Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 97.7%

Time: 9.1s

Alternatives: 16

Speedup: 9.6×

• 39.5% 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: 35.6% 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: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m \cdot \ell}{k\_m}\\ \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell}{k\_m} \cdot 2}{{\sin k\_m}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* (cos k_m) l) k_m)))
   (if (<= k_m 4.5e-156)
     (* (/ 2.0 (* (* k_m t) k_m)) (* t_1 (/ l k_m)))
     (* t_1 (/ (* (/ l k_m) 2.0) (* (pow (sin k_m) 2.0) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 4.5e-156) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = t_1 * (((l / k_m) * 2.0) / (pow(sin(k_m), 2.0) * t));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (cos(k_m) * l) / k_m
    if (k_m <= 4.5d-156) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
    else
        tmp = t_1 * (((l / k_m) * 2.0d0) / ((sin(k_m) ** 2.0d0) * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (Math.cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 4.5e-156) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = t_1 * (((l / k_m) * 2.0) / (Math.pow(Math.sin(k_m), 2.0) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (math.cos(k_m) * l) / k_m
	tmp = 0
	if k_m <= 4.5e-156:
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
	else:
		tmp = t_1 * (((l / k_m) * 2.0) / (math.pow(math.sin(k_m), 2.0) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) / k_m)
	tmp = 0.0
	if (k_m <= 4.5e-156)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(t_1 * Float64(l / k_m)));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l / k_m) * 2.0) / Float64((sin(k_m) ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (cos(k_m) * l) / k_m;
	tmp = 0.0;
	if (k_m <= 4.5e-156)
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	else
		tmp = t_1 * (((l / k_m) * 2.0) / ((sin(k_m) ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 4.5e-156], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l / k$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.49999999999999986e-156

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 75.4

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

   10. Applied rewrites 75.4%

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

   if 4.49999999999999986e-156 < k

   1. Initial program 26.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 98.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l/ N/A

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

       14. times-frac N/A

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

   11. Applied rewrites 98.3%

       \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 2}{{\sin k}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\left(\frac{{t}^{-1}}{\sin k\_m} \cdot \frac{2}{\sin k\_m}\right) \cdot \frac{\cos k\_m \cdot \ell}{k\_m}\right) \cdot \frac{\ell}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (*
  (* (* (/ (pow t -1.0) (sin k_m)) (/ 2.0 (sin k_m))) (/ (* (cos k_m) l) k_m))
  (/ l k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((pow(t, -1.0) / sin(k_m)) * (2.0 / sin(k_m))) * ((cos(k_m) * l) / k_m)) * (l / k_m);
}

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 = ((((t ** (-1.0d0)) / sin(k_m)) * (2.0d0 / sin(k_m))) * ((cos(k_m) * l) / k_m)) * (l / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((Math.pow(t, -1.0) / Math.sin(k_m)) * (2.0 / Math.sin(k_m))) * ((Math.cos(k_m) * l) / k_m)) * (l / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((math.pow(t, -1.0) / math.sin(k_m)) * (2.0 / math.sin(k_m))) * ((math.cos(k_m) * l) / k_m)) * (l / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64((t ^ -1.0) / sin(k_m)) * Float64(2.0 / sin(k_m))) * Float64(Float64(cos(k_m) * l) / k_m)) * Float64(l / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((((t ^ -1.0) / sin(k_m)) * (2.0 / sin(k_m))) * ((cos(k_m) * l) / k_m)) * (l / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(N[Power[t, -1.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 70.9

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

5. Applied rewrites 70.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. frac-times N/A

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

   12. pow2 N/A

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

   13. *-commutative N/A

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

   14. pow2 N/A

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 89.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 94.6%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-sin.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. metadata-eval N/A

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

    8. associate-*r/ N/A

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

    9. unpow2 N/A

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

    10. *-commutative N/A

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

    11. times-frac N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. inv-pow N/A

