Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.0% → 83.9%

Time: 8.6s

Alternatives: 19

Speedup: 6.8×

• 72.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% 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: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left|t\right| \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}} \cdot 2\\ \mathbf{elif}\;\left|t\right| \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(\left(t\_2 \cdot \left|t\right|\right) \cdot \mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k \cdot \left(t\_1 \cdot \tan k\right), \left(2 \cdot t\_1\right) \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(\left|t\right| \cdot \frac{\left|t\right| \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (sin k) l)) (t_2 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.7e-94)
     (*
      (/
       (* l l)
       (* (fabs t) (/ (* (pow k 2.0) (pow (sin k) 2.0)) (cos k))))
      2.0)
     (if (<= (fabs t) 4.4e+17)
       (/
        2.0
        (*
         (fabs t)
         (*
          (* t_2 (fabs t))
          (fma
           (/ k (* (fabs t) (fabs t)))
           (* k (* t_1 (tan k)))
           (* (* 2.0 t_1) (tan k))))))
       (/
        2.0
        (*
         (* (* t_2 (* (fabs t) (/ (* (fabs t) (sin k)) l))) (tan k))
         (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) / l;
	double t_2 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.7e-94) {
		tmp = ((l * l) / (fabs(t) * ((pow(k, 2.0) * pow(sin(k), 2.0)) / cos(k)))) * 2.0;
	} else if (fabs(t) <= 4.4e+17) {
		tmp = 2.0 / (fabs(t) * ((t_2 * fabs(t)) * fma((k / (fabs(t) * fabs(t))), (k * (t_1 * tan(k))), ((2.0 * t_1) * tan(k)))));
	} else {
		tmp = 2.0 / (((t_2 * (fabs(t) * ((fabs(t) * sin(k)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) / l)
	t_2 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.7e-94)
		tmp = Float64(Float64(Float64(l * l) / Float64(abs(t) * Float64(Float64((k ^ 2.0) * (sin(k) ^ 2.0)) / cos(k)))) * 2.0);
	elseif (abs(t) <= 4.4e+17)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(Float64(t_2 * abs(t)) * fma(Float64(k / Float64(abs(t) * abs(t))), Float64(k * Float64(t_1 * tan(k))), Float64(Float64(2.0 * t_1) * tan(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(abs(t) * Float64(Float64(abs(t) * sin(k)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.7e-94], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 4.4e+17], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[(t$95$2 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.6999999999999999e-94

   1. Initial program 54.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.4%

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 58.7%

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

   8. Applied rewrites 58.7%

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

   if 1.6999999999999999e-94 < t < 4.4e17

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 61.0%

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

   7. Applied rewrites 61.0%

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

   if 4.4e17 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

Alternative 2: 82.2% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left|t\right| \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{\left|t\right| \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 2.15e-69)
   (*
    (/
     (* l l)
     (* (fabs t) (/ (* (pow k 2.0) (pow (sin k) 2.0)) (cos k))))
    2.0)
   (/
    2.0
    (*
     (*
      (* (/ (fabs t) l) (* (fabs t) (/ (* (fabs t) (sin k)) l)))
      (tan k))
     (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.15e-69) {
		tmp = ((l * l) / (fabs(t) * ((pow(k, 2.0) * pow(sin(k), 2.0)) / cos(k)))) * 2.0;
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * ((fabs(t) * sin(k)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 2.15e-69) {
		tmp = ((l * l) / (Math.abs(t) * ((Math.pow(k, 2.0) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)))) * 2.0;
	} else {
		tmp = 2.0 / ((((Math.abs(t) / l) * (Math.abs(t) * ((Math.abs(t) * Math.sin(k)) / l))) * Math.tan(k)) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 2.15e-69:
		tmp = ((l * l) / (math.fabs(t) * ((math.pow(k, 2.0) * math.pow(math.sin(k), 2.0)) / math.cos(k)))) * 2.0
	else:
		tmp = 2.0 / ((((math.fabs(t) / l) * (math.fabs(t) * ((math.fabs(t) * math.sin(k)) / l))) * math.tan(k)) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.15e-69)
		tmp = Float64(Float64(Float64(l * l) / Float64(abs(t) * Float64(Float64((k ^ 2.0) * (sin(k) ^ 2.0)) / cos(k)))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(Float64(abs(t) * sin(k)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 2.15e-69)
		tmp = ((l * l) / (abs(t) * (((k ^ 2.0) * (sin(k) ^ 2.0)) / cos(k)))) * 2.0;
	else
		tmp = 2.0 / ((((abs(t) / l) * (abs(t) * ((abs(t) * sin(k)) / l))) * tan(k)) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.15e-69], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.15e-69

