Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 94.2%

Time: 8.7s

Alternatives: 14

Speedup: 4.4×

• 40.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 36.2% accurate, 1.0× speedup

Math

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

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: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 8 \cdot 10^{-130}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot t\_1\right)\right)\\ \mathbf{elif}\;k\_m \leq 0.00115:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\mathsf{fma}\left(0.3333333333333333, \frac{{k\_m}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k\_m}^{2}} \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{\mathsf{fma}\left(t, 0.5, t \cdot \left(-0.5 \cdot \cos \left(k\_m + k\_m\right)\right)\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 8e-130)
     (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) t_1)))
     (if (<= k_m 0.00115)
       (*
        2.0
        (*
         (cos k_m)
         (*
          (/
           (fma 0.3333333333333333 (/ (* (pow k_m 2.0) l) t) (/ l t))
           (pow k_m 2.0))
          t_1)))
       (*
        2.0
        (*
         (cos k_m)
         (*
          (/ l (* (fma t 0.5 (* t (* -0.5 (cos (+ k_m k_m))))) k_m))
          (/ l k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 8e-130) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * t_1));
	} else if (k_m <= 0.00115) {
		tmp = 2.0 * (cos(k_m) * ((fma(0.3333333333333333, ((pow(k_m, 2.0) * l) / t), (l / t)) / pow(k_m, 2.0)) * t_1));
	} else {
		tmp = 2.0 * (cos(k_m) * ((l / (fma(t, 0.5, (t * (-0.5 * cos((k_m + k_m))))) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 8e-130)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * t_1)));
	elseif (k_m <= 0.00115)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(fma(0.3333333333333333, Float64(Float64((k_m ^ 2.0) * l) / t), Float64(l / t)) / (k_m ^ 2.0)) * t_1)));
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64(fma(t, 0.5, Float64(t * Float64(-0.5 * cos(Float64(k_m + k_m))))) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8e-130], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.00115], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(N[(0.3333333333333333 * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] + N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[(t * 0.5 + N[(t * N[(-0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.0000000000000007e-130

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 8.0000000000000007e-130 < k < 0.00115

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\frac{1}{3} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{k \cdot k}\right)\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{3}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

       3. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{3}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{3}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{3}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

       6. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(\frac{1}{3}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

       7. lower-pow.f64 59.4

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(0.3333333333333333, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\ell}{k \cdot k}\right)\right) \]

   11. Applied rewrites 59.4%

       \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\mathsf{fma}\left(0.3333333333333333, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{k \cdot k}\right)\right) \]

   if 0.00115 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.6

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

   10. Applied rewrites 65.6%

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

Alternative 2: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{\sin k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e+99)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow (sin k_m) 2.0) t)) (/ l (* k_m k_m)))))
   (*
    2.0
    (*
     (cos k_m)
     (* (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)) (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e+99) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(sin(k_m), 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = 2.0 * (cos(k_m) * ((l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e+99)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((sin(k_m) ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.02e+99], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999998e99

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 86.7

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

   10. Applied rewrites 86.7%

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

   if 1.01999999999999998e99 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

Alternative 3: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{\mathsf{fma}\left(t, 0.5, t \cdot \left(-0.5 \cdot \cos \left(k\_m + k\_m\right)\right)\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    2.0
    (*
     (cos k_m)
     (*
      (/ l (* (fma t 0.5 (* t (* -0.5 (cos (+ k_m k_m))))) k_m))
      (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = 2.0 * (cos(k_m) * ((l / (fma(t, 0.5, (t * (-0.5 * cos((k_m + k_m))))) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64(fma(t, 0.5, Float64(t * Float64(-0.5 * cos(Float64(k_m + k_m))))) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.5e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[(t * 0.5 + N[(t * N[(-0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 8.500000000000001e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.6

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

   10. Applied rewrites 65.6%

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

Alternative 4: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    2.0
    (*
     (cos k_m)
     (* (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)) (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = 2.0 * (cos(k_m) * ((l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

