Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.9% → 93.2%

Time: 8.6s

Alternatives: 19

Speedup: 6.6×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.9% 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: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left|t\right|\right) \cdot k}{\left(-\ell\right) \cdot \cos k} \cdot \frac{-k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(t\_1, t\_1, 2\right) \cdot \left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 5.2e-76)
      (/
       2.0
       (*
        (/ (* (* (fma (cos (+ k k)) -0.5 0.5) (fabs t)) k) (* (- l) (cos k)))
        (/ (- k) l)))
      (/
       2.0
       (*
        (* (fma t_1 t_1 2.0) (* (/ (* (sin k) (fabs t)) l) (fabs t)))
        (* (/ (fabs t) l) (tan k))))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double tmp;
	if (fabs(t) <= 5.2e-76) {
		tmp = 2.0 / ((((fma(cos((k + k)), -0.5, 0.5) * fabs(t)) * k) / (-l * cos(k))) * (-k / l));
	} else {
		tmp = 2.0 / ((fma(t_1, t_1, 2.0) * (((sin(k) * fabs(t)) / l) * fabs(t))) * ((fabs(t) / l) * tan(k)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(t))
	tmp = 0.0
	if (abs(t) <= 5.2e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * abs(t)) * k) / Float64(Float64(-l) * cos(k))) * Float64(Float64(-k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(t_1, t_1, 2.0) * Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))) * Float64(Float64(abs(t) / l) * tan(k))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e-76], N[(2.0 / N[(N[(N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[((-l) * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-k) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.1999999999999999e-76

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-neg.f64 60.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 59.2%

      \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\cos k \cdot \left(\left(-\ell\right) \cdot \ell\right)}{-\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 5.1999999999999999e-76 < t

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 80.7%

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

Alternative 2: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left|t\right|\right) \cdot k}{\left(-\ell\right) \cdot \cos k} \cdot \frac{-k}{\ell}}\\ \mathbf{elif}\;\left|t\right| \leq 1.7 \cdot 10^{+105}:\\ \;\;\;\;\left(\frac{2}{\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k} \cdot \ell\right) \cdot \frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{\left|t\right|}{\ell}\right) \cdot \left(\frac{t\_1}{\ell} \cdot \left|t\right|\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 6.2e-76)
      (/
       2.0
       (*
        (/ (* (* (fma (cos (+ k k)) -0.5 0.5) (fabs t)) k) (* (- l) (cos k)))
        (/ (- k) l)))
      (if (<= (fabs t) 1.7e+105)
        (*
         (* (/ 2.0 (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k))) l)
         (/ l (* (* t_1 (fabs t)) (fabs t))))
        (/
         2.0
         (* (* (* (tan k) (/ (fabs t) l)) (* (/ t_1 l) (fabs t))) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 6.2e-76) {
		tmp = 2.0 / ((((fma(cos((k + k)), -0.5, 0.5) * fabs(t)) * k) / (-l * cos(k))) * (-k / l));
	} else if (fabs(t) <= 1.7e+105) {
		tmp = ((2.0 / (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))) * l) * (l / ((t_1 * fabs(t)) * fabs(t)));
	} else {
		tmp = 2.0 / (((tan(k) * (fabs(t) / l)) * ((t_1 / l) * fabs(t))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 6.2e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * abs(t)) * k) / Float64(Float64(-l) * cos(k))) * Float64(Float64(-k) / l)));
	elseif (abs(t) <= 1.7e+105)
		tmp = Float64(Float64(Float64(2.0 / Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k))) * l) * Float64(l / Float64(Float64(t_1 * abs(t)) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(abs(t) / l)) * Float64(Float64(t_1 / l) * abs(t))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.2e-76], N[(2.0 / N[(N[(N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[((-l) * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-k) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.7e+105], N[(N[(N[(2.0 / N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999939e-76

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-neg.f64 60.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 59.2%

      \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\cos k \cdot \left(\left(-\ell\right) \cdot \ell\right)}{-\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 6.19999999999999939e-76 < t < 1.7e105

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

   5. Applied rewrites 58.1%

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

   if 1.7e105 < t

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.4%

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

Alternative 3: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 8.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left|t\right|\right) \cdot k}{\left(-\ell\right) \cdot \cos k} \cdot \frac{-k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot \left|t\right|}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 8.4e-76)
      (/
       2.0
       (*
        (/ (* (* (fma (cos (+ k k)) -0.5 0.5) (fabs t)) k) (* (- l) (cos k)))
        (/ (- k) l)))
      (/
       2.0
       (*
        (* (fabs t) (* (/ (fabs t) l) (* (tan k) (/ (* (sin k) (fabs t)) l))))
        (fma t_1 t_1 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double tmp;
	if (fabs(t) <= 8.4e-76) {
		tmp = 2.0 / ((((fma(cos((k + k)), -0.5, 0.5) * fabs(t)) * k) / (-l * cos(k))) * (-k / l));
	} else {
		tmp = 2.0 / ((fabs(t) * ((fabs(t) / l) * (tan(k) * ((sin(k) * fabs(t)) / l)))) * fma(t_1, t_1, 2.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(t))
	tmp = 0.0
	if (abs(t) <= 8.4e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * abs(t)) * k) / Float64(Float64(-l) * cos(k))) * Float64(Float64(-k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(abs(t) / l) * Float64(tan(k) * Float64(Float64(sin(k) * abs(t)) / l)))) * fma(t_1, t_1, 2.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 8.4e-76], N[(2.0 / N[(N[(N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[((-l) * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-k) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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 < 8.39999999999999969e-76

