Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 84.7%

Time: 6.7s

Alternatives: 15

Speedup: 6.6×

• 72.5% 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: 55.2% 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: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ t_2 := \tan k \cdot \sin k\\ t_3 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 9 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \mathsf{fma}\left(k, k \cdot \frac{t\_2}{\ell \cdot \ell}, \left(2 \cdot \left(\left|t\right| \cdot \frac{\left|t\right|}{\ell \cdot \ell}\right)\right) \cdot t\_2\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_3}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t)))
        (t_2 (* (tan k) (sin k)))
        (t_3 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 9e+19)
      (/
       2.0
       (*
        (fabs t)
        (fma
         k
         (* k (/ t_2 (* l l)))
         (* (* 2.0 (* (fabs t) (/ (fabs t) (* l l)))) t_2))))
      (if (<= (fabs t) 2.7e+137)
        (*
         (/ l (* (* t_3 (fabs t)) (fabs t)))
         (* (/ l (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) 2.0))
        (/
         2.0
         (*
          (* (* (fabs t) (* (/ (fabs t) l) (/ t_3 l))) (tan k))
          (fma t_1 t_1 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double t_2 = tan(k) * sin(k);
	double t_3 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 9e+19) {
		tmp = 2.0 / (fabs(t) * fma(k, (k * (t_2 / (l * l))), ((2.0 * (fabs(t) * (fabs(t) / (l * l)))) * t_2)));
	} else if (fabs(t) <= 2.7e+137) {
		tmp = (l / ((t_3 * fabs(t)) * fabs(t))) * ((l / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * 2.0);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_3 / l))) * tan(k)) * 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))
	t_2 = Float64(tan(k) * sin(k))
	t_3 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 9e+19)
		tmp = Float64(2.0 / Float64(abs(t) * fma(k, Float64(k * Float64(t_2 / Float64(l * l))), Float64(Float64(2.0 * Float64(abs(t) * Float64(abs(t) / Float64(l * l)))) * t_2))));
	elseif (abs(t) <= 2.7e+137)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * abs(t)) * abs(t))) * Float64(Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_3 / l))) * tan(k)) * 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]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 9e+19], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(t$95$2 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.7e+137], N[(N[(l / N[(N[(t$95$3 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 9e19

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 70.9%

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

   if 9e19 < t < 2.70000000000000017e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   if 2.70000000000000017e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

Alternative 2: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ t_2 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{\left(t\_2 \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_2}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))) (t_2 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 6.2e-59)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 2.7e+137)
        (*
         (/ l (* (* t_2 (fabs t)) (fabs t)))
         (* (/ l (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) 2.0))
        (/
         2.0
         (*
          (* (* (fabs t) (* (/ (fabs t) l) (/ t_2 l))) (tan k))
          (fma t_1 t_1 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double t_2 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 6.2e-59) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 2.7e+137) {
		tmp = (l / ((t_2 * fabs(t)) * fabs(t))) * ((l / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * 2.0);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_2 / l))) * tan(k)) * 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))
	t_2 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 6.2e-59)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 2.7e+137)
		tmp = Float64(Float64(l / Float64(Float64(t_2 * abs(t)) * abs(t))) * Float64(Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_2 / l))) * tan(k)) * 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]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.2e-59], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.7e+137], N[(N[(l / N[(N[(t$95$2 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999998e-59

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 6.19999999999999998e-59 < t < 2.70000000000000017e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   if 2.70000000000000017e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

Alternative 3: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 6.2e-59)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 3e+137)
        (*
         (/ l (* (* t_1 (fabs t)) (fabs t)))
         (* (/ l (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) 2.0))
        (/
         2.0
         (* (* (* (fabs t) (* (/ (fabs t) l) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 6.2e-59) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 3e+137) {
		tmp = (l / ((t_1 * fabs(t)) * fabs(t))) * ((l / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * 2.0);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 6.2e-59)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 3e+137)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * abs(t))) * Float64(Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.2e-59], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+137], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999998e-59

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 6.19999999999999998e-59 < t < 3.0000000000000001e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   if 3.0000000000000001e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 4: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.02 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{\ell}{\left(\left(\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.02e-53)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 3e+137)
        (*
         (*
          (/
           l
           (*
            (* (* (* t_1 (fabs t)) (fabs t)) (tan k))
            (fma k (/ k (* (fabs t) (fabs t))) 2.0)))
          l)
         2.0)
        (/
         2.0
         (* (* (* (fabs t) (* (/ (fabs t) l) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.02e-53) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 3e+137) {
		tmp = ((l / ((((t_1 * fabs(t)) * fabs(t)) * tan(k)) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * l) * 2.0;
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.02e-53)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 3e+137)
		tmp = Float64(Float64(Float64(l / Float64(Float64(Float64(Float64(t_1 * abs(t)) * abs(t)) * tan(k)) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * l) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.02e-53], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+137], N[(N[(N[(l / N[(N[(N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.02000000000000002e-53