Alternative 5: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

   8. Applied rewrites 84.0%

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

Alternative 6: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\left(\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m} \cdot \ell\right) \cdot \left(\cos k\_m \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    (* (/ l (* (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m) k_m)) l)
    (* (cos k_m) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = ((l / (((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m)) * l) * (cos(k_m) * 2.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m)) * l) * Float64(cos(k_m) * 2.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

   8. Applied rewrites 78.8%

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

Alternative 7: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\ell + \ell\right) \cdot \frac{\cos k\_m}{\left(\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    (*
     (+ l l)
     (/ (cos k_m) (* (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m) k_m)))
    l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = ((l + l) * (cos(k_m) / (((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(l + l) * Float64(cos(k_m) / Float64(Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 78.7%

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

Alternative 8: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \ell}{\left(\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m} \cdot \left(\ell + \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7e-5)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    (/ (* (cos k_m) l) (* (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m) k_m))
    (+ l l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7e-5) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = ((cos(k_m) * l) / (((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m)) * (l + l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7e-5)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) / Float64(Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m)) * Float64(l + l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7e-5], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.9999999999999994e-5

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 6.9999999999999994e-5 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 78.8%

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

Alternative 9: 76.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(1 \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.25e+86)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 2.0) t)) (/ l (* k_m k_m)))))
   (*
    2.0
    (*
     1.0
     (* (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)) (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+86) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 2.0) * t)) * (l / (k_m * k_m))));
	} else {
		tmp = 2.0 * (1.0 * ((l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.25e+86)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 2.0) * t)) * Float64(l / Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 * Float64(1.0 * Float64(Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.25e+86], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 * N[(N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.24999999999999996e86

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 76.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-pow.f64 74.8

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

   11. Applied rewrites 74.8%

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

   if 2.24999999999999996e86 < k

   1. Initial program 36.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      9. lower-/.f64 74.0

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}}\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)} \cdot k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      4. times-frac N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k}\right)\right) \]

      7. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\cos \left(k + k\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \mathsf{neg}\left(\frac{1}{2}\right), \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      14. lower-/.f64 84.0

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}}\right)\right) \]

   8. Applied rewrites 84.0%

      \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(1 \cdot \left(\color{blue}{\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right) \cdot k}} \cdot \frac{\ell}{k}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.3%

       \[\leadsto 2 \cdot \left(1 \cdot \left(\color{blue}{\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \cdot \frac{\ell}{k}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \left(\frac{\ell}{{k\_m}^{3} \cdot t} \cdot \frac{\ell}{k\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(1 \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.25e+86)
   (* 2.0 (* (cos k_m) (* (/ l (* (pow k_m 3.0) t)) (/ l k_m))))
   (*
    2.0
    (*
     1.0
     (* (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)) (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+86) {
		tmp = 2.0 * (cos(k_m) * ((l / (pow(k_m, 3.0) * t)) * (l / k_m)));
	} else {
		tmp = 2.0 * (1.0 * ((l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m)) * (l / k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.25e+86)
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l / Float64((k_m ^ 3.0) * t)) * Float64(l / k_m))));
	else
		tmp = Float64(2.0 * Float64(1.0 * Float64(Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m)) * Float64(l / k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.25e+86], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / N[(N[Power[k$95$m, 3.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 * N[(N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.24999999999999996e86

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      9. lower-/.f64 74.0

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}}\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)} \cdot k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      4. times-frac N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k}\right)\right) \]

      7. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\cos \left(k + k\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \mathsf{neg}\left(\frac{1}{2}\right), \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      14. lower-/.f64 84.0

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}}\right)\right) \]

   8. Applied rewrites 84.0%

      \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{k}^{3} \cdot t} \cdot \frac{\ell}{k}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{k}^{3} \cdot t} \cdot \frac{\ell}{k}\right)\right) \]

       2. lower-pow.f64 72.4

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{k}^{3} \cdot t} \cdot \frac{\ell}{k}\right)\right) \]