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-neg.f64 60.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 59.2%

      \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\cos k \cdot \left(\left(-\ell\right) \cdot \ell\right)}{-\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 8.39999999999999969e-76 < t

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 75.4

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

   7. Applied rewrites 75.4%

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

Alternative 4: 89.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.36 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\left(\tan \left(\left|k\right|\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin \left(\left|k\right|\right) \cdot t}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), 0.5, -0.5\right) \cdot \left(t \cdot \left|k\right|\right)}{\left(-\ell\right) \cdot \cos \left(\left|k\right|\right)} \cdot \frac{\left|k\right|}{\ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.36e-6)
   (/
    2.0
    (* (* (* (tan (fabs k)) (/ t l)) (* (/ (* (sin (fabs k)) t) l) t)) 2.0))
   (/
    2.0
    (*
     (/
      (* (fma (cos (+ (fabs k) (fabs k))) 0.5 -0.5) (* t (fabs k)))
      (* (- l) (cos (fabs k))))
     (/ (fabs k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.36e-6) {
		tmp = 2.0 / (((tan(fabs(k)) * (t / l)) * (((sin(fabs(k)) * t) / l) * t)) * 2.0);
	} else {
		tmp = 2.0 / (((fma(cos((fabs(k) + fabs(k))), 0.5, -0.5) * (t * fabs(k))) / (-l * cos(fabs(k)))) * (fabs(k) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.36e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(abs(k)) * Float64(t / l)) * Float64(Float64(Float64(sin(abs(k)) * t) / l) * t)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), 0.5, -0.5) * Float64(t * abs(k))) / Float64(Float64(-l) * cos(abs(k)))) * Float64(abs(k) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.36e-6], N[(2.0 / N[(N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(t * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3599999999999999e-6

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.4%

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

   if 1.3599999999999999e-6 < k

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-neg.f64 60.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 59.2%

      \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\cos k \cdot \left(\left(-\ell\right) \cdot \ell\right)}{-\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 46.1%

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

Alternative 5: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t\_1 \cdot t}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|k\right| \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t\_1 \cdot t\_2\right) \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), 0.5, -0.5\right) \cdot \left(t \cdot \left|k\right|\right)\right) \cdot \frac{\left|k\right|}{\left(\cos \left(\left|k\right|\right) \cdot \ell\right) \cdot \left(-\ell\right)}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))))
   (if (<= (fabs k) 5e-36)
     (/ 2.0 (* (* (* t_2 (/ t l)) (* (/ (* t_1 t) l) t)) 2.0))
     (if (<= (fabs k) 2.5e+157)
       (/ 2.0 (/ (* (/ t l) (* (* t_1 t_2) (* (fabs k) (fabs k)))) l))
       (/
        2.0
        (*
         (* (fma (cos (+ (fabs k) (fabs k))) 0.5 -0.5) (* t (fabs k)))
         (/ (fabs k) (* (* (cos (fabs k)) l) (- l)))))))))

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) <= 5e-36) {
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	} else if (fabs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (fabs(k) * fabs(k)))) / l);
	} else {
		tmp = 2.0 / ((fma(cos((fabs(k) + fabs(k))), 0.5, -0.5) * (t * fabs(k))) * (fabs(k) / ((cos(fabs(k)) * l) * -l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 5e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t / l)) * Float64(Float64(Float64(t_1 * t) / l) * t)) * 2.0));
	elseif (abs(k) <= 2.5e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t_1 * t_2) * Float64(abs(k) * abs(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), 0.5, -0.5) * Float64(t * abs(k))) * Float64(abs(k) / Float64(Float64(cos(abs(k)) * l) * Float64(-l)))));
	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], 5e-36], N[(2.0 / N[(N[(N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.5e+157], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(t * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000004e-36

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.4%

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

   if 5.00000000000000004e-36 < k < 2.49999999999999988e157

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 64.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 2.49999999999999988e157 < k

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-neg.f64 60.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 59.2%

      \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\cos k \cdot \left(\left(-\ell\right) \cdot \ell\right)}{-\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 39.7%

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

Alternative 6: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t\_1 \cdot t}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|k\right| \leq 3 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t\_1 \cdot t\_2\right) \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}{\left(\cos \left(\left|k\right|\right) \cdot \ell\right) \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))))
   (if (<= (fabs k) 5e-36)
     (/ 2.0 (* (* (* t_2 (/ t l)) (* (/ (* t_1 t) l) t)) 2.0))
     (if (<= (fabs k) 3e+157)
       (/ 2.0 (/ (* (/ t l) (* (* t_1 t_2) (* (fabs k) (fabs k)))) l))
       (/
        2.0
        (/
         (*
          (* (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t) (fabs k))
          (fabs k))
         (* (* (cos (fabs k)) l) l)))))))