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 1.02000000000000002e-53 < t < 3.0000000000000001e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

   if 3.0000000000000001e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 5: 79.9% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{2}\right)\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{\sin k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 2.2e-61)
    (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
    (if (<= (fabs t) 3e+137)
      (*
       (/
        l
        (*
         (*
          (* (* k (+ 1.0 (* -0.16666666666666666 (pow k 2.0)))) (fabs t))
          (fabs t))
         (fabs t)))
       (* (/ l (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) 2.0))
      (/
       2.0
       (*
        (* (* (fabs t) (* (/ (fabs t) l) (/ (* (sin k) (fabs t)) l))) (tan k))
        2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.2e-61) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 3e+137) {
		tmp = (l / ((((k * (1.0 + (-0.16666666666666666 * pow(k, 2.0)))) * fabs(t)) * fabs(t)) * fabs(t))) * ((l / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * 2.0);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * ((sin(k) * fabs(t)) / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.2e-61)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 3e+137)
		tmp = Float64(Float64(l / Float64(Float64(Float64(Float64(k * Float64(1.0 + Float64(-0.16666666666666666 * (k ^ 2.0)))) * abs(t)) * abs(t)) * abs(t))) * Float64(Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(sin(k) * abs(t)) / l))) * tan(k)) * 2.0));
	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], 2.2e-61], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+137], N[(N[(l / N[(N[(N[(N[(k * N[(1.0 + N[(-0.16666666666666666 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.20000000000000009e-61

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 2.20000000000000009e-61 < t < 3.0000000000000001e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 57.7%

         \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{\color{blue}{2}}\right)\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2\right) \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot \left(1 + -0.16666666666666666 \cdot {k}^{2}\right)\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \left(\frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2\right) \]

   if 3.0000000000000001e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 6: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|}{\ell} \cdot \left(\left(1 - -1\right) \cdot \tan k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.5e+26)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 3e+137)
        (/
         2.0
         (/
          (* (/ (* (* t_1 (fabs t)) (fabs t)) l) (* (- 1.0 -1.0) (tan k)))
          l))
        (/
         2.0
         (* (* (* (fabs t) (* (/ (fabs t) l) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 3.5e+26) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 3e+137) {
		tmp = 2.0 / (((((t_1 * fabs(t)) * fabs(t)) / l) * ((1.0 - -1.0) * tan(k))) / l);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 3.5e+26) {
		tmp = 2.0 / (Math.abs(t) * (k * (k * ((Math.sin(k) * Math.tan(k)) / (l * l)))));
	} else if (Math.abs(t) <= 3e+137) {
		tmp = 2.0 / (((((t_1 * Math.abs(t)) * Math.abs(t)) / l) * ((1.0 - -1.0) * Math.tan(k))) / l);
	} else {
		tmp = 2.0 / (((Math.abs(t) * ((Math.abs(t) / l) * (t_1 / l))) * Math.tan(k)) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 3.5e+26:
		tmp = 2.0 / (math.fabs(t) * (k * (k * ((math.sin(k) * math.tan(k)) / (l * l)))))
	elif math.fabs(t) <= 3e+137:
		tmp = 2.0 / (((((t_1 * math.fabs(t)) * math.fabs(t)) / l) * ((1.0 - -1.0) * math.tan(k))) / l)
	else:
		tmp = 2.0 / (((math.fabs(t) * ((math.fabs(t) / l) * (t_1 / l))) * math.tan(k)) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 3.5e+26)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 3e+137)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_1 * abs(t)) * abs(t)) / l) * Float64(Float64(1.0 - -1.0) * tan(k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 3.5e+26)
		tmp = 2.0 / (abs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	elseif (abs(t) <= 3e+137)
		tmp = 2.0 / (((((t_1 * abs(t)) * abs(t)) / l) * ((1.0 - -1.0) * tan(k))) / l);
	else
		tmp = 2.0 / (((abs(t) * ((abs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.5e+26], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+137], N[(2.0 / N[(N[(N[(N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(1.0 - -1.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.4999999999999999e26