   11. Applied rewrites 72.4%

       \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{k}^{3} \cdot t} \cdot \frac{\ell}{k}\right)\right) \]

   if 2.24999999999999996e86 < k

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      9. lower-/.f64 74.0

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}}\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)} \cdot k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      4. times-frac N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k}\right)\right) \]

      7. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\left(\cos \left(k + k\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \mathsf{neg}\left(\frac{1}{2}\right), \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k}\right)\right) \]

      14. lower-/.f64 84.0

          \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k}}\right)\right) \]

   8. Applied rewrites 84.0%

      \[\leadsto 2 \cdot \left(\cos k \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}}\right)\right) \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(1 \cdot \left(\color{blue}{\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right) \cdot k}} \cdot \frac{\ell}{k}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.3%

       \[\leadsto 2 \cdot \left(1 \cdot \left(\color{blue}{\frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \cdot \frac{\ell}{k}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\ell}{{k\_m}^{4}}}{t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 5e+283)
   (* (* l (/ (/ l (pow k_m 4.0)) t)) 2.0)
   (* 2.0 (* (cos k_m) (/ (* l l) (* (* (* (- 0.5 0.5) t) k_m) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+283) {
		tmp = (l * ((l / pow(k_m, 4.0)) / t)) * 2.0;
	} else {
		tmp = 2.0 * (cos(k_m) * ((l * l) / ((((0.5 - 0.5) * t) * k_m) * k_m)));
	}
	return tmp;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+283) then
        tmp = (l * ((l / (k_m ** 4.0d0)) / t)) * 2.0d0
    else
        tmp = 2.0d0 * (cos(k_m) * ((l * l) / ((((0.5d0 - 0.5d0) * t) * k_m) * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+283) {
		tmp = (l * ((l / Math.pow(k_m, 4.0)) / t)) * 2.0;
	} else {
		tmp = 2.0 * (Math.cos(k_m) * ((l * l) / ((((0.5 - 0.5) * t) * k_m) * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 5e+283:
		tmp = (l * ((l / math.pow(k_m, 4.0)) / t)) * 2.0
	else:
		tmp = 2.0 * (math.cos(k_m) * ((l * l) / ((((0.5 - 0.5) * t) * k_m) * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+283)
		tmp = Float64(Float64(l * Float64(Float64(l / (k_m ^ 4.0)) / t)) * 2.0);
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l * l) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+283)
		tmp = (l * ((l / (k_m ^ 4.0)) / t)) * 2.0;
	else
		tmp = 2.0 * (cos(k_m) * ((l * l) / ((((0.5 - 0.5) * t) * k_m) * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+283], N[(N[(l * N[(N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000004e283

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 63.1

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 63.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

      3. lower-*.f64 63.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      7. associate-/l* N/A

         \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

      9. lower-/.f64 69.3

         \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   6. Applied rewrites 69.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

      3. associate-/r* N/A

         \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

      5. lower-/.f64 70.3

         \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

   8. Applied rewrites 70.3%

      \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

   if 5.0000000000000004e283 < (*.f64 l l)

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      9. lower-/.f64 74.0

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}}\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}}\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)}\right) \]

      14. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}}\right) \]

   6. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 36.7%

      \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \frac{\frac{\ell}{{k\_m}^{4}}}{t}\right) \cdot 2 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* l (/ (/ l (pow k_m 4.0)) t)) 2.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * ((l / pow(k_m, 4.0)) / t)) * 2.0;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * ((l / (k_m ** 4.0d0)) / t)) * 2.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * ((l / Math.pow(k_m, 4.0)) / t)) * 2.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * ((l / math.pow(k_m, 4.0)) / t)) * 2.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * Float64(Float64(l / (k_m ^ 4.0)) / t)) * 2.0)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * ((l / (k_m ^ 4.0)) / t)) * 2.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * N[(N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.1

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 63.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 69.3

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 69.3%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   3. associate-/r* N/A