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

    15. lower-pow.f64 N/A

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

    16. lift-sin.f64 N/A

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

    17. lower-/.f64 N/A

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

    18. lift-sin.f64 97.7

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

11. Applied rewrites 97.7%

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

Alternative 3: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m \cdot \ell}{k\_m}\\ \mathbf{if}\;k\_m \leq 2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\sin k\_m}^{2} \cdot t} \cdot t\_1\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* (cos k_m) l) k_m)))
   (if (<= k_m 2.75e-89)
     (* (/ 2.0 (* (* k_m t) k_m)) (* t_1 (/ l k_m)))
     (* (* (/ 2.0 (* (pow (sin k_m) 2.0) t)) t_1) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 2.75e-89) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = ((2.0 / (pow(sin(k_m), 2.0) * t)) * t_1) * (l / 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) :: t_1
    real(8) :: tmp
    t_1 = (cos(k_m) * l) / k_m
    if (k_m <= 2.75d-89) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
    else
        tmp = ((2.0d0 / ((sin(k_m) ** 2.0d0) * t)) * t_1) * (l / 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 t_1 = (Math.cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 2.75e-89) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = ((2.0 / (Math.pow(Math.sin(k_m), 2.0) * t)) * t_1) * (l / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (math.cos(k_m) * l) / k_m
	tmp = 0
	if k_m <= 2.75e-89:
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
	else:
		tmp = ((2.0 / (math.pow(math.sin(k_m), 2.0) * t)) * t_1) * (l / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) / k_m)
	tmp = 0.0
	if (k_m <= 2.75e-89)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(t_1 * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64((sin(k_m) ^ 2.0) * t)) * t_1) * Float64(l / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (cos(k_m) * l) / k_m;
	tmp = 0.0;
	if (k_m <= 2.75e-89)
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	else
		tmp = ((2.0 / ((sin(k_m) ^ 2.0) * t)) * t_1) * (l / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 2.75e-89], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.75000000000000006e-89

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.2

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

   5. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if 2.75000000000000006e-89 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 77.7

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

   5. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 99.4%

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

Alternative 4: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m \cdot \ell}{k\_m}\\ \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t} \cdot t\_1\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* (cos k_m) l) k_m)))
   (if (<= k_m 2.4e-5)
     (* (/ 2.0 (* (* k_m t) k_m)) (* t_1 (/ l k_m)))
     (* (* (/ 2.0 (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t)) t_1) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 2.4e-5) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t)) * t_1) * (l / 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) :: t_1
    real(8) :: tmp
    t_1 = (cos(k_m) * l) / k_m
    if (k_m <= 2.4d-5) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
    else
        tmp = ((2.0d0 / ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t)) * t_1) * (l / 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 t_1 = (Math.cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 2.4e-5) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = ((2.0 / ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t)) * t_1) * (l / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (math.cos(k_m) * l) / k_m
	tmp = 0
	if k_m <= 2.4e-5:
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m))
	else:
		tmp = ((2.0 / ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t)) * t_1) * (l / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) / k_m)
	tmp = 0.0
	if (k_m <= 2.4e-5)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(t_1 * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t)) * t_1) * Float64(l / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (cos(k_m) * l) / k_m;
	tmp = 0.0;
	if (k_m <= 2.4e-5)
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	else
		tmp = ((2.0 / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t)) * t_1) * (l / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 2.4e-5], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-5

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.8

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

   10. Applied rewrites 76.8%

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

   if 2.4000000000000001e-5 < k

   1. Initial program 23.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 99.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 99.5

