pow(k, 2.0) * (fabs(t) * sin(k))) * tan(k))) * l) * 2.0;
	} else if (fabs(t) <= 1.5e+102) {
		tmp = (2.0 / (((((sin(k) * fabs(t)) / l) * tan(k)) * fma((k / t_1), k, 2.0)) * t_1)) * l;
	} else {
		tmp = (2.0 / ((tan(k) * fabs(t)) * (sin(k) * t_2))) / (2.0 * t_2);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1e-42)
		tmp = Float64(Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * sin(k))) * tan(k))) * l) * 2.0);
	elseif (abs(t) <= 1.5e+102)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * tan(k)) * fma(Float64(k / t_1), k, 2.0)) * t_1)) * l);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * abs(t)) * Float64(sin(k) * t_2))) / Float64(2.0 * t_2));
	end
	return Float64(copysign(1.0, t) * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e-42], N[(N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.5e+102], N[(N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$1), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.00000000000000004e-42

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.64 63.9%

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

   8. Applied rewrites 63.9%

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

   if 1.00000000000000004e-42 < t < 1.4999999999999999e102

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 60.0%

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

   if 1.4999999999999999e102 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 69.9%

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

Alternative 5: 82.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 0.0068:\\ \;\;\;\;\left(\frac{\ell}{\left({k}^{2} \cdot \left(\left|t\right| \cdot \sin k\right)\right) \cdot \tan k} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\tan k \cdot \left|t\right|\right) \cdot \left(\sin k \cdot t\_1\right)}}{2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 0.0068)
      (* (* (/ l (* (* (pow k 2.0) (* (fabs t) (sin k))) (tan k))) l) 2.0)
      (/ (/ 2.0 (* (* (tan k) (fabs t)) (* (sin k) t_1))) (* 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 0.0068) {
		tmp = ((l / ((pow(k, 2.0) * (fabs(t) * sin(k))) * tan(k))) * l) * 2.0;
	} else {
		tmp = (2.0 / ((tan(k) * fabs(t)) * (sin(k) * t_1))) / (2.0 * t_1);
	}
	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 tmp;
	if (Math.abs(t) <= 0.0068) {
		tmp = ((l / ((Math.pow(k, 2.0) * (Math.abs(t) * Math.sin(k))) * Math.tan(k))) * l) * 2.0;
	} else {
		tmp = (2.0 / ((Math.tan(k) * Math.abs(t)) * (Math.sin(k) * t_1))) / (2.0 * t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) / l
	tmp = 0
	if math.fabs(t) <= 0.0068:
		tmp = ((l / ((math.pow(k, 2.0) * (math.fabs(t) * math.sin(k))) * math.tan(k))) * l) * 2.0
	else:
		tmp = (2.0 / ((math.tan(k) * math.fabs(t)) * (math.sin(k) * t_1))) / (2.0 * t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 0.0068)
		tmp = Float64(Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * sin(k))) * tan(k))) * l) * 2.0);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * abs(t)) * Float64(sin(k) * t_1))) / Float64(2.0 * t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) / l;
	tmp = 0.0;
	if (abs(t) <= 0.0068)
		tmp = ((l / (((k ^ 2.0) * (abs(t) * sin(k))) * tan(k))) * l) * 2.0;
	else
		tmp = (2.0 / ((tan(k) * abs(t)) * (sin(k) * t_1))) / (2.0 * t_1);
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 0.0068], N[(N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.00679999999999999962

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.64 63.9%

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

   8. Applied rewrites 63.9%

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

   if 0.00679999999999999962 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 69.9%

