Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 96.2%

Time: 7.4s

Alternatives: 14

Speedup: 5.8×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% 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: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k\_m \cdot \ell}{k\_m}}{0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5} \cdot \frac{\frac{\ell}{k\_m}}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (*
    2.0
    (*
     (/ (/ (* (cos k_m) l) k_m) (- 0.5 (* (cos (+ k_m k_m)) 0.5)))
     (/ (/ l k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((((cos(k_m) * l) / k_m) / (0.5 - (cos((k_m + k_m)) * 0.5))) * ((l / 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) :: tmp
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * ((((cos(k_m) * l) / k_m) / (0.5d0 - (cos((k_m + k_m)) * 0.5d0))) * ((l / 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((((Math.cos(k_m) * l) / k_m) / (0.5 - (Math.cos((k_m + k_m)) * 0.5))) * ((l / k_m) / t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * ((((math.cos(k_m) * l) / k_m) / (0.5 - (math.cos((k_m + k_m)) * 0.5))) * ((l / k_m) / t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(cos(k_m) * l) / k_m) / Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5))) * Float64(Float64(l / k_m) / t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * ((((cos(k_m) * l) / k_m) / (0.5 - (cos((k_m + k_m)) * 0.5))) * ((l / k_m) / t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 91.2

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

   8. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 85.9%

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

Alternative 2: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\frac{\ell}{k\_m}}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (*
    2.0
    (*
     (/ (* (cos k_m) l) k_m)
     (/ (/ l k_m) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((cos(k_m) * l) / k_m) * ((l / k_m) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * 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) :: tmp
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * (((cos(k_m) * l) / k_m) * ((l / k_m) / ((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * l) / k_m) * ((l / k_m) / ((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * (((math.cos(k_m) * l) / k_m) * ((l / k_m) / ((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(Float64(l / k_m) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * (((cos(k_m) * l) / k_m) * ((l / k_m) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 91.2

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

   8. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.6

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 85.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 85.9

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

      18. lift-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 85.9

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

   10. Applied rewrites 85.9%

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

Alternative 3: 92.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t\\ \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 8.5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \left(\frac{t\_1}{k\_m \cdot k\_m} \cdot \frac{\ell}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t\_1 \cdot \frac{\ell}{k\_m}}{k\_m \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)) (t_2 (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t)))
   (if (<= k_m 9e-5)
     (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
     (if (<= k_m 8.5e+127)
       (* 2.0 (* (/ t_1 (* k_m k_m)) (/ l t_2)))
       (* 2.0 (/ (* t_1 (/ l k_m)) (* k_m t_2)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double t_2 = (0.5 - (0.5 * cos((2.0 * k_m)))) * t;
	double tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else if (k_m <= 8.5e+127) {
		tmp = 2.0 * ((t_1 / (k_m * k_m)) * (l / t_2));
	} else {
		tmp = 2.0 * ((t_1 * (l / k_m)) / (k_m * t_2));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = cos(k_m) * l
    t_2 = (0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else if (k_m <= 8.5d+127) then
        tmp = 2.0d0 * ((t_1 / (k_m * k_m)) * (l / t_2))
    else
        tmp = 2.0d0 * ((t_1 * (l / k_m)) / (k_m * t_2))
    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 t_2 = (0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t;
	double tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else if (k_m <= 8.5e+127) {
		tmp = 2.0 * ((t_1 / (k_m * k_m)) * (l / t_2));
	} else {
		tmp = 2.0 * ((t_1 * (l / k_m)) / (k_m * t_2));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	t_2 = (0.5 - (0.5 * math.cos((2.0 * k_m)))) * t
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	elif k_m <= 8.5e+127:
		tmp = 2.0 * ((t_1 / (k_m * k_m)) * (l / t_2))
	else:
		tmp = 2.0 * ((t_1 * (l / k_m)) / (k_m * t_2))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	elseif (k_m <= 8.5e+127)
		tmp = Float64(2.0 * Float64(Float64(t_1 / Float64(k_m * k_m)) * Float64(l / t_2)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(l / k_m)) / Float64(k_m * t_2)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	t_2 = (0.5 - (0.5 * cos((2.0 * k_m)))) * t;
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	elseif (k_m <= 8.5e+127)
		tmp = 2.0 * ((t_1 / (k_m * k_m)) * (l / t_2));
	else
		tmp = 2.0 * ((t_1 * (l / k_m)) / (k_m * t_2));
	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]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.5e+127], N[(2.0 * N[(N[(t$95$1 / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k < 8.4999999999999997e127