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

   11. Applied rewrites 99.5%

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

Alternative 5: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(\frac{t\_1}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \frac{\ell}{k\_m}\right) \cdot 2}{k\_m \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 8.5e-5)
     (* (/ 2.0 (* (* k_m t) k_m)) (* (/ t_1 k_m) (/ l k_m)))
     (/
      (* (* t_1 (/ l k_m)) 2.0)
      (* k_m (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = (2.0 / ((k_m * t) * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = ((t_1 * (l / k_m)) * 2.0) / (k_m * ((0.5 - (0.5 * cos((2.0 * k_m)))) * t));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (k_m <= 8.5d-5) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * ((t_1 / k_m) * (l / k_m))
    else
        tmp = ((t_1 * (l / k_m)) * 2.0d0) / (k_m * ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * l;
	double tmp;
	if (k_m <= 8.5e-5) {
		tmp = (2.0 / ((k_m * t) * k_m)) * ((t_1 / k_m) * (l / k_m));
	} else {
		tmp = ((t_1 * (l / k_m)) * 2.0) / (k_m * ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = (2.0 / ((k_m * t) * k_m)) * ((t_1 / k_m) * (l / k_m))
	else:
		tmp = ((t_1 * (l / k_m)) * 2.0) / (k_m * ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 8.5e-5)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(Float64(t_1 / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(l / k_m)) * 2.0) / Float64(k_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = (2.0 / ((k_m * t) * k_m)) * ((t_1 / k_m) * (l / k_m));
	else
		tmp = ((t_1 * (l / k_m)) * 2.0) / (k_m * ((0.5 - (0.5 * cos((2.0 * k_m)))) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 8.5e-5], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.8

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

   10. Applied rewrites 76.8%

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

   if 8.500000000000001e-5 < k

   1. Initial program 23.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 93.7%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 93.7

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

   11. Applied rewrites 93.7%

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

Alternative 6: 85.1% accurate, 1.7× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.5e-5)
   (* (/ 2.0 (* (* k_m t) k_m)) (* (/ (* (cos k_m) l) k_m) (/ l k_m)))
   (*
    (/ 2.0 (* k_m (* k_m t)))
    (/ (* (cos k_m) (* l l)) (- 0.5 (* 0.5 (cos (* 2.0 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 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * ((cos(k_m) * (l * l)) / (0.5 - (0.5 * cos((2.0 * 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 (k_m <= 8.5d-5) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m))
    else
        tmp = (2.0d0 / (k_m * (k_m * t))) * ((cos(k_m) * (l * l)) / (0.5d0 - (0.5d0 * cos((2.0d0 * 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 (k_m <= 8.5e-5) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (((Math.cos(k_m) * l) / k_m) * (l / k_m));
	} else {
		tmp = (2.0 / (k_m * (k_m * t))) * ((Math.cos(k_m) * (l * l)) / (0.5 - (0.5 * Math.cos((2.0 * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.5e-5:
		tmp = (2.0 / ((k_m * t) * k_m)) * (((math.cos(k_m) * l) / k_m) * (l / k_m))
	else:
		tmp = (2.0 / (k_m * (k_m * t))) * ((math.cos(k_m) * (l * l)) / (0.5 - (0.5 * math.cos((2.0 * 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(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(cos(k_m) * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.5e-5)
		tmp = (2.0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
	else
		tmp = (2.0 / (k_m * (k_m * t))) * ((cos(k_m) * (l * l)) / (0.5 - (0.5 * cos((2.0 * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 8.500000000000001e-5

   1. Initial program 34.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.0

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

   5. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.8

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

   10. Applied rewrites 76.8%

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

   if 8.500000000000001e-5 < k

   1. Initial program 23.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 76.4

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

   5. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.7

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

   7. Applied rewrites 82.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 82.5

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

   9. Applied rewrites 82.5%

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

Alternative 7: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m \cdot \ell}{k\_m}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{elif}\;k\_m \leq 1.65 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot t\_1\right) \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{k\_m}\right) \cdot 2}{k\_m \cdot \left({\sin k\_m}^{2} \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* (cos k_m) l) k_m)))
   (if (<= k_m 1.8e-97)
     (* (/ 2.0 (* (* k_m t) k_m)) (* t_1 (/ l k_m)))
     (if (<= k_m 1.65e+129)
       (*
        (* (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m)) t_1)
        (/ l k_m))
       (/ (* (* l (/ l k_m)) 2.0) (* k_m (* (pow (sin k_m) 2.0) t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) / k_m;
	double tmp;
	if (k_m <= 1.8e-97) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else if (k_m <= 1.65e+129) {
		tmp = (((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * t_1) * (l / k_m);
	} else {
		tmp = ((l * (l / k_m)) * 2.0) / (k_m * (pow(sin(k_m), 2.0) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) / k_m)
	tmp = 0.0
	if (k_m <= 1.8e-97)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(t_1 * Float64(l / k_m)));
	elseif (k_m <= 1.65e+129)
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * t_1) * Float64(l / k_m));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / k_m)) * 2.0) / Float64(k_m * Float64((sin(k_m) ^ 2.0) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-97], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.65e+129], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.79999999999999999e-97