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

Alternative 6: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 0.048:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \left(\tan k \cdot t\_1\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot t\_1\right)}}{2 \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 0.048)
     (/
      2.0
      (*
       (* (* (/ (* k t) (fabs l)) t) (* (tan k) t_1))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ (/ 2.0 (* (* (tan k) t) (* (sin k) t_1))) (* 2.0 t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 0.048) {
		tmp = 2.0 / (((((k * t) / fabs(l)) * t) * (tan(k) * t_1)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / ((tan(k) * t) * (sin(k) * t_1))) / (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 = t / abs(l)
    if (abs(l) <= 0.048d0) then
        tmp = 2.0d0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = (2.0d0 / ((tan(k) * t) * (sin(k) * t_1))) / (2.0d0 * t_1)
    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) <= 0.048) {
		tmp = 2.0 / (((((k * t) / Math.abs(l)) * t) * (Math.tan(k) * t_1)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / ((Math.tan(k) * t) * (Math.sin(k) * t_1))) / (2.0 * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 0.048:
		tmp = 2.0 / (((((k * t) / math.fabs(l)) * t) * (math.tan(k) * t_1)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = (2.0 / ((math.tan(k) * t) * (math.sin(k) * t_1))) / (2.0 * t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 0.048)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * Float64(tan(k) * t_1)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(tan(k) * t) * Float64(sin(k) * t_1))) / Float64(2.0 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 0.048)
		tmp = 2.0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = (2.0 / ((tan(k) * t) * (sin(k) * t_1))) / (2.0 * t_1);
	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], 0.048], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 0.048000000000000001

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.3%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 0.048000000000000001 < l

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 69.9%

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

Alternative 7: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \left(\tan k \cdot t\_1\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \left(\sin k \cdot t\_1\right)\right) \cdot 2\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 1.22e+54)
     (/
      2.0
      (*
       (* (* (/ (* k t) (fabs l)) t) (* (tan k) t_1))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ 2.0 (* t_1 (* (* (* (tan k) t) (* (sin k) t_1)) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 1.22e+54) {
		tmp = 2.0 / (((((k * t) / fabs(l)) * t) * (tan(k) * t_1)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (t_1 * (((tan(k) * t) * (sin(k) * t_1)) * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 1.22d+54) then
        tmp = 2.0d0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (t_1 * (((tan(k) * t) * (sin(k) * t_1)) * 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) <= 1.22e+54) {
		tmp = 2.0 / (((((k * t) / Math.abs(l)) * t) * (Math.tan(k) * t_1)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (t_1 * (((Math.tan(k) * t) * (Math.sin(k) * t_1)) * 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 1.22e+54:
		tmp = 2.0 / (((((k * t) / math.fabs(l)) * t) * (math.tan(k) * t_1)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (t_1 * (((math.tan(k) * t) * (math.sin(k) * t_1)) * 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 1.22e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * Float64(tan(k) * t_1)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(tan(k) * t) * Float64(sin(k) * t_1)) * 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) <= 1.22e+54)
		tmp = 2.0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (t_1 * (((tan(k) * t) * (sin(k) * t_1)) * 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], 1.22e+54], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.22e54

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.3%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 1.22e54 < l

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 69.9%

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

   10. Applied rewrites 69.9%

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

Alternative 8: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \left(\tan k \cdot t\_1\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{\sin k \cdot t}{\left|\ell\right|} \cdot t\right)\right) \cdot t\_1\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 1.22e+54)
     (/
      2.0
      (*
       (* (* (/ (* k t) (fabs l)) t) (* (tan k) t_1))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ 2.0 (* (* (* (tan k) (* (/ (* (sin k) t) (fabs l)) t)) t_1) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 1.22e+54) {
		tmp = 2.0 / (((((k * t) / fabs(l)) * t) * (tan(k) * t_1)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((tan(k) * (((sin(k) * t) / fabs(l)) * t)) * t_1) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 1.22d+54) then
        tmp = 2.0d0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((tan(k) * (((sin(k) * t) / abs(l)) * t)) * t_1) * 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) <= 1.22e+54) {
		tmp = 2.0 / (((((k * t) / Math.abs(l)) * t) * (Math.tan(k) * t_1)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (((Math.sin(k) * t) / Math.abs(l)) * t)) * t_1) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 1.22e+54:
		tmp = 2.0 / (((((k * t) / math.fabs(l)) * t) * (math.tan(k) * t_1)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((math.tan(k) * (((math.sin(k) * t) / math.fabs(l)) * t)) * t_1) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 1.22e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * Float64(tan(k) * t_1)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(Float64(Float64(sin(k) * t) / abs(l)) * t)) * t_1) * 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) <= 1.22e+54)
		tmp = 2.0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((tan(k) * (((sin(k) * t) / abs(l)) * t)) * t_1) * 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], 1.22e+54], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.22e54