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) <= 5e-36) {
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	} else if (fabs(k) <= 3e+157) {
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (fabs(k) * fabs(k)))) / l);
	} else {
		tmp = 2.0 / (((((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t) * fabs(k)) * fabs(k)) / ((cos(fabs(k)) * l) * l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(abs(k))
    t_2 = tan(abs(k))
    if (abs(k) <= 5d-36) then
        tmp = 2.0d0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0d0)
    else if (abs(k) <= 3d+157) then
        tmp = 2.0d0 / (((t / l) * ((t_1 * t_2) * (abs(k) * abs(k)))) / l)
    else
        tmp = 2.0d0 / (((((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k)) / ((cos(abs(k)) * l) * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double t_2 = Math.tan(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 5e-36) {
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	} else if (Math.abs(k) <= 3e+157) {
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (Math.abs(k) * Math.abs(k)))) / l);
	} else {
		tmp = 2.0 / (((((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t) * Math.abs(k)) * Math.abs(k)) / ((Math.cos(Math.abs(k)) * l) * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	t_2 = math.tan(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 5e-36:
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0)
	elif math.fabs(k) <= 3e+157:
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (math.fabs(k) * math.fabs(k)))) / l)
	else:
		tmp = 2.0 / (((((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t) * math.fabs(k)) * math.fabs(k)) / ((math.cos(math.fabs(k)) * l) * l))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 5e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t / l)) * Float64(Float64(Float64(t_1 * t) / l) * t)) * 2.0));
	elseif (abs(k) <= 3e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t_1 * t_2) * Float64(abs(k) * abs(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t) * abs(k)) * abs(k)) / Float64(Float64(cos(abs(k)) * l) * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	t_2 = tan(abs(k));
	tmp = 0.0;
	if (abs(k) <= 5e-36)
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	elseif (abs(k) <= 3e+157)
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (abs(k) * abs(k)))) / l);
	else
		tmp = 2.0 / (((((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k)) / ((cos(abs(k)) * l) * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5e-36], N[(2.0 / N[(N[(N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 3e+157], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000004e-36

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.4%

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

   if 5.00000000000000004e-36 < k < 3.0000000000000001e157

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 64.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 3.0000000000000001e157 < k

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

   6. Applied rewrites 59.2%

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

Alternative 7: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t\_1 \cdot t}{\ell} \cdot t\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|k\right| \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t\_1 \cdot t\_2\right) \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\_2\right) \cdot \left(t\_1 \cdot \left|k\right|\right)\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))))
   (if (<= (fabs k) 5e-36)
     (/ 2.0 (* (* (* t_2 (/ t l)) (* (/ (* t_1 t) l) t)) 2.0))
     (if (<= (fabs k) 2.5e+157)
       (/ 2.0 (/ (* (/ t l) (* (* t_1 t_2) (* (fabs k) (fabs k)))) l))
       (/ 2.0 (* (* (* (/ t (* l l)) t_2) (* t_1 (fabs k))) (fabs k)))))))

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) <= 5e-36) {
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	} else if (fabs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (fabs(k) * fabs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_2) * (t_1 * fabs(k))) * fabs(k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(abs(k))
    t_2 = tan(abs(k))
    if (abs(k) <= 5d-36) then
        tmp = 2.0d0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0d0)
    else if (abs(k) <= 2.5d+157) then
        tmp = 2.0d0 / (((t / l) * ((t_1 * t_2) * (abs(k) * abs(k)))) / l)
    else
        tmp = 2.0d0 / ((((t / (l * l)) * t_2) * (t_1 * abs(k))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double t_2 = Math.tan(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 5e-36) {
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	} else if (Math.abs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (Math.abs(k) * Math.abs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_2) * (t_1 * Math.abs(k))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	t_2 = math.tan(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 5e-36:
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0)
	elif math.fabs(k) <= 2.5e+157:
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (math.fabs(k) * math.fabs(k)))) / l)
	else:
		tmp = 2.0 / ((((t / (l * l)) * t_2) * (t_1 * math.fabs(k))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	tmp = 0.0
	if (abs(k) <= 5e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t / l)) * Float64(Float64(Float64(t_1 * t) / l) * t)) * 2.0));
	elseif (abs(k) <= 2.5e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t_1 * t_2) * Float64(abs(k) * abs(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / Float64(l * l)) * t_2) * Float64(t_1 * abs(k))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	t_2 = tan(abs(k));
	tmp = 0.0;
	if (abs(k) <= 5e-36)
		tmp = 2.0 / (((t_2 * (t / l)) * (((t_1 * t) / l) * t)) * 2.0);
	elseif (abs(k) <= 2.5e+157)
		tmp = 2.0 / (((t / l) * ((t_1 * t_2) * (abs(k) * abs(k)))) / l);
	else
		tmp = 2.0 / ((((t / (l * l)) * t_2) * (t_1 * abs(k))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5e-36], N[(2.0 / N[(N[(N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.5e+157], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$1 * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000004e-36

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 78.7

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.4%

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

   if 5.00000000000000004e-36 < k < 2.49999999999999988e157

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 64.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 2.49999999999999988e157 < k