   1. Initial program 35.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.6

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

   10. Applied rewrites 76.6%

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

   if 1.79999999999999999e-97 < k < 1.64999999999999995e129

   1. Initial program 26.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 91.6

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

   5. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 96.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r/ N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. div-add-rev N/A

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

       6. lower-/.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 66.8

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 66.8%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   if 1.64999999999999995e129 < k

   1. Initial program 21.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 63.1

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

   5. Applied rewrites 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 92.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 66.9%

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

Alternative 8: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.75 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{\sin k\_m}^{2} \cdot t} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.75e-89)
   (* (/ 2.0 (* (* k_m t) k_m)) (* (/ (* (cos k_m) l) k_m) (/ l k_m)))
   (* (* (/ 2.0 (* (pow (sin k_m) 2.0) t)) (/ l k_m)) (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.75e-89) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
	} else {
		tmp = ((2.0 / (pow(sin(k_m), 2.0) * t)) * (l / k_m)) * (l / 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 (k_m <= 2.75d-89) then
        tmp = (2.0d0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m))
    else
        tmp = ((2.0d0 / ((sin(k_m) ** 2.0d0) * t)) * (l / k_m)) * (l / 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 (k_m <= 2.75e-89) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (((Math.cos(k_m) * l) / k_m) * (l / k_m));
	} else {
		tmp = ((2.0 / (Math.pow(Math.sin(k_m), 2.0) * t)) * (l / k_m)) * (l / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.75e-89:
		tmp = (2.0 / ((k_m * t) * k_m)) * (((math.cos(k_m) * l) / k_m) * (l / k_m))
	else:
		tmp = ((2.0 / (math.pow(math.sin(k_m), 2.0) * t)) * (l / k_m)) * (l / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.75e-89)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64((sin(k_m) ^ 2.0) * t)) * Float64(l / k_m)) * Float64(l / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.75e-89)
		tmp = (2.0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
	else
		tmp = ((2.0 / ((sin(k_m) ^ 2.0) * t)) * (l / k_m)) * (l / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.75e-89], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.75000000000000006e-89

   1. Initial program 35.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 68.2

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

   5. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 76.3

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

   10. Applied rewrites 76.3%

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

   if 2.75000000000000006e-89 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 77.7

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

   5. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. frac-times N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 66.3%

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

Alternative 9: 75.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m \cdot \ell}{k\_m}\\ \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(t\_1 \cdot \frac{\ell}{k\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{t}}{k\_m \cdot k\_m} \cdot t\_1\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* (cos k_m) l) k_m)))
   (if (<= l 1.25e+191)
     (* (/ 2.0 (* (* k_m t) k_m)) (* t_1 (/ l k_m)))
     (*
      (* (/ (/ (fma 0.6666666666666666 (* k_m k_m) 2.0) t) (* k_m k_m)) t_1)
      (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (cos(k_m) * l) / k_m;
	double tmp;
	if (l <= 1.25e+191) {
		tmp = (2.0 / ((k_m * t) * k_m)) * (t_1 * (l / k_m));
	} else {
		tmp = (((fma(0.6666666666666666, (k_m * k_m), 2.0) / t) / (k_m * k_m)) * t_1) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(cos(k_m) * l) / k_m)
	tmp = 0.0
	if (l <= 1.25e+191)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(t_1 * Float64(l / k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.6666666666666666, Float64(k_m * k_m), 2.0) / t) / Float64(k_m * k_m)) * t_1) * 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[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[l, 1.25e+191], N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.25000000000000005e191

   1. Initial program 31.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 72.1

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 88.9%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-*.f64 73.6

         \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   10. Applied rewrites 73.6%

       \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

   if 1.25000000000000005e191 < l

   1. Initial program 33.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 57.6

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      11. frac-times N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