   1. Initial program 54.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.4%

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 58.7%

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

   8. Applied rewrites 58.7%

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

   if 2.15e-69 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left|t\right| \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ k (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2.15e-69)
     (*
      (/
       (* l l)
       (* (fabs t) (/ (* (pow k 2.0) (pow (sin k) 2.0)) (cos k))))
      2.0)
     (/
      2.0
      (*
       (*
        (* (/ (* (sin k) (fabs t)) l) (* (/ (fabs t) l) (fabs t)))
        (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) <= 2.15e-69) {
		tmp = ((l * l) / (fabs(t) * ((pow(k, 2.0) * pow(sin(k), 2.0)) / cos(k)))) * 2.0;
	} else {
		tmp = 2.0 / (((((sin(k) * fabs(t)) / l) * ((fabs(t) / l) * fabs(t))) * 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) <= 2.15e-69)
		tmp = Float64(Float64(Float64(l * l) / Float64(abs(t) * Float64(Float64((k ^ 2.0) * (sin(k) ^ 2.0)) / cos(k)))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * Float64(Float64(abs(t) / l) * abs(t))) * 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], 2.15e-69], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $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 < 2.15e-69

   1. Initial program 54.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.4%

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 58.7%

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

   8. Applied rewrites 58.7%

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

   if 2.15e-69 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-+l+ N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-*.f64 N/A

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

      17. frac-times N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-fma.f64 74.4%

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

   7. Applied rewrites 74.4%

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

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot \left(\left|t\right| \cdot {\sin k}^{2}\right)}{\cos k}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ k (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2.15e-69)
     (*
      (/
       (* l l)
       (/ (* (pow k 2.0) (* (fabs t) (pow (sin k) 2.0))) (cos k)))
      2.0)
     (/
      2.0
      (*
       (*
        (* (/ (* (sin k) (fabs t)) l) (* (/ (fabs t) l) (fabs t)))
        (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) <= 2.15e-69) {
		tmp = ((l * l) / ((pow(k, 2.0) * (fabs(t) * pow(sin(k), 2.0))) / cos(k))) * 2.0;
	} else {
		tmp = 2.0 / (((((sin(k) * fabs(t)) / l) * ((fabs(t) / l) * fabs(t))) * 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) <= 2.15e-69)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64((k ^ 2.0) * Float64(abs(t) * (sin(k) ^ 2.0))) / cos(k))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * Float64(Float64(abs(t) / l) * abs(t))) * 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], 2.15e-69], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $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 < 2.15e-69

   1. Initial program 54.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 45.4%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 59.4%

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

   6. Applied rewrites 59.4%

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

   if 2.15e-69 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-+l+ N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-*.f64 N/A

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

      17. frac-times N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-fma.f64 74.4%

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

   7. Applied rewrites 74.4%

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

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot t\_1\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|t\right| \leq 6.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\left|t\right| \leq 8.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\frac{\ell}{\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(\left|t\right| \cdot \frac{\left|t\right| \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.9e-211)
     (/
      2.0
      (*
       (*
        (fabs t)
        (*
         (fabs t)
         (*
          (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
          t_1)))
       2.0))
     (if (<= (fabs t) 6.9e-47)
       (/
        2.0
        (*
         (* (* t_1 (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
         (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))
       (if (<= (fabs t) 8.5e+113)
         (*
          (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t)))
          (*
           (/ l (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k)))
           2.0))
         (/
          2.0
          (*
           (* (* t_1 (* (fabs t) (/ (* (fabs t) (sin k)) l))) (tan k))
           (+ 1.0 1.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.9e-211) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * t_1))) * 2.0);
	} else if (fabs(t) <= 6.9e-47) {
		tmp = 2.0 / (((t_1 * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	} else if (fabs(t) <= 8.5e+113) {
		tmp = (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * ((l / (fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k))) * 2.0);
	} else {
		tmp = 2.0 / (((t_1 * (fabs(t) * ((fabs(t) * sin(k)) / l))) * tan(k)) * (1.0 + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.9e-211)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * t_1))) * 2.0));
	elseif (abs(t) <= 6.9e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	elseif (abs(t) <= 8.5e+113)
		tmp = Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * Float64(Float64(l / Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(abs(t) * Float64(Float64(abs(t) * sin(k)) / l))) * tan(k)) * Float64(1.0 + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.9e-211], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6.9e-47], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 8.5e+113], N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.9000000000000001e-211