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 86.7

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

   8. Applied rewrites 76.2%

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

   if 8.4999999999999997e127 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 87.3

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

   8. Applied rewrites 79.3%

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

Alternative 4: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (*
    2.0
    (*
     (/ (* (cos k_m) l) (* k_m k_m))
     (/ l (* (- 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((cos(k_m) * l) / (k_m * k_m)) * (l / ((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) :: tmp
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * (((cos(k_m) * l) / (k_m * k_m)) * (l / ((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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * l) / (k_m * k_m)) * (l / ((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):
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * (((math.cos(k_m) * l) / (k_m * k_m)) * (l / ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * l) / Float64(k_m * k_m)) * Float64(l / 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)
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * (((cos(k_m) * l) / (k_m * k_m)) * (l / ((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_] := If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 86.7

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

   8. Applied rewrites 76.2%

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

Alternative 5: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)\right) \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (*
    2.0
    (*
     (* (cos k_m) l)
     (/ l (* (* (* k_m 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((cos(k_m) * l) * (l / (((k_m * 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) :: tmp
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * ((cos(k_m) * l) * (l / (((k_m * 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * l) * (l / (((k_m * 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):
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * ((math.cos(k_m) * l) * (l / (((k_m * 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)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * l) * Float64(l / Float64(Float64(Float64(k_m * k_m) * 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)
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * ((cos(k_m) * l) * (l / (((k_m * 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_] := If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 74.7%

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

Alternative 6: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k\_m}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e-5)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (*
    2.0
    (*
     l
     (*
      l
      (/
       (cos k_m)
       (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* k_m k_m)) t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (l * (l * (cos(k_m) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * 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) :: tmp
    if (k_m <= 9d-5) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * (l * (l * (cos(k_m) / (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * (k_m * 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 tmp;
	if (k_m <= 9e-5) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (l * (l * (Math.cos(k_m) / (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * (k_m * k_m)) * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9e-5:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * (l * (l * (math.cos(k_m) / (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * (k_m * k_m)) * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k_m) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m * k_m)) * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e-5)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * (l * (l * (cos(k_m) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * k_m)) * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e-5], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000057e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 9.00000000000000057e-5 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 74.7

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 74.7

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

   10. Applied rewrites 74.7%

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

Alternative 7: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{t}{\ell \cdot \frac{\ell}{k\_m \cdot k\_m}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.8e-10)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (/ 2.0 (* (* (tan k_m) (sin k_m)) (/ t (* l (/ l (* k_m k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.8e-10) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 / ((tan(k_m) * sin(k_m)) * (t / (l * (l / (k_m * k_m)))));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.8d-10) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 / ((tan(k_m) * sin(k_m)) * (t / (l * (l / (k_m * k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.8e-10) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 / ((Math.tan(k_m) * Math.sin(k_m)) * (t / (l * (l / (k_m * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.8e-10:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 / ((math.tan(k_m) * math.sin(k_m)) * (t / (l * (l / (k_m * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.8e-10)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * sin(k_m)) * Float64(t / Float64(l * Float64(l / Float64(k_m * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.8e-10)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 / ((tan(k_m) * sin(k_m)) * (t / (l * (l / (k_m * 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, 3.8e-10], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / N[(l * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7999999999999998e-10

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 3.7999999999999998e-10 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.0

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 75.0

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 75.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. times-frac N/A