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

Alternative 8: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right)\\ t_2 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(t\_1 \cdot \frac{t\_2 \cdot t}{\ell}\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;\left|k\right| \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t\_2 \cdot t\_1\right) \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\_1\right) \cdot \left(t\_2 \cdot \left|k\right|\right)\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (tan (fabs k))) (t_2 (sin (fabs k))))
   (if (<= (fabs k) 5e-36)
     (/ 2.0 (* (* t (* (/ t l) (* t_1 (/ (* t_2 t) l)))) 2.0))
     (if (<= (fabs k) 2.5e+157)
       (/ 2.0 (/ (* (/ t l) (* (* t_2 t_1) (* (fabs k) (fabs k)))) l))
       (/ 2.0 (* (* (* (/ t (* l l)) t_1) (* t_2 (fabs k))) (fabs k)))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k));
	double t_2 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 5e-36) {
		tmp = 2.0 / ((t * ((t / l) * (t_1 * ((t_2 * t) / l)))) * 2.0);
	} else if (fabs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (fabs(k) * fabs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * fabs(k))) * fabs(k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = tan(abs(k))
    t_2 = sin(abs(k))
    if (abs(k) <= 5d-36) then
        tmp = 2.0d0 / ((t * ((t / l) * (t_1 * ((t_2 * t) / l)))) * 2.0d0)
    else if (abs(k) <= 2.5d+157) then
        tmp = 2.0d0 / (((t / l) * ((t_2 * t_1) * (abs(k) * abs(k)))) / l)
    else
        tmp = 2.0d0 / ((((t / (l * l)) * t_1) * (t_2 * abs(k))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(Math.abs(k));
	double t_2 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 5e-36) {
		tmp = 2.0 / ((t * ((t / l) * (t_1 * ((t_2 * t) / l)))) * 2.0);
	} else if (Math.abs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (Math.abs(k) * Math.abs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * Math.abs(k))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(math.fabs(k))
	t_2 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 5e-36:
		tmp = 2.0 / ((t * ((t / l) * (t_1 * ((t_2 * t) / l)))) * 2.0)
	elif math.fabs(k) <= 2.5e+157:
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (math.fabs(k) * math.fabs(k)))) / l)
	else:
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * math.fabs(k))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = tan(abs(k))
	t_2 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 5e-36)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(t / l) * Float64(t_1 * Float64(Float64(t_2 * t) / l)))) * 2.0));
	elseif (abs(k) <= 2.5e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t_2 * t_1) * Float64(abs(k) * abs(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / Float64(l * l)) * t_1) * Float64(t_2 * abs(k))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(abs(k));
	t_2 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 5e-36)
		tmp = 2.0 / ((t * ((t / l) * (t_1 * ((t_2 * t) / l)))) * 2.0);
	elseif (abs(k) <= 2.5e+157)
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (abs(k) * abs(k)))) / l);
	else
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * abs(k))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5e-36], N[(2.0 / N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$2 * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.5e+157], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$2 * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000004e-36

   1. Initial program 55.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.3

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

   3. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.8%

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

   if 5.00000000000000004e-36 < k < 2.49999999999999988e157

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 64.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 2.49999999999999988e157 < k

   1. Initial program 55.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot k} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot k} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      11. lower-*.f64 63.9