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.3%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 1.22e54 < l

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

Alternative 9: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 1.3 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \left(\tan k \cdot t\_1\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot t\_1\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot t\_1\right) \cdot 2\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 1.3e+48)
     (/
      2.0
      (*
       (* (* (/ (* k t) (fabs l)) t) (* (tan k) t_1))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ 2.0 (* (* (sin k) t_1) (* (* (* (tan k) t) t_1) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 1.3e+48) {
		tmp = 2.0 / (((((k * t) / fabs(l)) * t) * (tan(k) * t_1)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((sin(k) * t_1) * (((tan(k) * t) * t_1) * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 1.3d+48) then
        tmp = 2.0d0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((sin(k) * t_1) * (((tan(k) * t) * t_1) * 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) <= 1.3e+48) {
		tmp = 2.0 / (((((k * t) / Math.abs(l)) * t) * (Math.tan(k) * t_1)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((Math.sin(k) * t_1) * (((Math.tan(k) * t) * t_1) * 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 1.3e+48:
		tmp = 2.0 / (((((k * t) / math.fabs(l)) * t) * (math.tan(k) * t_1)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((math.sin(k) * t_1) * (((math.tan(k) * t) * t_1) * 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 1.3e+48)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * Float64(tan(k) * t_1)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * t_1) * Float64(Float64(Float64(tan(k) * t) * t_1) * 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) <= 1.3e+48)
		tmp = 2.0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((sin(k) * t_1) * (((tan(k) * t) * t_1) * 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], 1.3e+48], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.29999999999999998e48

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.3%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 1.29999999999999998e48 < l

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 70.6%

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

Alternative 10: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(t\_2 \cdot \left(\frac{\left(k \cdot k\right) \cdot \left|t\right|}{\ell} \cdot 2\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{\frac{t\_1}{\ell} \cdot \left(\left(k \cdot \frac{\sin k \cdot \left|t\right|}{\ell}\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_1}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot t\_2\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) (fabs t))) (t_2 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 2.7e-162)
      (/ 2.0 (* (fabs t) (* t_2 (* (/ (* (* k k) (fabs t)) l) 2.0))))
      (if (<= (fabs t) 2.9e+34)
        (/
         2.0
         (*
          (/ t_1 l)
          (* (* k (/ (* (sin k) (fabs t)) l)) (fma k (/ k t_1) 2.0))))
        (/
         2.0
         (* (* (* (tan k) (* (/ (* k (fabs t)) l) (fabs t))) t_2) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 2.7e-162) {
		tmp = 2.0 / (fabs(t) * (t_2 * ((((k * k) * fabs(t)) / l) * 2.0)));
	} else if (fabs(t) <= 2.9e+34) {
		tmp = 2.0 / ((t_1 / l) * ((k * ((sin(k) * fabs(t)) / l)) * fma(k, (k / t_1), 2.0)));
	} else {
		tmp = 2.0 / (((tan(k) * (((k * fabs(t)) / l) * fabs(t))) * t_2) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 2.7e-162)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(t_2 * Float64(Float64(Float64(Float64(k * k) * abs(t)) / l) * 2.0))));
	elseif (abs(t) <= 2.9e+34)
		tmp = Float64(2.0 / Float64(Float64(t_1 / l) * Float64(Float64(k * Float64(Float64(sin(k) * abs(t)) / l)) * fma(k, Float64(k / t_1), 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(Float64(Float64(k * abs(t)) / l) * abs(t))) * t_2) * 2.0));
	end
	return Float64(copysign(1.0, t) * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.7e-162], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(t$95$2 * N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.9e+34], N[(2.0 / N[(N[(t$95$1 / l), $MachinePrecision] * N[(N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.69999999999999984e-162

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.1%

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

      13. lift-+.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 57.6%

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

   8. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 62.9%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   10. Applied rewrites 62.9%

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

   if 2.69999999999999984e-162 < t < 2.9000000000000001e34

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.1%

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

      13. lift-+.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 54.2%

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

   if 2.9000000000000001e34 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 67.6%