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

   8. Applied rewrites 83.4%

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

Alternative 8: 75.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4400:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{1 \cdot \ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{\sin k\_m}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4400.0)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (* 2.0 (/ (* (/ (* 1.0 l) k_m) (/ l k_m)) (* (pow (sin k_m) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4400.0) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((((1.0 * l) / k_m) * (l / k_m)) / (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) :: tmp
    if (k_m <= 4400.0d0) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * ((((1.0d0 * l) / k_m) * (l / k_m)) / ((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 tmp;
	if (k_m <= 4400.0) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((((1.0 * l) / k_m) * (l / k_m)) / (Math.pow(Math.sin(k_m), 2.0) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4400.0:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * ((((1.0 * l) / k_m) * (l / k_m)) / (math.pow(math.sin(k_m), 2.0) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4400.0)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(1.0 * l) / k_m) * Float64(l / k_m)) / Float64((sin(k_m) ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4400.0)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * ((((1.0 * l) / k_m) * (l / k_m)) / ((sin(k_m) ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4400.0], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(1.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $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 < 4400

   1. Initial program 36.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 62.8

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

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

      10. sqr-pow N/A

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

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 62.8

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 62.8

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if 4400 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      5. lower-/.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t} \cdot {\sin k}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. lower-*.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      10. pow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      11. lift-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      14. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \color{blue}{{\sin k}^{2}}} \]

      16. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

      17. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

   6. Applied rewrites 75.0%

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2}} \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\color{blue}{\sin k}}^{2} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{{\color{blue}{\sin k}}^{2} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}}{{\sin k}^{\color{blue}{2}} \cdot t} \]

      6. times-frac N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{\sin k}^{2}} \cdot t} \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{\sin k}^{2}} \cdot t} \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{\sin k}}^{2} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin \color{blue}{k}}^{2} \cdot t} \]

      10. lower-/.f64 91.2

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{\color{blue}{2}} \cdot t} \]

   8. Applied rewrites 91.2%

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{\sin k}^{2}} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{\frac{1 \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.2%

       \[\leadsto 2 \cdot \frac{\frac{1 \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 14200000000000:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k\_m \cdot \left(\ell \cdot \ell\right)}{k\_m \cdot k\_m}}{{k\_m}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 14200000000000.0)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (* 2.0 (/ (/ (* (cos k_m) (* l l)) (* k_m k_m)) (* (pow k_m 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 14200000000000.0) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((cos(k_m) * (l * l)) / (k_m * k_m)) / (pow(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) :: tmp
    if (k_m <= 14200000000000.0d0) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * (((cos(k_m) * (l * l)) / (k_m * k_m)) / ((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 tmp;
	if (k_m <= 14200000000000.0) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * (((Math.cos(k_m) * (l * l)) / (k_m * k_m)) / (Math.pow(k_m, 2.0) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 14200000000000.0:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * (((math.cos(k_m) * (l * l)) / (k_m * k_m)) / (math.pow(k_m, 2.0) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 14200000000000.0)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) * Float64(l * l)) / Float64(k_m * k_m)) / Float64((k_m ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 14200000000000.0)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * (((cos(k_m) * (l * l)) / (k_m * k_m)) / ((k_m ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 14200000000000.0], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e13

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 62.8

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 62.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      5. lower-*.f64 62.8

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

      10. sqr-pow N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      15. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      16. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      17. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      18. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      20. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      21. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot t\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      18. lower-*.f64 73.4

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 73.4%

      \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

   if 1.42e13 < k

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      5. lower-/.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t} \cdot {\sin k}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. lower-*.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      10. pow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      11. lift-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      14. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \color{blue}{{\sin k}^{2}}} \]

      16. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

      17. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

   6. Applied rewrites 75.0%

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2} \cdot t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{k}^{2} \cdot \color{blue}{t}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{k}^{2} \cdot t} \]

      2. lower-pow.f64 67.2

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{k}^{2} \cdot t} \]