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

   8. Applied rewrites 63.9%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right)\\ t_2 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\left(\frac{\left|k\right| \cdot t}{\ell} \cdot \left(\frac{t\_2 \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{\left|k\right|}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\left|k\right| \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t\_2 \cdot t\_1\right) \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\_1\right) \cdot \left(t\_2 \cdot \left|k\right|\right)\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (tan (fabs k))) (t_2 (sin (fabs k))))
   (if (<= (fabs k) 3.8e-62)
     (/
      2.0
      (*
       (* (/ (* (fabs k) t) l) (* (/ (* t_2 t) l) t))
       (+ (+ 1.0 (pow (/ (fabs k) t) 2.0)) 1.0)))
     (if (<= (fabs k) 2.5e+157)
       (/ 2.0 (/ (* (/ t l) (* (* t_2 t_1) (* (fabs k) (fabs k)))) l))
       (/ 2.0 (* (* (* (/ t (* l l)) t_1) (* t_2 (fabs k))) (fabs k)))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k));
	double t_2 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 3.8e-62) {
		tmp = 2.0 / ((((fabs(k) * t) / l) * (((t_2 * t) / l) * t)) * ((1.0 + pow((fabs(k) / t), 2.0)) + 1.0));
	} else if (fabs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (fabs(k) * fabs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * fabs(k))) * fabs(k));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = tan(abs(k))
    t_2 = sin(abs(k))
    if (abs(k) <= 3.8d-62) then
        tmp = 2.0d0 / ((((abs(k) * t) / l) * (((t_2 * t) / l) * t)) * ((1.0d0 + ((abs(k) / t) ** 2.0d0)) + 1.0d0))
    else if (abs(k) <= 2.5d+157) then
        tmp = 2.0d0 / (((t / l) * ((t_2 * t_1) * (abs(k) * abs(k)))) / l)
    else
        tmp = 2.0d0 / ((((t / (l * l)) * t_1) * (t_2 * abs(k))) * abs(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(Math.abs(k));
	double t_2 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 3.8e-62) {
		tmp = 2.0 / ((((Math.abs(k) * t) / l) * (((t_2 * t) / l) * t)) * ((1.0 + Math.pow((Math.abs(k) / t), 2.0)) + 1.0));
	} else if (Math.abs(k) <= 2.5e+157) {
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (Math.abs(k) * Math.abs(k)))) / l);
	} else {
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * Math.abs(k))) * Math.abs(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(math.fabs(k))
	t_2 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 3.8e-62:
		tmp = 2.0 / ((((math.fabs(k) * t) / l) * (((t_2 * t) / l) * t)) * ((1.0 + math.pow((math.fabs(k) / t), 2.0)) + 1.0))
	elif math.fabs(k) <= 2.5e+157:
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (math.fabs(k) * math.fabs(k)))) / l)
	else:
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * math.fabs(k))) * math.fabs(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = tan(abs(k))
	t_2 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.8e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) / l) * Float64(Float64(Float64(t_2 * t) / l) * t)) * Float64(Float64(1.0 + (Float64(abs(k) / t) ^ 2.0)) + 1.0)));
	elseif (abs(k) <= 2.5e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t_2 * t_1) * Float64(abs(k) * abs(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / Float64(l * l)) * t_1) * Float64(t_2 * abs(k))) * abs(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(abs(k));
	t_2 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 3.8e-62)
		tmp = 2.0 / ((((abs(k) * t) / l) * (((t_2 * t) / l) * t)) * ((1.0 + ((abs(k) / t) ^ 2.0)) + 1.0));
	elseif (abs(k) <= 2.5e+157)
		tmp = 2.0 / (((t / l) * ((t_2 * t_1) * (abs(k) * abs(k)))) / l);
	else
		tmp = 2.0 / ((((t / (l * l)) * t_1) * (t_2 * abs(k))) * abs(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.8e-62], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$2 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(N[Abs[k], $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.5e+157], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$2 * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.80000000000000006e-62

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.3

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 78.7

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\color{blue}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 72.3

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 3.80000000000000006e-62 < k < 2.49999999999999988e157

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left(k \cdot k\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\tan k \cdot \color{blue}{\sin k}\right) \cdot \left(k \cdot k\right)\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left(k \cdot k\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\left(\color{blue}{\tan k} \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}{\color{blue}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}{\color{blue}{\ell}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}{\ell}} \]

      11. lower-*.f64 64.6

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}{\ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot k\right)\right)}{\ell}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot k\right)\right)}{\ell}} \]

      14. lower-*.f64 64.6

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot k\right)\right)}{\ell}} \]

   8. Applied rewrites 64.6%

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot k\right)\right)}{\color{blue}{\ell}}} \]

   if 2.49999999999999988e157 < k

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k\right) \cdot k} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \sin k\right) \cdot k\right) \cdot k} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

      11. lower-*.f64 63.9

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot k} \]

   8. Applied rewrites 63.9%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\sin k \cdot k\right)\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\left(\frac{\left|k\right| \cdot t}{\ell} \cdot \left(\frac{t\_1 \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{\left|k\right|}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan \left(\left|k\right|\right) \cdot t\_1\right)\right) \cdot \left(\left|k\right| \cdot \frac{\left|k\right|}{\ell \cdot \ell}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))))
   (if (<= (fabs k) 1.75e-63)
     (/
      2.0
      (*
       (* (/ (* (fabs k) t) l) (* (/ (* t_1 t) l) t))
       (+ (+ 1.0 (pow (/ (fabs k) t) 2.0)) 1.0)))
     (/
      2.0
      (* (* t (* (tan (fabs k)) t_1)) (* (fabs k) (/ (fabs k) (* l l))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 1.75e-63) {
		tmp = 2.0 / ((((fabs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + pow((fabs(k) / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t * (tan(fabs(k)) * t_1)) * (fabs(k) * (fabs(k) / (l * l))));
	}
	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 = sin(abs(k))
    if (abs(k) <= 1.75d-63) then
        tmp = 2.0d0 / ((((abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0d0 + ((abs(k) / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((t * (tan(abs(k)) * t_1)) * (abs(k) * (abs(k) / (l * l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.75e-63) {
		tmp = 2.0 / ((((Math.abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + Math.pow((Math.abs(k) / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t * (Math.tan(Math.abs(k)) * t_1)) * (Math.abs(k) * (Math.abs(k) / (l * l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.75e-63:
		tmp = 2.0 / ((((math.fabs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + math.pow((math.fabs(k) / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((t * (math.tan(math.fabs(k)) * t_1)) * (math.fabs(k) * (math.fabs(k) / (l * l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.75e-63)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) / l) * Float64(Float64(Float64(t_1 * t) / l) * t)) * Float64(Float64(1.0 + (Float64(abs(k) / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(tan(abs(k)) * t_1)) * Float64(abs(k) * Float64(abs(k) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.75e-63)
		tmp = 2.0 / ((((abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + ((abs(k) / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((t * (tan(abs(k)) * t_1)) * (abs(k) * (abs(k) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.75e-63], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$1 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(N[Abs[k], $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75000000000000002e-63