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

Alternative 11: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 3.1 \cdot 10^{-152}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \left(\tan k \cdot t\_1\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \left(\frac{k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}{\left|\ell\right|} \cdot t\right)\right) \cdot t\_1\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 3.1e-152)
     (/
      2.0
      (*
       (* (* (/ (* k t) (fabs l)) t) (* (tan k) t_1))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/
      2.0
      (*
       (*
        (*
         (tan k)
         (*
          (/ (* k (+ t (* -0.16666666666666666 (* (pow k 2.0) t)))) (fabs l))
          t))
        t_1)
       2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 3.1e-152) {
		tmp = 2.0 / (((((k * t) / fabs(l)) * t) * (tan(k) * t_1)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((tan(k) * (((k * (t + (-0.16666666666666666 * (pow(k, 2.0) * t)))) / fabs(l)) * t)) * t_1) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 3.1d-152) then
        tmp = 2.0d0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((tan(k) * (((k * (t + ((-0.16666666666666666d0) * ((k ** 2.0d0) * t)))) / abs(l)) * t)) * t_1) * 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) <= 3.1e-152) {
		tmp = 2.0 / (((((k * t) / Math.abs(l)) * t) * (Math.tan(k) * t_1)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (((k * (t + (-0.16666666666666666 * (Math.pow(k, 2.0) * t)))) / Math.abs(l)) * t)) * t_1) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 3.1e-152:
		tmp = 2.0 / (((((k * t) / math.fabs(l)) * t) * (math.tan(k) * t_1)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((math.tan(k) * (((k * (t + (-0.16666666666666666 * (math.pow(k, 2.0) * t)))) / math.fabs(l)) * t)) * t_1) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 3.1e-152)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * Float64(tan(k) * t_1)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(Float64(Float64(k * Float64(t + Float64(-0.16666666666666666 * Float64((k ^ 2.0) * t)))) / abs(l)) * t)) * t_1) * 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) <= 3.1e-152)
		tmp = 2.0 / (((((k * t) / abs(l)) * t) * (tan(k) * t_1)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((tan(k) * (((k * (t + (-0.16666666666666666 * ((k ^ 2.0) * t)))) / abs(l)) * t)) * t_1) * 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], 3.1e-152], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k * N[(t + N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.0999999999999998e-152

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 78.3%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 3.0999999999999998e-152 < l

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.64 72.8%

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \left(\frac{k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]

   11. Applied rewrites 72.8%

       \[\leadsto \frac{2}{\left(\left(\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.45 \cdot 10^{-180}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(t\_1 \cdot \left(\frac{\left(k \cdot k\right) \cdot \left|t\right|}{\ell} \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot \left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot t\_1\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.45e-180)
      (/ 2.0 (* (fabs t) (* t_1 (* (/ (* (* k k) (fabs t)) l) 2.0))))
      (/
       2.0
       (*
        (* (* k (* (/ (* (sin k) (fabs t)) l) (fabs t))) t_1)
        (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.45e-180) {
		tmp = 2.0 / (fabs(t) * (t_1 * ((((k * k) * fabs(t)) / l) * 2.0)));
	} else {
		tmp = 2.0 / (((k * (((sin(k) * fabs(t)) / l) * fabs(t))) * t_1) * ((1.0 + pow((k / fabs(t)), 2.0)) + 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;
	double tmp;
	if (Math.abs(t) <= 1.45e-180) {
		tmp = 2.0 / (Math.abs(t) * (t_1 * ((((k * k) * Math.abs(t)) / l) * 2.0)));
	} else {
		tmp = 2.0 / (((k * (((Math.sin(k) * Math.abs(t)) / l) * Math.abs(t))) * t_1) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) / l
	tmp = 0
	if math.fabs(t) <= 1.45e-180:
		tmp = 2.0 / (math.fabs(t) * (t_1 * ((((k * k) * math.fabs(t)) / l) * 2.0)))
	else:
		tmp = 2.0 / (((k * (((math.sin(k) * math.fabs(t)) / l) * math.fabs(t))) * t_1) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.45e-180)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(t_1 * Float64(Float64(Float64(Float64(k * k) * abs(t)) / l) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))) * t_1) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) / l;
	tmp = 0.0;
	if (abs(t) <= 1.45e-180)
		tmp = 2.0 / (abs(t) * (t_1 * ((((k * k) * abs(t)) / l) * 2.0)));
	else
		tmp = 2.0 / (((k * (((sin(k) * abs(t)) / l) * abs(t))) * t_1) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e-180], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(t$95$1 * N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4499999999999999e-180