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 3.4999999999999999e26 < t < 3.0000000000000001e137

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 63.0%

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

   if 3.0000000000000001e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 7: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.2e-37)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 3e+137)
        (* (/ l (* (* t_1 (fabs t)) (fabs t))) (/ l k))
        (/
         2.0
         (* (* (* (fabs t) (* (/ (fabs t) l) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.2e-37) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 3e+137) {
		tmp = (l / ((t_1 * fabs(t)) * fabs(t))) * (l / k);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.2e-37) {
		tmp = 2.0 / (Math.abs(t) * (k * (k * ((Math.sin(k) * Math.tan(k)) / (l * l)))));
	} else if (Math.abs(t) <= 3e+137) {
		tmp = (l / ((t_1 * Math.abs(t)) * Math.abs(t))) * (l / k);
	} else {
		tmp = 2.0 / (((Math.abs(t) * ((Math.abs(t) / l) * (t_1 / l))) * Math.tan(k)) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.2e-37:
		tmp = 2.0 / (math.fabs(t) * (k * (k * ((math.sin(k) * math.tan(k)) / (l * l)))))
	elif math.fabs(t) <= 3e+137:
		tmp = (l / ((t_1 * math.fabs(t)) * math.fabs(t))) * (l / k)
	else:
		tmp = 2.0 / (((math.fabs(t) * ((math.fabs(t) / l) * (t_1 / l))) * math.tan(k)) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.2e-37)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 3e+137)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * abs(t))) * Float64(l / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.2e-37)
		tmp = 2.0 / (abs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	elseif (abs(t) <= 3e+137)
		tmp = (l / ((t_1 * abs(t)) * abs(t))) * (l / k);
	else
		tmp = 2.0 / (((abs(t) * ((abs(t) / l) * (t_1 / l))) * tan(k)) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.2e-37], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+137], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.19999999999999995e-37

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 1.19999999999999995e-37 < t < 3.0000000000000001e137

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if 3.0000000000000001e137 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 8: 78.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(k \cdot \left(k \cdot \frac{\sin k \cdot \tan k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{elif}\;\left|t\right| \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 5.5e-53)
      (/ 2.0 (* (fabs t) (* k (* k (/ (* (sin k) (tan k)) (* l l))))))
      (if (<= (fabs t) 5.6e+102)
        (* (/ l (* (* (* (fabs t) (fabs t)) (fabs t)) k)) (/ l k))
        (/
         2.0
         (*
          (* (* (fabs t) (* (/ (fabs t) l) (/ (* k (fabs t)) l))) (tan k))
          (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) <= 5.5e-53) {
		tmp = 2.0 / (fabs(t) * (k * (k * ((sin(k) * tan(k)) / (l * l)))));
	} else if (fabs(t) <= 5.6e+102) {
		tmp = (l / (((fabs(t) * fabs(t)) * fabs(t)) * k)) * (l / k);
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * ((k * fabs(t)) / l))) * tan(k)) * 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) <= 5.5e-53)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(k * Float64(k * Float64(Float64(sin(k) * tan(k)) / Float64(l * l))))));
	elseif (abs(t) <= 5.6e+102)
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(t) * abs(t)) * abs(t)) * k)) * Float64(l / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 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], 5.5e-53], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.6e+102], N[(N[(l / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.50000000000000023e-53

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 63.2%

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

   if 5.50000000000000023e-53 < t < 5.60000000000000037e102

   1. Initial program 55.2%

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

   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.f64 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.f64 N/A

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

      5. lower-pow.f64 51.2%

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 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 55.5%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.8%

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

   6. Applied rewrites 59.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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. lower-/.f64 61.2%

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

   8. Applied rewrites 61.2%

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

   if 5.60000000000000037e102 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

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

Alternative 9: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-100)
      (/
       2.0
       (* (fabs t) (/ (pow k 4.0) (* (/ (pow l 1.0) (/ 1.0 l)) (cos k)))))
      (/
       2.0
       (*
        (* (* (fabs t) (* (/ (fabs t) l) (/ (* k (fabs t)) l))) (tan k))
        (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) <= 1.25e-100) {
		tmp = 2.0 / (fabs(t) * (pow(k, 4.0) / ((pow(l, 1.0) / (1.0 / l)) * cos(k))));
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * ((k * fabs(t)) / l))) * tan(k)) * 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) <= 1.25e-100)
		tmp = Float64(2.0 / Float64(abs(t) * Float64((k ^ 4.0) / Float64(Float64((l ^ 1.0) / Float64(1.0 / l)) * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 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], 1.25e-100], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[(N[Power[l, 1.0], $MachinePrecision] / N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.25e-100