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.3

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 78.7

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\color{blue}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 72.3

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 1.75000000000000002e-63 < k

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{\ell \cdot \ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{\ell \cdot \ell}}} \]

   6. Applied rewrites 62.2%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 1.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\left(\frac{\left|k\right| \cdot t}{\ell} \cdot \left(\frac{t\_1 \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{\left|k\right|}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left|k\right| \cdot \left(\left|k\right| \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan \left(\left|k\right|\right) \cdot t\_1\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))))
   (if (<= (fabs k) 1.6e-63)
     (/
      2.0
      (*
       (* (/ (* (fabs k) t) l) (* (/ (* t_1 t) l) t))
       (+ (+ 1.0 (pow (/ (fabs k) t) 2.0)) 1.0)))
     (/
      2.0
      (* (fabs k) (* (fabs k) (* (/ t (* l l)) (* (tan (fabs k)) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 1.6e-63) {
		tmp = 2.0 / ((((fabs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + pow((fabs(k) / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (fabs(k) * (fabs(k) * ((t / (l * l)) * (tan(fabs(k)) * t_1))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k))
    if (abs(k) <= 1.6d-63) then
        tmp = 2.0d0 / ((((abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0d0 + ((abs(k) / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (abs(k) * (abs(k) * ((t / (l * l)) * (tan(abs(k)) * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.6e-63) {
		tmp = 2.0 / ((((Math.abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + Math.pow((Math.abs(k) / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (Math.abs(k) * (Math.abs(k) * ((t / (l * l)) * (Math.tan(Math.abs(k)) * t_1))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.6e-63:
		tmp = 2.0 / ((((math.fabs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + math.pow((math.fabs(k) / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (math.fabs(k) * (math.fabs(k) * ((t / (l * l)) * (math.tan(math.fabs(k)) * t_1))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.6e-63)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) / l) * Float64(Float64(Float64(t_1 * t) / l) * t)) * Float64(Float64(1.0 + (Float64(abs(k) / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(abs(k) * Float64(abs(k) * Float64(Float64(t / Float64(l * l)) * Float64(tan(abs(k)) * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.6e-63)
		tmp = 2.0 / ((((abs(k) * t) / l) * (((t_1 * t) / l) * t)) * ((1.0 + ((abs(k) / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (abs(k) * (abs(k) * ((t / (l * l)) * (tan(abs(k)) * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.6e-63], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$1 * t), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(N[Abs[k], $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999994e-63

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.3

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 78.7

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\color{blue}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 72.3

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 1.59999999999999994e-63 < k

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos \color{blue}{k}}\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}\right)} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}\right)\right)} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}\right)\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}\right)\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\frac{\sin k}{\cos k}}\right)\right)\right)} \]

      18. lift-sin.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \frac{\sin k}{\cos \color{blue}{k}}\right)\right)\right)} \]

      19. lift-cos.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\sin k \cdot \frac{\sin k}{\cos k}\right)\right)\right)} \]

   6. Applied rewrites 63.9%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 77.8% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.32 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\left|t\right|}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.32e-100)
    (/ 2.0 (* (* (/ (* k k) l) (/ (fabs t) l)) (* k k)))
    (/
     2.0
     (*
      (* (/ (* k (fabs t)) l) (* (/ (* (sin k) (fabs t)) l) (fabs t)))
      (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.32e-100) {
		tmp = 2.0 / ((((k * k) / l) * (fabs(t) / l)) * (k * k));
	} else {
		tmp = 2.0 / ((((k * fabs(t)) / l) * (((sin(k) * fabs(t)) / l) * fabs(t))) * ((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) <= 1.32e-100) {
		tmp = 2.0 / ((((k * k) / l) * (Math.abs(t) / l)) * (k * k));
	} else {
		tmp = 2.0 / ((((k * Math.abs(t)) / l) * (((Math.sin(k) * Math.abs(t)) / l) * Math.abs(t))) * ((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) <= 1.32e-100:
		tmp = 2.0 / ((((k * k) / l) * (math.fabs(t) / l)) * (k * k))
	else:
		tmp = 2.0 / ((((k * math.fabs(t)) / l) * (((math.sin(k) * math.fabs(t)) / l) * math.fabs(t))) * ((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) <= 1.32e-100)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(abs(t) / l)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * abs(t)) / l) * Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))) * 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) <= 1.32e-100)
		tmp = 2.0 / ((((k * k) / l) * (abs(t) / l)) * (k * k));
	else
		tmp = 2.0 / ((((k * abs(t)) / l) * (((sin(k) * abs(t)) / l) * abs(t))) * ((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], 1.32e-100], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $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 < 1.31999999999999994e-100

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-pow.f64 53.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       8. lower-/.f64 58.6

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       11. lift-*.f64 58.6

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 58.6%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   if 1.31999999999999994e-100 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.3

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\tan k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 78.7