      14. pow2 N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   7. Applied rewrites 90.5%

      \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\color{blue}{\frac{\cos k \cdot \ell}{k}} \cdot \frac{\ell}{k}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   9. Applied rewrites 99.5%

      \[\leadsto \left(\frac{2}{{\sin k}^{2} \cdot t} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{2}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       2. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       3. associate-*r/ N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2 \cdot 1}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2}}{t} + \frac{2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       5. div-add-rev N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       6. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{2}{3} \cdot {k}^{2} + 2}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, {k}^{2}, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       8. pow2 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{{k}^{2}} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       10. pow2 N/A

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(\frac{2}{3}, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

       11. lift-*.f64 61.9

           \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]

   12. Applied rewrites 61.9%

       \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{t}}{k \cdot k} \cdot \frac{\cos k \cdot \ell}{k}\right) \cdot \frac{\ell}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(k\_m \cdot t\right) \cdot k\_m} \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* (* k_m t) k_m)) (* (/ (* (cos k_m) l) k_m) (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
}

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 = (2.0d0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * t) * k_m)) * (((Math.cos(k_m) * l) / k_m) * (l / k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / ((k_m * t) * k_m)) * (((math.cos(k_m) * l) / k_m) * (l / k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(Float64(k_m * t) * k_m)) * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(l / k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / ((k_m * t) * k_m)) * (((cos(k_m) * l) / k_m) * (l / k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 70.9

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot \left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{\sin k}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\color{blue}{\sin k}}^{2}} \]

   8. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin \color{blue}{k}}^{2}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   10. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   11. frac-times N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{t}\right) \cdot {\sin k}^{2}} \]

   14. pow2 N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot {\sin \color{blue}{k}}^{2}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   16. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

7. Applied rewrites 89.0%

   \[\leadsto \frac{2}{{\sin k}^{2} \cdot t} \cdot \color{blue}{\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right)} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

   5. lift-*.f64 72.5

      \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \]

10. Applied rewrites 72.5%

    \[\leadsto \frac{2}{\left(k \cdot t\right) \cdot k} \cdot \left(\frac{\cos k \cdot \ell}{\color{blue}{k}} \cdot \frac{\ell}{k}\right) \]
11. Add Preprocessing

Alternative 11: 70.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{k\_m \cdot k\_m} \cdot \left({k\_m}^{-2} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.4e-148)
   (* (/ 2.0 (* k_m k_m)) (* (pow k_m -2.0) (* (/ l t) l)))
   (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.4e-148) {
		tmp = (2.0 / (k_m * k_m)) * (pow(k_m, -2.0) * ((l / t) * l));
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / 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 (t <= 1.4d-148) then
        tmp = (2.0d0 / (k_m * k_m)) * ((k_m ** (-2.0d0)) * ((l / t) * l))
    else
        tmp = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / 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 (t <= 1.4e-148) {
		tmp = (2.0 / (k_m * k_m)) * (Math.pow(k_m, -2.0) * ((l / t) * l));
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.4e-148:
		tmp = (2.0 / (k_m * k_m)) * (math.pow(k_m, -2.0) * ((l / t) * l))
	else:
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.4e-148)
		tmp = Float64(Float64(2.0 / Float64(k_m * k_m)) * Float64((k_m ^ -2.0) * Float64(Float64(l / t) * l)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.4e-148)
		tmp = (2.0 / (k_m * k_m)) * ((k_m ^ -2.0) * ((l / t) * l));
	else
		tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.4e-148], N[(N[(2.0 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4e-148

   1. Initial program 29.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

      8. lift-*.f64 56.8

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 \cdot 1}{{k}^{4}} \cdot \frac{\color{blue}{\ell} \cdot \ell}{t} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 \cdot 1}{{k}^{\left(2 + 2\right)}} \cdot \frac{\ell \cdot \ell}{t} \]

      5. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot 1}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

      6. times-frac N/A

         \[\leadsto \left(\frac{2}{{k}^{2}} \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{2}{{k}^{2}} \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{2}{{k}^{2}} \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\color{blue}{\ell} \cdot \ell}{t} \]