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.6%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos k}} \]

   10. Applied rewrites 53.6%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]
   11. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos k}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\left(1 - -1\right)} \cdot \cos k}} \]

       3. pow-sub N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{{\ell}^{-1}} \cdot \cos k}} \]

       4. lower-unsound-pow.f32 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{{\ell}^{-1}} \cdot \cos k}} \]

       5. lower-pow.f32 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{{\ell}^{-1}} \cdot \cos k}} \]

       6. inv-pow N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}} \]

       7. lower-unsound-/.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}} \]

       8. lower-unsound-pow.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}} \]

       9. lower-/.f64 53.6%

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}} \]

   12. Applied rewrites 53.6%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\frac{{\ell}^{1}}{\frac{1}{\ell}} \cdot \cos k}} \]

   if 1.25e-100 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

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

Alternative 10: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.25 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.25e-100)
      (/ 2.0 (* (fabs t) (/ (pow k 4.0) (* (pow l 2.0) (cos k)))))
      (/
       2.0
       (*
        (* (* (fabs t) (* (/ (fabs t) l) (/ (* k (fabs t)) l))) (tan k))
        (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) <= 1.25e-100) {
		tmp = 2.0 / (fabs(t) * (pow(k, 4.0) / (pow(l, 2.0) * cos(k))));
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * ((k * fabs(t)) / l))) * tan(k)) * 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) <= 1.25e-100)
		tmp = Float64(2.0 / Float64(abs(t) * Float64((k ^ 4.0) / Float64((l ^ 2.0) * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 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], 1.25e-100], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.25e-100

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.6%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos k}} \]

   10. Applied rewrites 53.6%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

   if 1.25e-100 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

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

Alternative 11: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 4.5e-203)
      (/ 2.0 (* (fabs t) (/ (pow k 4.0) (pow l 2.0))))
      (/
       2.0
       (*
        (* (* (fabs t) (* (/ (fabs t) l) (/ (* k (fabs t)) l))) (tan k))
        (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) <= 4.5e-203) {
		tmp = 2.0 / (fabs(t) * (pow(k, 4.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((fabs(t) * ((fabs(t) / l) * ((k * fabs(t)) / l))) * tan(k)) * 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) <= 4.5e-203)
		tmp = Float64(2.0 / Float64(abs(t) * Float64((k ^ 4.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 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], 4.5e-203], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 4.5000000000000002e-203

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 59.4%

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      3. lower-pow.f64 52.4%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

   10. Applied rewrites 52.4%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]

   if 4.5000000000000002e-203 < t

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.8%

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

   3. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.3%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.3%

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

   5. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 75.3%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

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

Alternative 12: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.25 \cdot 10^{+159}:\\ \;\;\;\;\frac{\ell}{\left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{\left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(\left|k\right|\right)}^{4}}{{\ell}^{2}}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.25e+159)
   (* (/ l (* (* (* (sin (fabs k)) t) t) t)) (/ l (fabs k)))
   (/ 2.0 (* t (/ (pow (fabs k) 4.0) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.25e+159) {
		tmp = (l / (((sin(fabs(k)) * t) * t) * t)) * (l / fabs(k));
	} else {
		tmp = 2.0 / (t * (pow(fabs(k), 4.0) / pow(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) <= 1.25d+159) then
        tmp = (l / (((sin(abs(k)) * t) * t) * t)) * (l / abs(k))
    else
        tmp = 2.0d0 / (t * ((abs(k) ** 4.0d0) / (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) <= 1.25e+159) {
		tmp = (l / (((Math.sin(Math.abs(k)) * t) * t) * t)) * (l / Math.abs(k));
	} else {
		tmp = 2.0 / (t * (Math.pow(Math.abs(k), 4.0) / Math.pow(l, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.25e+159:
		tmp = (l / (((math.sin(math.fabs(k)) * t) * t) * t)) * (l / math.fabs(k))
	else:
		tmp = 2.0 / (t * (math.pow(math.fabs(k), 4.0) / math.pow(l, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.25e+159)
		tmp = Float64(Float64(l / Float64(Float64(Float64(sin(abs(k)) * t) * t) * t)) * Float64(l / abs(k)));
	else
		tmp = Float64(2.0 / Float64(t * Float64((abs(k) ^ 4.0) / (l ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.25e+159)
		tmp = (l / (((sin(abs(k)) * t) * t) * t)) * (l / abs(k));
	else
		tmp = 2.0 / (t * ((abs(k) ^ 4.0) / (l ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.25e+159], N[(N[(l / N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[Power[N[Abs[k], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000001e159