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.1%

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

      13. lift-+.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 57.6%

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

   8. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 62.9%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   10. Applied rewrites 62.9%

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

   if 1.4499999999999999e-180 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 70.7%

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

Alternative 13: 72.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 3.3 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{\left(\left(\tan \left(\left|k\right|\right) \cdot \left(\frac{\left|k\right| \cdot t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\left(\left|k\right| \cdot \left|k\right|\right) \cdot t}{\ell} \cdot 2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 3.3e+58)
   (/ 2.0 (* (* (* (tan (fabs k)) (* (/ (* (fabs k) t) l) t)) (/ t l)) 2.0))
   (/ 2.0 (* t (* (/ t l) (* (/ (* (* (fabs k) (fabs k)) t) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 3.3e+58) {
		tmp = 2.0 / (((tan(fabs(k)) * (((fabs(k) * t) / l) * t)) * (t / l)) * 2.0);
	} else {
		tmp = 2.0 / (t * ((t / l) * ((((fabs(k) * fabs(k)) * t) / l) * 2.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 3.3d+58) then
        tmp = 2.0d0 / (((tan(abs(k)) * (((abs(k) * t) / l) * t)) * (t / l)) * 2.0d0)
    else
        tmp = 2.0d0 / (t * ((t / l) * ((((abs(k) * abs(k)) * t) / l) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 3.3e+58) {
		tmp = 2.0 / (((Math.tan(Math.abs(k)) * (((Math.abs(k) * t) / l) * t)) * (t / l)) * 2.0);
	} else {
		tmp = 2.0 / (t * ((t / l) * ((((Math.abs(k) * Math.abs(k)) * t) / l) * 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 3.3e+58:
		tmp = 2.0 / (((math.tan(math.fabs(k)) * (((math.fabs(k) * t) / l) * t)) * (t / l)) * 2.0)
	else:
		tmp = 2.0 / (t * ((t / l) * ((((math.fabs(k) * math.fabs(k)) * t) / l) * 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 3.3e+58)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(abs(k)) * Float64(Float64(Float64(abs(k) * t) / l) * t)) * Float64(t / l)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(t / l) * Float64(Float64(Float64(Float64(abs(k) * abs(k)) * t) / l) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 3.3e+58)
		tmp = 2.0 / (((tan(abs(k)) * (((abs(k) * t) / l) * t)) * (t / l)) * 2.0);
	else
		tmp = 2.0 / (t * ((t / l) * ((((abs(k) * abs(k)) * t) / l) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 3.3e+58], N[(2.0 / N[(N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.29999999999999983e58

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 67.6%

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

   if 3.29999999999999983e58 < k

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.1%

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

      13. lift-+.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 57.6%

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

   8. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 62.9%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   10. Applied rewrites 62.9%

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

Alternative 14: 72.2% accurate, 1.5× speedup

Math

\[\frac{2}{\left(\left(\tan k \cdot \left(\frac{\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{2}\right)\right) \cdot t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (*
    (*
     (tan k)
     (* (/ (* (* k (+ 1.0 (* -0.16666666666666666 (pow k 2.0)))) t) l) t))
    (/ t l))
   2.0)))

C

double code(double t, double l, double k) {
	return 2.0 / (((tan(k) * ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * t) / l) * t)) * (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 = 2.0d0 / (((tan(k) * ((((k * (1.0d0 + ((-0.16666666666666666d0) * (k ** 2.0d0)))) * t) / l) * t)) * (t / l)) * 2.0d0)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((Math.tan(k) * ((((k * (1.0 + (-0.16666666666666666 * Math.pow(k, 2.0)))) * t) / l) * t)) * (t / l)) * 2.0);
}

Python

def code(t, l, k):
	return 2.0 / (((math.tan(k) * ((((k * (1.0 + (-0.16666666666666666 * math.pow(k, 2.0)))) * t) / l) * t)) * (t / l)) * 2.0)