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\color{blue}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-*.f64 72.3

         \[\leadsto \frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.65 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot t\_1\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(t\_1 \cdot \left(k \cdot \frac{\sin k \cdot \left|t\right|}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.65e-99)
      (/ 2.0 (* (* (/ (* k k) l) t_1) (* k k)))
      (/
       2.0
       (*
        (* (fabs t) (* t_1 (* k (/ (* (sin k) (fabs t)) l))))
        (+ (+ 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) <= 1.65e-99) {
		tmp = 2.0 / ((((k * k) / l) * t_1) * (k * k));
	} else {
		tmp = 2.0 / ((fabs(t) * (t_1 * (k * ((sin(k) * fabs(t)) / l)))) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(t) / l;
	double tmp;
	if (Math.abs(t) <= 1.65e-99) {
		tmp = 2.0 / ((((k * k) / l) * t_1) * (k * k));
	} else {
		tmp = 2.0 / ((Math.abs(t) * (t_1 * (k * ((Math.sin(k) * Math.abs(t)) / l)))) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(t) / l
	tmp = 0
	if math.fabs(t) <= 1.65e-99:
		tmp = 2.0 / ((((k * k) / l) * t_1) * (k * k))
	else:
		tmp = 2.0 / ((math.fabs(t) * (t_1 * (k * ((math.sin(k) * math.fabs(t)) / l)))) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.65e-99)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * t_1) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(t_1 * Float64(k * Float64(Float64(sin(k) * abs(t)) / l)))) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(t) / l;
	tmp = 0.0;
	if (abs(t) <= 1.65e-99)
		tmp = 2.0 / ((((k * k) / l) * t_1) * (k * k));
	else
		tmp = 2.0 / ((abs(t) * (t_1 * (k * ((sin(k) * abs(t)) / l)))) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.65e-99], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(t$95$1 * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $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 < 1.64999999999999993e-99

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-pow.f64 53.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       8. lower-/.f64 58.6

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       11. lift-*.f64 58.6

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 58.6%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   if 1.64999999999999993e-99 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 68.3

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 68.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 75.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{k} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{k} \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.0% accurate, 3.6× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\left|t\right|}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 2.1e-31)
    (/ 2.0 (* (* (/ (* k k) l) (/ (fabs t) l)) (* k k)))
    (* (/ l (* (* (* (fabs t) (fabs t)) (fabs t)) k)) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.1e-31) {
		tmp = 2.0 / ((((k * k) / l) * (fabs(t) / l)) * (k * k));
	} else {
		tmp = (l / (((fabs(t) * fabs(t)) * fabs(t)) * k)) * (l / k);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 2.1e-31) {
		tmp = 2.0 / ((((k * k) / l) * (Math.abs(t) / l)) * (k * k));
	} else {
		tmp = (l / (((Math.abs(t) * Math.abs(t)) * Math.abs(t)) * k)) * (l / k);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 2.1e-31:
		tmp = 2.0 / ((((k * k) / l) * (math.fabs(t) / l)) * (k * k))
	else:
		tmp = (l / (((math.fabs(t) * math.fabs(t)) * math.fabs(t)) * k)) * (l / k)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.1e-31)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(abs(t) / l)) * Float64(k * k)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(t) * abs(t)) * abs(t)) * k)) * Float64(l / k));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 2.1e-31)
		tmp = 2.0 / ((((k * k) / l) * (abs(t) / l)) * (k * k));
	else
		tmp = (l / (((abs(t) * abs(t)) * abs(t)) * k)) * (l / k);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.1e-31], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.09999999999999991e-31

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-pow.f64 53.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       8. lower-/.f64 58.6

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

       11. lift-*.f64 58.6

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 58.6%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   if 2.09999999999999991e-31 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.4

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.5

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 60.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lower-/.f64 61.1

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 61.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.4% accurate, 3.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.8e-31)
    (/ 2.0 (* (* (fabs t) (/ (* k k) (* l l))) (* k k)))
    (* (/ l (* (* (* (fabs t) (fabs t)) (fabs t)) k)) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.8e-31) {
		tmp = 2.0 / ((fabs(t) * ((k * k) / (l * l))) * (k * k));
	} else {
		tmp = (l / (((fabs(t) * fabs(t)) * fabs(t)) * k)) * (l / k);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.8e-31) {
		tmp = 2.0 / ((Math.abs(t) * ((k * k) / (l * l))) * (k * k));
	} else {
		tmp = (l / (((Math.abs(t) * Math.abs(t)) * Math.abs(t)) * k)) * (l / k);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.8e-31:
		tmp = 2.0 / ((math.fabs(t) * ((k * k) / (l * l))) * (k * k))
	else:
		tmp = (l / (((math.fabs(t) * math.fabs(t)) * math.fabs(t)) * k)) * (l / k)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.8e-31)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(k * k) / Float64(l * l))) * Float64(k * k)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(t) * abs(t)) * abs(t)) * k)) * Float64(l / k));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.8e-31)
		tmp = 2.0 / ((abs(t) * ((k * k) / (l * l))) * (k * k));
	else
		tmp = (l / (((abs(t) * abs(t)) * abs(t)) * k)) * (l / 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], 1.8e-31], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000002e-31

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-pow.f64 53.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

       6. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{k}^{2}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{k}^{2}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{k}^{2}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \frac{{k}^{2}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(t \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