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow3 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.0%

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

   3. Applied rewrites 60.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 57.6%

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

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 64.0%

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

   11. Applied rewrites 64.0%

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

   if 1.9000000000000001e-211 < t < 6.8999999999999999e-47

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.3%

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

   10. Applied rewrites 70.3%

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

   if 6.8999999999999999e-47 < t < 8.5000000000000001e113

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 56.6%

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

   if 8.5000000000000001e113 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 67.4%

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

Alternative 6: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ t_2 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot t\_2\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left(t\_2 \cdot \left|t\right|\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ k (fabs t))) (t_2 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2e-211)
     (/
      2.0
      (*
       (*
        (fabs t)
        (*
         (fabs t)
         (*
          (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
          t_2)))
       2.0))
     (/
      2.0
      (*
       (* (* (/ (* (sin k) (fabs t)) l) (* t_2 (fabs t))) (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 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 2e-211) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * t_2))) * 2.0);
	} else {
		tmp = 2.0 / (((((sin(k) * fabs(t)) / l) * (t_2 * fabs(t))) * 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(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 2e-211)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * t_2))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * Float64(t_2 * abs(t))) * 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[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2e-211], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$2 * N[Abs[t], $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 < 2.0000000000000002e-211

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow3 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.0%

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

   3. Applied rewrites 60.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 57.6%

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

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 64.0%

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

   11. Applied rewrites 64.0%

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

   if 2.0000000000000002e-211 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-+l+ N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-*.f64 N/A

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

      17. frac-times N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lower-fma.f64 74.4%

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

   7. Applied rewrites 74.4%

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

Alternative 7: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot t\_1\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|t\right| \leq 6.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\left|t\right| \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \left(\ell + \ell\right)}{\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(\left|t\right| \cdot \frac{\left|t\right| \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.9e-211)
     (/
      2.0
      (*
       (*
        (fabs t)
        (*
         (fabs t)
         (*
          (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
          t_1)))
       2.0))
     (if (<= (fabs t) 6.9e-47)
       (/
        2.0
        (*
         (* (* t_1 (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
         (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))
       (if (<= (fabs t) 5e+80)
         (/
          (*
           (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t)))
           (+ l l))
          (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k)))
         (/
          2.0
          (*
           (* (* t_1 (* (fabs t) (/ (* (fabs t) (sin k)) l))) (tan k))
           (+ 1.0 1.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.9e-211) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * t_1))) * 2.0);
	} else if (fabs(t) <= 6.9e-47) {
		tmp = 2.0 / (((t_1 * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	} else if (fabs(t) <= 5e+80) {
		tmp = ((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * (l + l)) / (fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k));
	} else {
		tmp = 2.0 / (((t_1 * (fabs(t) * ((fabs(t) * sin(k)) / l))) * tan(k)) * (1.0 + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.9e-211)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * t_1))) * 2.0));
	elseif (abs(t) <= 6.9e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	elseif (abs(t) <= 5e+80)
		tmp = Float64(Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * Float64(l + l)) / Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(abs(t) * Float64(Float64(abs(t) * sin(k)) / l))) * tan(k)) * Float64(1.0 + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.9e-211], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6.9e-47], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5e+80], N[(N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.9000000000000001e-211

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow3 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.0%

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

   3. Applied rewrites 60.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 57.6%

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

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 64.0%

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

   11. Applied rewrites 64.0%

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

   if 1.9000000000000001e-211 < t < 6.8999999999999999e-47

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.3%

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

   10. Applied rewrites 70.3%

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

   if 6.8999999999999999e-47 < t < 4.9999999999999996e80

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 55.0%

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

   if 4.9999999999999996e80 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 67.4%