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * t) / l) * t)) * Float64(t / l)) * 2.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((tan(k) * ((((k * (1.0 + (-0.16666666666666666 * (k ^ 2.0)))) * t) / l) * t)) * (t / l)) * 2.0);
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[(k * N[(1.0 + N[(-0.16666666666666666 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. 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.64 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 67.4%

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

3. Applied rewrites 67.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. associate-*r* N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

5. Applied rewrites 77.5%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 69.9%

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

9. Taylor expanded in k around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-pow.64 72.5%

       \[\leadsto \frac{2}{\left(\left(\tan k \cdot \left(\frac{\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]

11. Applied rewrites 72.5%

    \[\leadsto \frac{2}{\left(\left(\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{2}\right)\right)} \cdot t}{\ell} \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot 2} \]
12. Add Preprocessing

Alternative 15: 69.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\frac{\left(k \cdot k\right) \cdot \left|t\right|}{\ell} \cdot 2\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k}}{t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.4e-57)
      (/
       2.0
       (* (fabs t) (* (/ (fabs t) l) (* (/ (* (* k k) (fabs t)) l) 2.0))))
      (if (<= (fabs t) 1e+142)
        (* (/ (/ l (* (* (fabs t) (fabs t)) k)) t_1) l)
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 1.4e-57) {
		tmp = 2.0 / (fabs(t) * ((fabs(t) / l) * ((((k * k) * fabs(t)) / l) * 2.0)));
	} else if (fabs(t) <= 1e+142) {
		tmp = ((l / ((fabs(t) * fabs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 1.4e-57) {
		tmp = 2.0 / (Math.abs(t) * ((Math.abs(t) / l) * ((((k * k) * Math.abs(t)) / l) * 2.0)));
	} else if (Math.abs(t) <= 1e+142) {
		tmp = ((l / ((Math.abs(t) * Math.abs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 1.4e-57:
		tmp = 2.0 / (math.fabs(t) * ((math.fabs(t) / l) * ((((k * k) * math.fabs(t)) / l) * 2.0)))
	elif math.fabs(t) <= 1e+142:
		tmp = ((l / ((math.fabs(t) * math.fabs(t)) * k)) / t_1) * l
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 1.4e-57)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(k * k) * abs(t)) / l) * 2.0))));
	elseif (abs(t) <= 1e+142)
		tmp = Float64(Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) / t_1) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 1.4e-57)
		tmp = 2.0 / (abs(t) * ((abs(t) / l) * ((((k * k) * abs(t)) / l) * 2.0)));
	elseif (abs(t) <= 1e+142)
		tmp = ((l / ((abs(t) * abs(t)) * k)) / t_1) * l;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.4e-57], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+142], N[(N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4e-57

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

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

   3. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.1%

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

      13. lift-+.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 57.6%

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

   8. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 62.9%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   10. Applied rewrites 62.9%

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

   if 1.4e-57 < t < 1.00000000000000005e142

   1. Initial program 54.3%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-pow.64 49.7%

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

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.2%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.0%

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

      14. lift-pow.64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.0%

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      5. lower-/.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

      8. lower-*.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   10. Applied rewrites 64.0%

       \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   if 1.00000000000000005e142 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\ell}{\left|t\right| \cdot \left(\left|t\right| \cdot \left(\frac{\left(k \cdot k\right) \cdot \left|t\right|}{\ell} \cdot 2\right)\right)} \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k}}{t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.4e-57)
      (*
       (/ l (* (fabs t) (* (fabs t) (* (/ (* (* k k) (fabs t)) l) 2.0))))
       2.0)
      (if (<= (fabs t) 1e+142)
        (* (/ (/ l (* (* (fabs t) (fabs t)) k)) t_1) l)
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 1.4e-57) {
		tmp = (l / (fabs(t) * (fabs(t) * ((((k * k) * fabs(t)) / l) * 2.0)))) * 2.0;
	} else if (fabs(t) <= 1e+142) {
		tmp = ((l / ((fabs(t) * fabs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 1.4e-57) {
		tmp = (l / (Math.abs(t) * (Math.abs(t) * ((((k * k) * Math.abs(t)) / l) * 2.0)))) * 2.0;
	} else if (Math.abs(t) <= 1e+142) {
		tmp = ((l / ((Math.abs(t) * Math.abs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 1.4e-57:
		tmp = (l / (math.fabs(t) * (math.fabs(t) * ((((k * k) * math.fabs(t)) / l) * 2.0)))) * 2.0
	elif math.fabs(t) <= 1e+142:
		tmp = ((l / ((math.fabs(t) * math.fabs(t)) * k)) / t_1) * l
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 1.4e-57)
		tmp = Float64(Float64(l / Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(Float64(k * k) * abs(t)) / l) * 2.0)))) * 2.0);
	elseif (abs(t) <= 1e+142)
		tmp = Float64(Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) / t_1) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 1.4e-57)
		tmp = (l / (abs(t) * (abs(t) * ((((k * k) * abs(t)) / l) * 2.0)))) * 2.0;
	elseif (abs(t) <= 1e+142)
		tmp = ((l / ((abs(t) * abs(t)) * k)) / t_1) * l;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.4e-57], N[(N[(l / N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+142], N[(N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4e-57