       12. lift-*.f64 53.8

           \[\leadsto \frac{2}{\left(t \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 53.8%

       \[\leadsto \frac{2}{\left(t \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   if 1.80000000000000002e-31 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.4

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.5

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 60.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lower-/.f64 61.1

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 61.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.4% accurate, 3.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.8e-31)
    (/ 2.0 (* (* (* (/ (fabs t) (* l l)) (* k k)) k) k))
    (* (/ l (* (* (* (fabs t) (fabs t)) (fabs t)) k)) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.8e-31) {
		tmp = 2.0 / ((((fabs(t) / (l * l)) * (k * k)) * k) * k);
	} else {
		tmp = (l / (((fabs(t) * fabs(t)) * fabs(t)) * k)) * (l / k);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.8e-31) {
		tmp = 2.0 / ((((Math.abs(t) / (l * l)) * (k * k)) * k) * k);
	} else {
		tmp = (l / (((Math.abs(t) * Math.abs(t)) * Math.abs(t)) * k)) * (l / k);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.8e-31:
		tmp = 2.0 / ((((math.fabs(t) / (l * l)) * (k * k)) * k) * k)
	else:
		tmp = (l / (((math.fabs(t) * math.fabs(t)) * math.fabs(t)) * k)) * (l / k)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.8e-31)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / Float64(l * l)) * Float64(k * k)) * k) * k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(t) * abs(t)) * abs(t)) * k)) * Float64(l / k));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.8e-31)
		tmp = 2.0 / ((((abs(t) / (l * l)) * (k * k)) * k) * k);
	else
		tmp = (l / (((abs(t) * abs(t)) * abs(t)) * k)) * (l / 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], 1.8e-31], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000002e-31

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.0

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-pow.f64 53.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot k\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(k \cdot \color{blue}{k}\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot \color{blue}{k}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot \color{blue}{k}} \]

   11. Applied rewrites 54.8%

       \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k}} \]

   if 1.80000000000000002e-31 < t

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.4

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.5

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 60.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lower-/.f64 61.1

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 61.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left|\ell\right|}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\left|\ell\right|}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\ell\right|}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \left|\ell\right|\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 4e-116)
   (* (/ (fabs l) (* (* (* t t) t) k)) (/ (fabs l) k))
   (* (/ (fabs l) (* (* t (* t (* t k))) k)) (fabs l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 4e-116) {
		tmp = (fabs(l) / (((t * t) * t) * k)) * (fabs(l) / k);
	} else {
		tmp = (fabs(l) / ((t * (t * (t * k))) * k)) * fabs(l);
	}
	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(l) <= 4d-116) then
        tmp = (abs(l) / (((t * t) * t) * k)) * (abs(l) / k)
    else
        tmp = (abs(l) / ((t * (t * (t * k))) * k)) * abs(l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 4e-116) {
		tmp = (Math.abs(l) / (((t * t) * t) * k)) * (Math.abs(l) / k);
	} else {
		tmp = (Math.abs(l) / ((t * (t * (t * k))) * k)) * Math.abs(l);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 4e-116:
		tmp = (math.fabs(l) / (((t * t) * t) * k)) * (math.fabs(l) / k)
	else:
		tmp = (math.fabs(l) / ((t * (t * (t * k))) * k)) * math.fabs(l)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 4e-116)
		tmp = Float64(Float64(abs(l) / Float64(Float64(Float64(t * t) * t) * k)) * Float64(abs(l) / k));
	else
		tmp = Float64(Float64(abs(l) / Float64(Float64(t * Float64(t * Float64(t * k))) * k)) * abs(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 4e-116)
		tmp = (abs(l) / (((t * t) * t) * k)) * (abs(l) / k);
	else
		tmp = (abs(l) / ((t * (t * (t * k))) * k)) * abs(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 4e-116], N[(N[(N[Abs[l], $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[l], $MachinePrecision] / N[(N[(t * N[(t * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4e-116

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.4

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.5

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 60.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lower-/.f64 61.1

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 61.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 4e-116 < l

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.4

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.5

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 60.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 60.0

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. lower-*.f64 63.7

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 63.4% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* t (* t (* t k))) k)) l))

C

double code(double t, double l, double k) {
	return (l / ((t * (t * (t * k))) * k)) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / ((t * (t * (t * k))) * k)) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((t * (t * (t * k))) * k)) * l;
}

Python

def code(t, l, k):
	return (l / ((t * (t * (t * k))) * k)) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(t * Float64(t * Float64(t * k))) * k)) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((t * (t * (t * k))) * k)) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(t * N[(t * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 55.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.4

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.4%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.5

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 60.0

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 60.0

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   8. lower-*.f64 63.7

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

10. Applied rewrites 63.7%

    \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]
11. Add Preprocessing

Alternative 19: 63.2% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* k (* t t)) (* t k))) l))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (t * k))) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / ((k * (t * t)) * (t * k))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (t * k))) * l;
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * (t * k))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * Float64(t * k))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * (t * k))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 55.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.4

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.4%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.5

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 60.0

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 60.0

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   9. lower-*.f64 63.2

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

10. Applied rewrites 63.2%

    \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(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))))