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

Alternative 8: 77.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 46000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot t\_1}{\ell}\right)\right) \cdot t\_2\right) \cdot \left(1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\frac{t\_1}{\ell} \cdot t\_2\right) \cdot \mathsf{fma}\left(\frac{\frac{\left|k\right|}{t}}{t}, \left|k\right|, 2\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))))
  (if (<= (fabs k) 46000000000000.0)
    (/ 2.0 (* (* (* (/ t l) (* t (/ (* t t_1) l))) t_2) (+ 1.0 1.0)))
    (/
     2.0
     (*
      t
      (*
       (* (/ t l) t)
       (*
        (* (/ t_1 l) t_2)
        (fma (/ (/ (fabs k) t) t) (fabs k) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double t_2 = tan(fabs(k));
	double tmp;
	if (fabs(k) <= 46000000000000.0) {
		tmp = 2.0 / ((((t / l) * (t * ((t * t_1) / l))) * t_2) * (1.0 + 1.0));
	} else {
		tmp = 2.0 / (t * (((t / l) * t) * (((t_1 / l) * t_2) * fma(((fabs(k) / t) / t), fabs(k), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 46000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(t * t_1) / l))) * t_2) * Float64(1.0 + 1.0)));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t / l) * t) * Float64(Float64(Float64(t_1 / l) * t_2) * fma(Float64(Float64(abs(k) / t) / t), abs(k), 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 46000000000000.0], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(t * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(t$95$1 / l), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[(N[Abs[k], $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision] * N[Abs[k], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e13

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.5%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 67.4%

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

   if 4.6e13 < k

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

   5. Applied rewrites 62.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 68.6%

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

   7. Applied rewrites 68.6%

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

Alternative 9: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(t \cdot \frac{k \cdot t}{\left|\ell\right|}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(t \cdot \frac{t \cdot \sin k}{\left|\ell\right|}\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ t (fabs l))))
  (if (<= (fabs l) 3.5e-74)
    (/
     2.0
     (*
      (* (* t_1 (* t (/ (* k t) (fabs l)))) (tan k))
      (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
    (/
     2.0
     (*
      (* (* t_1 (* t (/ (* t (sin k)) (fabs l)))) (tan k))
      (+ 1.0 1.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 3.5e-74) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) / fabs(l)))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_1 * (t * ((t * sin(k)) / fabs(l)))) * tan(k)) * (1.0 + 1.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 3.5d-74) then
        tmp = 2.0d0 / (((t_1 * (t * ((k * t) / abs(l)))) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((t_1 * (t * ((t * sin(k)) / abs(l)))) * tan(k)) * (1.0d0 + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 3.5e-74) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) / Math.abs(l)))) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_1 * (t * ((t * Math.sin(k)) / Math.abs(l)))) * Math.tan(k)) * (1.0 + 1.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 3.5e-74:
		tmp = 2.0 / (((t_1 * (t * ((k * t) / math.fabs(l)))) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((t_1 * (t * ((t * math.sin(k)) / math.fabs(l)))) * math.tan(k)) * (1.0 + 1.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 3.5e-74)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * Float64(Float64(k * t) / abs(l)))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * Float64(Float64(t * sin(k)) / abs(l)))) * tan(k)) * Float64(1.0 + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 3.5e-74)
		tmp = 2.0 / (((t_1 * (t * ((k * t) / abs(l)))) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((t_1 * (t * ((t * sin(k)) / abs(l)))) * tan(k)) * (1.0 + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 3.5e-74], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $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], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.5000000000000001e-74

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 65.6%

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

   3. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\color{blue}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 70.3%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 70.3%

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 3.5000000000000001e-74 < l

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 65.6%

          \[\leadsto \frac{2}{\left(\frac{\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 65.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell}} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot t\right) \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.4%