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if 1.25000000000000001e159 < k

   1. Initial program 55.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. lower-cos.f64 59.4%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      3. lower-pow.f64 52.4%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

   10. Applied rewrites 52.4%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;\left|t\right| \leq 10^{+89}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_1\right) \cdot \left(k \cdot \left|t\right|\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.2e-53)
      (/ 2.0 (* (fabs t) (/ (pow k 4.0) (pow l 2.0))))
      (if (<= (fabs t) 1e+89)
        (* (/ l (* (* t_1 (fabs t)) k)) (/ l k))
        (* (/ l (* (* k t_1) (* k (fabs t)))) l))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 3.2e-53) {
		tmp = 2.0 / (fabs(t) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (fabs(t) <= 1e+89) {
		tmp = (l / ((t_1 * fabs(t)) * k)) * (l / k);
	} else {
		tmp = (l / ((k * t_1) * (k * fabs(t)))) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 3.2e-53) {
		tmp = 2.0 / (Math.abs(t) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (Math.abs(t) <= 1e+89) {
		tmp = (l / ((t_1 * Math.abs(t)) * k)) * (l / k);
	} else {
		tmp = (l / ((k * t_1) * (k * Math.abs(t)))) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 3.2e-53:
		tmp = 2.0 / (math.fabs(t) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	elif math.fabs(t) <= 1e+89:
		tmp = (l / ((t_1 * math.fabs(t)) * k)) * (l / k)
	else:
		tmp = (l / ((k * t_1) * (k * math.fabs(t)))) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 3.2e-53)
		tmp = Float64(2.0 / Float64(abs(t) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (abs(t) <= 1e+89)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * k)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_1) * Float64(k * abs(t)))) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) * abs(t);
	tmp = 0.0;
	if (abs(t) <= 3.2e-53)
		tmp = 2.0 / (abs(t) * ((k ^ 4.0) / (l ^ 2.0)));
	elseif (abs(t) <= 1e+89)
		tmp = (l / ((t_1 * abs(t)) * k)) * (l / k);
	else
		tmp = (l / ((k * t_1) * (k * abs(t)))) * 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] * 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], 3.2e-53], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+89], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$1), $MachinePrecision] * N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.2000000000000001e-53

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}, \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

   4. Applied rewrites 63.4%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(2, \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}, \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. lower-cos.f64 59.4%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

      3. lower-pow.f64 52.4%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

   10. Applied rewrites 52.4%

       \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{\color{blue}{2}}}} \]

   if 3.2000000000000001e-53 < t < 9.99999999999999995e88

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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.f64 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.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 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 55.5%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.8%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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. lower-/.f64 61.2%

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 61.2%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 9.99999999999999995e88 < t

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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.f64 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.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 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 55.5%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.8%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.8%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 63.0%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 63.0%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.0% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* k (* t t)) (* k t))) l))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * 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 / ((k * (t * t)) * (k * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (k * t))) * l;
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * (k * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * Float64(k * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * (k * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

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.f64 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.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.2%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 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 55.5%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 59.8%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.8%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 59.8%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.8%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. lower-*.f64 63.0%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 63.0%

    \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 15: 58.4% accurate, 6.6× speedup

Math

\[\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* t (* (* t t) (* k k)))) l))

C

double code(double t, double l, double k) {
	return (l / (t * ((t * t) * (k * 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 * ((t * t) * (k * k)))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (t * ((t * t) * (k * k)))) * l;
}

Python

def code(t, l, k):
	return (l / (t * ((t * t) * (k * k)))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(t * Float64(Float64(t * t) * Float64(k * k)))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (t * ((t * t) * (k * k)))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(t * N[(N[(t * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

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.f64 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.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.2%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 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 55.5%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 59.8%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.8%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 59.8%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.8%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. unpow2 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot {k}^{2}} \cdot \ell \]

   7. associate-*l* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot {k}^{2}\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot {k}^{2}\right)} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot {k}^{2}\right)} \cdot \ell \]

   10. unpow2 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   11. lower-*.f64 58.4%

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

10. Applied rewrites 58.4%

    \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025185 
(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))))