   9. Applied rewrites 67.2%

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{k}^{2} \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(0.5 - 0.5\right)\right) \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 2e+208)
   (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t)))
   (* 2.0 (* (* l l) (/ (cos k_m) (* (* (* k_m k_m) (- 0.5 0.5)) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e+208) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((l * l) * (cos(k_m) / (((k_m * k_m) * (0.5 - 0.5)) * 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) :: tmp
    if ((l * l) <= 2d+208) then
        tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
    else
        tmp = 2.0d0 * ((l * l) * (cos(k_m) / (((k_m * k_m) * (0.5d0 - 0.5d0)) * 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 tmp;
	if ((l * l) <= 2e+208) {
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	} else {
		tmp = 2.0 * ((l * l) * (Math.cos(k_m) / (((k_m * k_m) * (0.5 - 0.5)) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 2e+208:
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
	else:
		tmp = 2.0 * ((l * l) * (math.cos(k_m) / (((k_m * k_m) * (0.5 - 0.5)) * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e+208)
		tmp = Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * l) * Float64(cos(k_m) / Float64(Float64(Float64(k_m * k_m) * Float64(0.5 - 0.5)) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 2e+208)
		tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
	else
		tmp = 2.0 * ((l * l) * (cos(k_m) / (((k_m * k_m) * (0.5 - 0.5)) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+208], N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e208

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      5. lower-pow.f64 62.8

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. Applied rewrites 62.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      5. lower-*.f64 62.8

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      8. lift-*.f64 62.8

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

      10. sqr-pow N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      15. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      16. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      17. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      18. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      20. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      21. lower-*.f64 62.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot t\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      18. lower-*.f64 73.4

          \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 73.4%

      \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

   if 2e208 < (*.f64 l l)

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      5. lower-/.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{\color{blue}{t} \cdot {\sin k}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. lower-*.f64 75.0

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      10. pow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      11. lift-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      14. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot {\sin k}^{2}} \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \color{blue}{{\sin k}^{2}}} \]

      16. *-commutative N/A

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

      17. lower-*.f64 75.0

          \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot \color{blue}{t}} \]

   6. Applied rewrites 75.0%

      \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2} \cdot t}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{\color{blue}{{\sin k}^{2}} \cdot t} \]

      3. associate-/l/ N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      6. associate-/l* N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\color{blue}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot \color{blue}{t}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot {\sin k}^{2}\right) \cdot \color{blue}{t}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot {\sin k}^{2}\right) \cdot \color{blue}{t}}\right) \]

   8. Applied rewrites 67.4%

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(k \cdot k\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right) \cdot t}}\right) \]

   9. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot \left(\frac{1}{2} - \frac{1}{2}\right)\right) \cdot t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 34.6%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot \left(0.5 - 0.5\right)\right) \cdot t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell + \ell}{k\_m \cdot k\_m} \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (+ l l) (* k_m k_m)) (/ l (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) * Float64(l / Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l + l) / (k_m * k_m)) * (l / ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.8

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   5. lower-*.f64 62.8

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   8. lift-*.f64 62.8

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

   10. sqr-pow N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   15. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   17. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   18. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   19. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   20. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   21. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

6. Applied rewrites 62.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left({k}^{2} \cdot t\right)} \]

   10. times-frac N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

   13. count-2-rev N/A

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

   16. pow2 N/A

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   18. lower-*.f64 73.4

       \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

8. Applied rewrites 73.4%

   \[\leadsto \frac{\ell + \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
9. Add Preprocessing

Alternative 12: 70.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(t \cdot \left(k\_m \cdot k\_m\right)\right) \cdot k\_m\right) \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (+ l l) (/ l (* (* (* t (* k_m k_m)) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l + l) * (l / (((t * (k_m * k_m)) * 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 = (l + l) * (l / (((t * (k_m * k_m)) * k_m) * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l + l) * (l / (((t * (k_m * k_m)) * k_m) * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l + l) * (l / (((t * (k_m * k_m)) * k_m) * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l + l) * Float64(l / Float64(Float64(Float64(t * Float64(k_m * k_m)) * k_m) * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l + l) * (l / (((t * (k_m * k_m)) * k_m) * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.8