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(t \cdot \frac{k \cdot t}{\left|\ell\right|}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\left|\ell\right| \cdot \frac{\left|\ell\right|}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(t\_1 \cdot t\right) \cdot \left(\left(\frac{\sin k}{\left|\ell\right|} \cdot \tan k\right) \cdot 2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ t (fabs l))))
  (if (<= (fabs l) 1.6e-61)
    (/
     2.0
     (*
      (* (* t_1 (* t (/ (* k t) (fabs l)))) (tan k))
      (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
    (if (<= (fabs l) 2.4e+131)
      (* (fabs l) (/ (fabs l) (* (* (* k t) t) (* t k))))
      (/
       2.0
       (*
        t
        (* (* t_1 t) (* (* (/ (sin k) (fabs l)) (tan k)) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 1.6e-61) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) / fabs(l)))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else if (fabs(l) <= 2.4e+131) {
		tmp = fabs(l) * (fabs(l) / (((k * t) * t) * (t * k)));
	} else {
		tmp = 2.0 / (t * ((t_1 * t) * (((sin(k) / fabs(l)) * tan(k)) * 2.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(l)
    if (abs(l) <= 1.6d-61) then
        tmp = 2.0d0 / (((t_1 * (t * ((k * t) / abs(l)))) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else if (abs(l) <= 2.4d+131) then
        tmp = abs(l) * (abs(l) / (((k * t) * t) * (t * k)))
    else
        tmp = 2.0d0 / (t * ((t_1 * t) * (((sin(k) / abs(l)) * tan(k)) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 1.6e-61) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) / Math.abs(l)))) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else if (Math.abs(l) <= 2.4e+131) {
		tmp = Math.abs(l) * (Math.abs(l) / (((k * t) * t) * (t * k)));
	} else {
		tmp = 2.0 / (t * ((t_1 * t) * (((Math.sin(k) / Math.abs(l)) * Math.tan(k)) * 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 1.6e-61:
		tmp = 2.0 / (((t_1 * (t * ((k * t) / math.fabs(l)))) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	elif math.fabs(l) <= 2.4e+131:
		tmp = math.fabs(l) * (math.fabs(l) / (((k * t) * t) * (t * k)))
	else:
		tmp = 2.0 / (t * ((t_1 * t) * (((math.sin(k) / math.fabs(l)) * math.tan(k)) * 2.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 1.6e-61)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * Float64(Float64(k * t) / abs(l)))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	elseif (abs(l) <= 2.4e+131)
		tmp = Float64(abs(l) * Float64(abs(l) / Float64(Float64(Float64(k * t) * t) * Float64(t * k))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(t_1 * t) * Float64(Float64(Float64(sin(k) / abs(l)) * tan(k)) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 1.6e-61)
		tmp = 2.0 / (((t_1 * (t * ((k * t) / abs(l)))) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	elseif (abs(l) <= 2.4e+131)
		tmp = abs(l) * (abs(l) / (((k * t) * t) * (t * k)));
	else
		tmp = 2.0 / (t * ((t_1 * t) * (((sin(k) / abs(l)) * tan(k)) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 1.6e-61], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $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], If[LessEqual[N[Abs[l], $MachinePrecision], 2.4e+131], N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[(N[(k * t), $MachinePrecision] * t), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(t$95$1 * t), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.6000000000000001e-61

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 65.6%

          \[\leadsto \frac{2}{\left(\frac{\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 65.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell}} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot t\right) \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\color{blue}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 70.3%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 70.3%

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 1.6000000000000001e-61 < l < 2.3999999999999999e131

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      6. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      9. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   10. Applied rewrites 66.3%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   if 2.3999999999999999e131 < l

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 60.0%

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right)} \cdot 2} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell} \cdot 2\right)\right)}} \]

   8. Applied rewrites 62.4%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot 2\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot t\_1\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.9e-211)
     (/
      2.0
      (*
       (*
        (fabs t)
        (*
         (fabs t)
         (*
          (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
          t_1)))
       2.0))
     (/
      2.0
      (*
       (* (* t_1 (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
       (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.9e-211) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * t_1))) * 2.0);
	} else {
		tmp = 2.0 / (((t_1 * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.9e-211)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * t_1))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.9e-211], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9000000000000001e-211

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 60.0%

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}\right)\right) \cdot 2} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\color{blue}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      6. lower-/.f64 57.6%

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)} \cdot 2} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

       3. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)} \cdot 2} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right) \cdot 2} \]

       5. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)}\right) \cdot 2} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

       9. lower-*.f64 64.0%

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

   11. Applied rewrites 64.0%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right)\right)\right)} \cdot 2} \]

   if 1.9000000000000001e-211 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 65.6%

          \[\leadsto \frac{2}{\left(\frac{\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 65.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell}} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot t\right) \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 75.5%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\color{blue}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 70.3%