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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.64 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 67.4%

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. lower-*.f64 68.1%

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      12. lower-*.f64 68.1%

          \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

   5. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \color{blue}{\frac{{k}^{2} \cdot t}{\ell}}\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \]

      4. lower-pow.64 57.6%

         \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \]

   8. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot t}{\ell} \cdot \left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot 2} \]

   10. Applied rewrites 62.0%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot 2\right)\right)} \cdot 2} \]

   if 1.4e-57 < t < 1.00000000000000005e142

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      5. lower-/.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

      8. lower-*.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   10. Applied rewrites 64.0%

       \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   if 1.00000000000000005e142 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k}}{k}}{\left|t\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e+142)
      (* (/ (/ (/ l (* (* (fabs t) (fabs t)) k)) k) (fabs t)) l)
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 1e+142) {
		tmp = (((l / ((fabs(t) * fabs(t)) * k)) / k) / fabs(t)) * l;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 1e+142) {
		tmp = (((l / ((Math.abs(t) * Math.abs(t)) * k)) / k) / Math.abs(t)) * l;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 1e+142:
		tmp = (((l / ((math.fabs(t) * math.fabs(t)) * k)) / k) / math.fabs(t)) * l
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 1e+142)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) / k) / abs(t)) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 1e+142)
		tmp = (((l / ((abs(t) * abs(t)) * k)) / k) / abs(t)) * l;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e+142], N[(N[(N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000005e142

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k \cdot t} \cdot \ell \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k}}{t} \cdot \ell \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k}}{t} \cdot \ell \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k}}{t} \cdot \ell \]

      9. lower-/.f64 63.9%

         \[\leadsto \frac{\frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k}}{t} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{k}}{t} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{k}}{t} \cdot \ell \]

      12. lower-*.f64 63.9%

          \[\leadsto \frac{\frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{k}}{t} \cdot \ell \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{k}}{t} \cdot \ell \]

   if 1.00000000000000005e142 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k}}{t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e+142)
      (* (/ (/ l (* (* (fabs t) (fabs t)) k)) t_1) l)
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 1e+142) {
		tmp = ((l / ((fabs(t) * fabs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 1e+142) {
		tmp = ((l / ((Math.abs(t) * Math.abs(t)) * k)) / t_1) * l;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 1e+142:
		tmp = ((l / ((math.fabs(t) * math.fabs(t)) * k)) / t_1) * l
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 1e+142)
		tmp = Float64(Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) / t_1) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 1e+142)
		tmp = ((l / ((abs(t) * abs(t)) * k)) / t_1) * l;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e+142], N[(N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000005e142

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      5. lower-/.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot \left(t \cdot t\right)}}{t \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

      8. lower-*.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   10. Applied rewrites 64.0%

       \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{t \cdot k} \cdot \ell \]

   if 1.00000000000000005e142 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{+142}:\\ \;\;\;\;\frac{\frac{\ell}{t\_1}}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e+142)
      (* (/ (/ l t_1) (* (* (fabs t) (fabs t)) k)) l)
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 1e+142) {
		tmp = ((l / t_1) / ((fabs(t) * fabs(t)) * k)) * l;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 1e+142) {
		tmp = ((l / t_1) / ((Math.abs(t) * Math.abs(t)) * k)) * l;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 1e+142:
		tmp = ((l / t_1) / ((math.fabs(t) * math.fabs(t)) * k)) * l
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 1e+142)
		tmp = Float64(Float64(Float64(l / t_1) / Float64(Float64(abs(t) * abs(t)) * k)) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 1e+142)
		tmp = ((l / t_1) / ((abs(t) * abs(t)) * k)) * l;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e+142], N[(N[(N[(l / t$95$1), $MachinePrecision] / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000005e142