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   5. lower-*.f64 62.8

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   8. lift-*.f64 62.8

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

   10. sqr-pow N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   15. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   17. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   18. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   19. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   20. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   21. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

6. Applied rewrites 62.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   7. count-2-rev N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. lower-/.f64 68.7

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   11. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\color{blue}{k} \cdot k\right)\right)} \]

   14. associate-*l* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   16. cube-unmult N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\color{blue}{3}}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\left(\frac{6}{\color{blue}{2}}\right)}\right)} \]

   18. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left({k}^{\left(\frac{6}{2}\right)} \cdot \color{blue}{k}\right)} \]

   19. associate-*r* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

8. Applied rewrites 69.9%

   \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k} \]

   3. associate-*r* N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k} \]

   5. lower-*.f64 70.7

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k} \]

10. Applied rewrites 70.7%

    \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k} \]
11. Add Preprocessing

Alternative 13: 69.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right)\right) \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (+ l l) (/ l (* (* t (* (* k_m k_m) k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l + l) * (l / ((t * ((k_m * k_m) * 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 = (l + l) * (l / ((t * ((k_m * k_m) * k_m)) * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l + l) * (l / ((t * ((k_m * k_m) * k_m)) * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l + l) * (l / ((t * ((k_m * k_m) * k_m)) * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l + l) * Float64(l / Float64(Float64(t * Float64(Float64(k_m * k_m) * k_m)) * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l + l) * (l / ((t * ((k_m * k_m) * k_m)) * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.8

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   5. lower-*.f64 62.8

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   8. lift-*.f64 62.8

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

   10. sqr-pow N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   15. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   17. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   18. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   19. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   20. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   21. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

6. Applied rewrites 62.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   7. count-2-rev N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. lower-/.f64 68.7

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   11. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\color{blue}{k} \cdot k\right)\right)} \]

   14. associate-*l* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   16. cube-unmult N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\color{blue}{3}}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\left(\frac{6}{\color{blue}{2}}\right)}\right)} \]

   18. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left({k}^{\left(\frac{6}{2}\right)} \cdot \color{blue}{k}\right)} \]

   19. associate-*r* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

8. Applied rewrites 69.9%

   \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k}} \]
9. Add Preprocessing

Alternative 14: 68.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (+ l l) (/ l (* t (* (* (* k_m k_m) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l + l) * (l / (t * (((k_m * k_m) * 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 = (l + l) * (l / (t * (((k_m * k_m) * k_m) * k_m)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l + l) * (l / (t * (((k_m * k_m) * k_m) * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l + l) * (l / (t * (((k_m * k_m) * k_m) * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l + l) * Float64(l / Float64(t * Float64(Float64(Float64(k_m * k_m) * k_m) * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l + l) * (l / (t * (((k_m * k_m) * k_m) * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(t * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.8

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   5. lower-*.f64 62.8

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   8. lift-*.f64 62.8

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t} \]

   10. sqr-pow N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

   13. metadata-eval N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   15. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   16. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

   17. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   18. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   19. lift-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

   20. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   21. lower-*.f64 62.8

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

6. Applied rewrites 62.8%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   7. count-2-rev N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\color{blue}{\ell}}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. lower-/.f64 68.7

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   11. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\color{blue}{k} \cdot k\right)\right)} \]

   14. associate-*l* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   16. cube-unmult N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\color{blue}{3}}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(k \cdot {k}^{\left(\frac{6}{\color{blue}{2}}\right)}\right)} \]

   18. *-commutative N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left({k}^{\left(\frac{6}{2}\right)} \cdot \color{blue}{k}\right)} \]

   19. associate-*r* N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot {k}^{\left(\frac{6}{2}\right)}\right) \cdot \color{blue}{k}} \]

8. Applied rewrites 69.9%

   \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot \color{blue}{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(t \cdot \left(\left(k \cdot k\right) \cdot k\right)\right) \cdot k} \]

   3. associate-*l* N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)}} \]

   5. lower-*.f64 68.7

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot \color{blue}{k}\right)} \]

10. Applied rewrites 68.7%

    \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))