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 70.3%

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.76 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot t\_1\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|t\right| \leq 1.12 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \left(\left(t\_1 \cdot \left|t\right|\right) \cdot \left(\left(\frac{k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left(\left|t\right| \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (/ (fabs t) l)))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.76e-102)
     (/
      2.0
      (*
       (*
        (fabs t)
        (*
         (fabs t)
         (*
          (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
          t_1)))
       2.0))
     (if (<= (fabs t) 1.12e-16)
       (/
        2.0
        (*
         (fabs t)
         (*
          (* t_1 (fabs t))
          (*
           (* (/ k l) (tan k))
           (fma (/ k (* (fabs t) (fabs t))) k 2.0)))))
       (* l (/ l (* (* (* k (fabs t)) (fabs t)) (* (fabs t) k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.76e-102) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * t_1))) * 2.0);
	} else if (fabs(t) <= 1.12e-16) {
		tmp = 2.0 / (fabs(t) * ((t_1 * fabs(t)) * (((k / l) * tan(k)) * fma((k / (fabs(t) * fabs(t))), k, 2.0))));
	} else {
		tmp = l * (l / (((k * fabs(t)) * fabs(t)) * (fabs(t) * k)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.76e-102)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * t_1))) * 2.0));
	elseif (abs(t) <= 1.12e-16)
		tmp = Float64(2.0 / Float64(abs(t) * Float64(Float64(t_1 * abs(t)) * Float64(Float64(Float64(k / l) * tan(k)) * fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0)))));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * abs(t)) * abs(t)) * Float64(abs(t) * k))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.76e-102], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.12e-16], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.7599999999999999e-102

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 60.0%

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}\right)\right) \cdot 2} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\color{blue}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      6. lower-/.f64 57.6%

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)} \cdot 2} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

       3. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)} \cdot 2} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right) \cdot 2} \]

       5. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)}\right) \cdot 2} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

       9. lower-*.f64 64.0%

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

   11. Applied rewrites 64.0%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right)\right)\right)} \cdot 2} \]

   if 1.7599999999999999e-102 < t < 1.12e-16

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \sin k}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 65.6%

          \[\leadsto \frac{2}{\left(\frac{\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 65.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k}{\ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 62.3%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\color{blue}{\frac{k}{\ell}} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 55.8%

         \[\leadsto \frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\frac{k}{\color{blue}{\ell}} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right)\right)} \]

   8. Applied rewrites 55.8%

      \[\leadsto \frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left(\color{blue}{\frac{k}{\ell}} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right)\right)} \]

   if 1.12e-16 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      6. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      9. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   10. Applied rewrites 66.3%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.7% accurate, 2.6× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{\left|t\right|}{\ell}\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left(\left|t\right| \cdot k\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 7.4e-19)
   (/
    2.0
    (*
     (*
      (fabs t)
      (*
       (fabs t)
       (*
        (* (/ (fma 0.16666666666666666 (* k k) 1.0) l) (* k k))
        (/ (fabs t) l))))
     2.0))
   (* l (/ l (* (* (* k (fabs t)) (fabs t)) (* (fabs t) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 7.4e-19) {
		tmp = 2.0 / ((fabs(t) * (fabs(t) * (((fma(0.16666666666666666, (k * k), 1.0) / l) * (k * k)) * (fabs(t) / l)))) * 2.0);
	} else {
		tmp = l * (l / (((k * fabs(t)) * fabs(t)) * (fabs(t) * k)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 7.4e-19)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(abs(t) * Float64(Float64(Float64(fma(0.16666666666666666, Float64(k * k), 1.0) / l) * Float64(k * k)) * Float64(abs(t) / l)))) * 2.0));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * abs(t)) * abs(t)) * Float64(abs(t) * k))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 7.4e-19], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.4000000000000001e-19

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \frac{\sin k \cdot \tan k}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 60.0%

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{\tan k \cdot \sin k}{\ell}\right) \cdot \color{blue}{2}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)}\right)\right) \cdot 2} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\color{blue}{\ell}}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

      6. lower-/.f64 57.6%

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

   9. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)}\right) \cdot 2} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)} \cdot 2} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right) \cdot 2} \]

       3. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)} \cdot 2} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right) \cdot 2} \]

       5. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)\right)} \cdot 2} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right)\right)\right)}\right) \cdot 2} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

       9. lower-*.f64 64.0%

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot \color{blue}{\left(\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2}}{\ell}, \frac{1}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)}\right)\right) \cdot 2} \]