      \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

   5. lower-/.f64 70.3

      \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]

8. Applied rewrites 70.3%

   \[\leadsto \left(\ell \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right) \cdot 2 \]
9. Add Preprocessing

Alternative 13: 70.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell + \ell\right) \cdot {k\_m}^{-4}}{t} \cdot \ell \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (/ (* (+ l l) (pow k_m -4.0)) t) l))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((l + l) * pow(k_m, -4.0)) / t) * l;
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((l + l) * (k_m ** (-4.0d0))) / t) * l
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((l + l) * Math.pow(k_m, -4.0)) / t) * l;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l + l) * math.pow(k_m, -4.0)) / t) * l

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l + l) * (k_m ^ -4.0)) / t) * l)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l + l) * (k_m ^ -4.0)) / t) * l;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l + l), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.1

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 63.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 69.3

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 69.3%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \color{blue}{\ell} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \color{blue}{\ell} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \ell \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\ell \cdot 2}{{k}^{4} \cdot t} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{2 \cdot \ell}{{k}^{4} \cdot t} \cdot \ell \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{{k}^{4} \cdot t} \cdot \ell \]

   10. count-2-rev N/A

       \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

   11. lower-+.f64 69.3

       \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

8. Applied rewrites 69.3%

   \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\ell + \ell}{{k}^{4}}}{t} \cdot \ell \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\ell + \ell}{{k}^{4}}}{t} \cdot \ell \]

   5. mult-flip N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}}{t} \cdot \ell \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}}{t} \cdot \ell \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}}{t} \cdot \ell \]

   8. pow-flip N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \ell \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \ell \]

   10. metadata-eval 70.2

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot {k}^{-4}}{t} \cdot \ell \]

10. Applied rewrites 70.2%

    \[\leadsto \frac{\left(\ell + \ell\right) \cdot {k}^{-4}}{t} \cdot \ell \]
11. Add Preprocessing

Alternative 14: 68.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\left(\ell + \ell\right) \cdot {k\_m}^{-4}\right) \cdot \frac{\ell}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* (+ l l) (pow k_m -4.0)) (/ l t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l + l) * pow(k_m, -4.0)) * (l / t);
}

Fortran

k_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l + l) * (k_m ** (-4.0d0))) * (l / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l + l) * Math.pow(k_m, -4.0)) * (l / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) * math.pow(k_m, -4.0)) * (l / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l + l) * (k_m ^ -4.0)) * Float64(l / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l + l) * (k_m ^ -4.0)) * (l / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l + l), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 63.1

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 63.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 63.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 69.3

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 69.3%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \color{blue}{\ell} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \color{blue}{\ell} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot 2\right) \cdot \ell \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\ell \cdot 2}{{k}^{4} \cdot t} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{2 \cdot \ell}{{k}^{4} \cdot t} \cdot \ell \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{{k}^{4} \cdot t} \cdot \ell \]

   10. count-2-rev N/A

       \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

   11. lower-+.f64 69.3

       \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

8. Applied rewrites 69.3%

   \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \color{blue}{\ell} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4} \cdot t} \cdot \ell \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{{k}^{4} \cdot t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{4} \cdot \color{blue}{t}} \]

   5. times-frac N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell + \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

   7. mult-flip N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}\right) \cdot \frac{\color{blue}{\ell}}{t} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}\right) \cdot \frac{\color{blue}{\ell}}{t} \]

   9. lift-pow.f64 N/A

      \[\leadsto \left(\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}\right) \cdot \frac{\ell}{t} \]

   10. pow-flip N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot \frac{\ell}{t} \]

   11. lower-pow.f64 N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot \frac{\ell}{t} \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{-4}\right) \cdot \frac{\ell}{t} \]

   13. lower-/.f64 68.7

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{-4}\right) \cdot \frac{\ell}{\color{blue}{t}} \]

10. Applied rewrites 68.7%

    \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{-4}\right) \cdot \color{blue}{\frac{\ell}{t}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))))