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lower-/.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 64.0%

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 64.0%

       \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

   if 1.00000000000000005e142 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 66.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \left(\left|t\right| \cdot \left|t\right|\right)\right) \cdot \left|t\right|} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) k)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 2e+113)
      (* (/ l (* (* k (* (fabs t) (fabs t))) (fabs t))) (/ l k))
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * k;
	double tmp;
	if (fabs(t) <= 2e+113) {
		tmp = (l / ((k * (fabs(t) * fabs(t))) * fabs(t))) * (l / k);
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * k;
	double tmp;
	if (Math.abs(t) <= 2e+113) {
		tmp = (l / ((k * (Math.abs(t) * Math.abs(t))) * Math.abs(t))) * (l / k);
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * k
	tmp = 0
	if math.fabs(t) <= 2e+113:
		tmp = (l / ((k * (math.fabs(t) * math.fabs(t))) * math.fabs(t))) * (l / k)
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * k)
	tmp = 0.0
	if (abs(t) <= 2e+113)
		tmp = Float64(Float64(l / Float64(Float64(k * Float64(abs(t) * abs(t))) * abs(t))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * k;
	tmp = 0.0;
	if (abs(t) <= 2e+113)
		tmp = (l / ((k * (abs(t) * abs(t))) * abs(t))) * (l / k);
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2e+113], N[(N[(l / N[(N[(k * N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2e113

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      11. pow3 N/A

          \[\leadsto \frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k} \]

      12. lift-pow.64 N/A

          \[\leadsto \frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k} \]

      13. *-commutative N/A

          \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \]

      14. lift-pow.64 N/A

          \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \]

      15. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

      17. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      20. lower-/.f64 63.2%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 2e113 < t

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.64 49.7%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.0%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 62.5%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 65.7%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 65.7%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 65.7% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* t k) (* t k)) t)) l))

C

double code(double t, double l, double k) {
	return (l / (((t * k) * (t * k)) * t)) * l;
}

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 / (((t * k) * (t * k)) * t)) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((t * k) * (t * k)) * t)) * l;
}

Python

def code(t, l, k):
	return (l / (((t * k) * (t * k)) * t)) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(t * k) * Float64(t * k)) * t)) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((t * k) * (t * k)) * t)) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(t * k), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.64 49.7%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 54.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.0%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.0%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lower-*.f64 62.5%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

8. Applied rewrites 62.5%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(\left(t \cdot k\right) \cdot t\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(\left(t \cdot k\right) \cdot t\right)} \cdot \ell \]

   8. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t} \cdot \ell \]

   10. lower-*.f64 65.4%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t} \cdot \ell \]

10. Applied rewrites 65.4%

    \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t} \cdot \ell \]
11. Add Preprocessing

Alternative 22: 65.4% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* t k) t) (* t k))) l))

C

double code(double t, double l, double k) {
	return (l / (((t * k) * t) * (t * k))) * l;
}

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 / (((t * k) * t) * (t * k))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((t * k) * t) * (t * k))) * l;
}

Python

def code(t, l, k):
	return (l / (((t * k) * t) * (t * k))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(t * k) * t) * Float64(t * k))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((t * k) * t) * (t * k))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(t * k), $MachinePrecision] * t), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.64 49.7%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 54.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.0%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.0%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lower-*.f64 62.5%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

8. Applied rewrites 62.5%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   6. lower-*.f64 65.7%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

10. Applied rewrites 65.7%

    \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 23: 62.5% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* k (* t t)) (* t k))) l))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (t * k))) * l;
}

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 * k))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (t * k))) * l;
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * (t * k))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * Float64(t * k))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * (t * k))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.64 49.7%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 49.7%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 54.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 59.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.0%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.0%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lower-*.f64 62.5%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

8. Applied rewrites 62.5%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))