   11. Applied rewrites 64.0%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot \left(\left(\frac{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right)\right)\right)} \cdot 2} \]

   if 7.4000000000000001e-19 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      6. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      9. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   10. Applied rewrites 66.3%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 70.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 10^{+299}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (/
      2.0
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     1e+299)
  (* (/ l (* (* t t) k)) (/ l (* k t)))
  (* l (/ (/ (/ l (* (* k k) t)) t) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))) <= 1e+299) {
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	} else {
		tmp = l * (((l / ((k * k) * t)) / t) / t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))) <= 1d+299) then
        tmp = (l / ((t * t) * k)) * (l / (k * t))
    else
        tmp = l * (((l / ((k * k) * t)) / t) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= 1e+299) {
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	} else {
		tmp = l * (((l / ((k * k) * t)) / t) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= 1e+299:
		tmp = (l / ((t * t) * k)) * (l / (k * t))
	else:
		tmp = l * (((l / ((k * k) * t)) / t) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= 1e+299)
		tmp = Float64(Float64(l / Float64(Float64(t * t) * k)) * Float64(l / Float64(k * t)));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(Float64(k * k) * t)) / t) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0))) <= 1e+299)
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	else
		tmp = l * (((l / ((k * k) * t)) / t) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 1e+299], N[(N[(l / N[(N[(t * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.0000000000000001e299

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{t \cdot k} \]

      8. lower-/.f64 65.1%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{t \cdot \color{blue}{k}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{t}} \]

      11. lower-*.f64 65.1%

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 1.0000000000000001e299 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t \cdot \color{blue}{t}} \]

      5. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{t} \]

      8. lower-/.f64 63.3%

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{t} \]

   8. Applied rewrites 63.3%

      \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq \infty:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<=
     (/
      2.0
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     INFINITY)
  (* (/ l (* (* t t) k)) (/ l (* k t)))
  (* l (/ (/ l (* (* (* k k) t) t)) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))) <= ((double) INFINITY)) {
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	} else {
		tmp = l * ((l / (((k * k) * t) * t)) / t);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= Double.POSITIVE_INFINITY) {
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	} else {
		tmp = l * ((l / (((k * k) * t) * t)) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= math.inf:
		tmp = (l / ((t * t) * k)) * (l / (k * t))
	else:
		tmp = l * ((l / (((k * k) * t) * t)) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= Inf)
		tmp = Float64(Float64(l / Float64(Float64(t * t) * k)) * Float64(l / Float64(k * t)));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(Float64(Float64(k * k) * t) * t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0))) <= Inf)
		tmp = (l / ((t * t) * k)) * (l / (k * t));
	else
		tmp = l * ((l / (((k * k) * t) * t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], Infinity], N[(N[(l / N[(N[(t * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{t \cdot k} \]

      8. lower-/.f64 65.1%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{t \cdot \color{blue}{k}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{t}} \]

      11. lower-*.f64 65.1%

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

      8. lower-*.f64 63.2%

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 6 \cdot 10^{-125}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(\left|k\right| \cdot t\right) \cdot t\right) \cdot \left(t \cdot \left|k\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot t}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 6e-125)
  (* l (/ l (* (* (* (fabs k) t) t) (* t (fabs k)))))
  (* l (/ (/ l (* (* (* (fabs k) (fabs k)) t) t)) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 6e-125) {
		tmp = l * (l / (((fabs(k) * t) * t) * (t * fabs(k))));
	} else {
		tmp = l * ((l / (((fabs(k) * fabs(k)) * t) * t)) / t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 6d-125) then
        tmp = l * (l / (((abs(k) * t) * t) * (t * abs(k))))
    else
        tmp = l * ((l / (((abs(k) * abs(k)) * t) * t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 6e-125) {
		tmp = l * (l / (((Math.abs(k) * t) * t) * (t * Math.abs(k))));
	} else {
		tmp = l * ((l / (((Math.abs(k) * Math.abs(k)) * t) * t)) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 6e-125:
		tmp = l * (l / (((math.fabs(k) * t) * t) * (t * math.fabs(k))))
	else:
		tmp = l * ((l / (((math.fabs(k) * math.fabs(k)) * t) * t)) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 6e-125)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(abs(k) * t) * t) * Float64(t * abs(k)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(Float64(Float64(abs(k) * abs(k)) * t) * t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 6e-125)
		tmp = l * (l / (((abs(k) * t) * t) * (t * abs(k))));
	else
		tmp = l * ((l / (((abs(k) * abs(k)) * t) * t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 6e-125], N[(l * N[(l / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(t * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999998e-125