      9. pow2 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\ell \cdot \ell}{t} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\frac{2}{k \cdot k} \cdot \frac{1}{{k}^{2}}\right) \cdot \frac{\ell \cdot \ell}{t} \]

      11. pow-flip N/A

          \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{\left(\mathsf{neg}\left(2\right)\right)}\right) \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot \ell}{t} \]

      13. lower-pow.f64 56.8

          \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   7. Applied rewrites 56.8%

      \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot \ell}{t} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\color{blue}{\ell} \cdot \ell}{t} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot \ell}{t} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\frac{2}{k \cdot k} \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left({k}^{-2} \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left({k}^{-2} \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{{k}^{-2}} \cdot \frac{\ell \cdot \ell}{t}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{\color{blue}{-2}} \cdot \frac{\ell \cdot \ell}{t}\right) \]

      11. pow2 N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \frac{{\ell}^{2}}{t}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \color{blue}{\frac{{\ell}^{2}}{t}}\right) \]

      13. pow2 N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \frac{\ell \cdot \ell}{t}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\ell}\right)\right) \]

      17. lift-/.f64 64.6

          \[\leadsto \frac{2}{k \cdot k} \cdot \left({k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\right) \]

   9. Applied rewrites 64.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left({k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \ell\right)\right)} \]

   if 1.4e-148 < t

   1. Initial program 35.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

      16. lift-sin.f64 68.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

      6. lower-/.f64 70.9

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied rewrites 70.9%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* (* k_m k_m) t)) (* (/ l k_m) (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
}

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 = (2.0d0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l / k_m) * Float64(l / k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / ((k_m * k_m) * t)) * ((l / k_m) * (l / k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 70.9

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
7. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k}}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   6. lower-/.f64 67.6

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

8. Applied rewrites 67.6%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]
9. Add Preprocessing

Alternative 13: 66.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{k\_m \cdot \left(k\_m \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 2.0 (* k_m (* k_m t))) (/ (* l l) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
}

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 = (2.0d0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(k_m * Float64(k_m * t))) * Float64(Float64(l * l) / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / (k_m * (k_m * t))) * ((l * l) / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \color{blue}{\cos k}}{{\sin k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos \color{blue}{k}}{{\sin k}^{2}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{\sin k}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\color{blue}{\sin k}}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin \color{blue}{k}}^{2}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

   15. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{\color{blue}{2}}} \]

   16. lift-sin.f64 70.9

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \left(\color{blue}{\ell} \cdot \ell\right)}{{\sin k}^{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

   5. lower-*.f64 73.6

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}{{\sin k}^{2}} \]

7. Applied rewrites 73.6%

   \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}} \]

   2. pow2 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{{k}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

   5. lower-*.f64 61.0

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k} \]

10. Applied rewrites 61.0%

    \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]
11. Add Preprocessing

Alternative 14: 20.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (/ (* -0.11666666666666667 (* l l)) t))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.11666666666666667 * (l * 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 = ((-0.11666666666666667d0) * (l * l)) / t
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.11666666666666667 * (l * l)) / t;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.11666666666666667 * (l * l)) / t

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.11666666666666667 * Float64(l * l)) / t)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.11666666666666667 * (l * l)) / t;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(-0.11666666666666667 * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 17.5

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 17.5%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{t} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot \left(\ell \cdot \ell\right)}{t} \]

   9. lift-*.f64 17.5

      \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]

10. Applied rewrites 17.5%

    \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]
11. Add Preprocessing

Alternative 15: 20.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (/ (* l l) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * 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 = (-0.11666666666666667d0) * ((l * l) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * ((l * l) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * ((l * l) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(Float64(l * l) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * ((l * l) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 17.5

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 17.5%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 16: 18.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* -0.11666666666666667 (* l (/ l t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (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 = (-0.11666666666666667d0) * (l * (l / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.11666666666666667 * (l * (l / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.11666666666666667 * (l * (l / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.11666666666666667 * Float64(l * Float64(l / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.11666666666666667 * (l * (l / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.11666666666666667 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 17.5

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 17.5%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 14.9

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 14.9%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(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))))