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      6. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      9. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   10. Applied rewrites 66.3%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   if 5.9999999999999998e-125 < k

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

      8. lower-*.f64 63.2%

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 69.2% accurate, 4.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left(\left|t\right| \cdot k\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5e+16)
   (* l (/ l (* (* (* (* k k) (fabs t)) (fabs t)) (fabs t))))
   (* l (/ l (* (* (* k (fabs t)) (fabs t)) (* (fabs t) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 5e+16) {
		tmp = l * (l / ((((k * k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = l * (l / (((k * fabs(t)) * fabs(t)) * (fabs(t) * k)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 5e+16) {
		tmp = l * (l / ((((k * k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t)));
	} else {
		tmp = l * (l / (((k * Math.abs(t)) * Math.abs(t)) * (Math.abs(t) * k)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 5e+16:
		tmp = l * (l / ((((k * k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t)))
	else:
		tmp = l * (l / (((k * math.fabs(t)) * math.fabs(t)) * (math.fabs(t) * k)))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 5e+16)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(Float64(k * k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * abs(t)) * abs(t)) * Float64(abs(t) * k))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 5e+16)
		tmp = l * (l / ((((k * k) * abs(t)) * abs(t)) * abs(t)));
	else
		tmp = l * (l / (((k * abs(t)) * abs(t)) * (abs(t) * k)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5e+16], N[(l * N[(l / N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5e16

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. lower-*.f64 62.1%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \]

   8. Applied rewrites 62.1%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

   if 5e16 < t

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   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 50.3%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.0%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      16. lower-*.f64 58.1%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      5. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      10. lower-*.f64 63.2%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      6. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

      9. lower-*.f64 66.3%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   10. Applied rewrites 66.3%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.3% accurate, 6.8× speedup

Math

\[\ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* l (/ l (* (* (* k t) t) (* t k)))))

C

double code(double t, double l, double k) {
	return l * (l / (((k * t) * t) * (t * k)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l / (((k * t) * t) * (t * k)))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l / (((k * t) * t) * (t * k)));
}

Python

def code(t, l, k):
	return l * (l / (((k * t) * t) * (t * k)))

Julia

function code(t, l, k)
	return Float64(l * Float64(l / Float64(Float64(Float64(k * t) * t) * Float64(t * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l / (((k * t) * t) * (t * k)));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l / N[(N[(N[(k * t), $MachinePrecision] * t), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

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 50.3%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.0%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. cube-mult N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   15. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   16. lower-*.f64 58.1%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

6. Applied rewrites 58.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   5. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   10. lower-*.f64 63.2%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

8. Applied rewrites 63.2%

   \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

   5. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   6. lower-*.f64 66.3%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

   9. lower-*.f64 66.3%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(t \cdot k\right)} \]

10. Applied rewrites 66.3%

    \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]
11. Add Preprocessing

Alternative 19: 65.9% accurate, 6.8× speedup

Math

\[\ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* l (/ l (* t (* (* k t) (* k t))))))

C

double code(double t, double l, double k) {
	return l * (l / (t * ((k * t) * (k * t))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l / (t * ((k * t) * (k * t))))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l / (t * ((k * t) * (k * t))));
}

Python

def code(t, l, k):
	return l * (l / (t * ((k * t) * (k * t))))

Julia

function code(t, l, k)
	return Float64(l * Float64(l / Float64(t * Float64(Float64(k * t) * Float64(k * t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l / (t * ((k * t) * (k * t))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l / N[(t * N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

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 50.3%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.0%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. cube-mult N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   15. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   16. lower-*.f64 58.1%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

6. Applied rewrites 58.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   5. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   10. lower-*.f64 63.2%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

8. Applied rewrites 63.2%

   \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \]

   4. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot k\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \]

   6. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}} \]

   8. lower-*.f64 65.9%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   10. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   11. lower-*.f64 65.9%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   14. lower-*.f64 65.9%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

10. Applied rewrites 65.9%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))