Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 94.7%

Time: 7.9s

Alternatives: 11

Speedup: 5.8×

• 40.1% 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: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell + \ell}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-6)
   (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m)))
   (*
    (/ (* (cos k_m) l) k_m)
    (/ (+ l l) (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = ((cos(k_m) * l) / k_m) * ((l + l) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * 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 <= 1.7d-6) then
        tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
    else
        tmp = ((cos(k_m) * l) / k_m) * ((l + l) / (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * 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 <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = ((Math.cos(k_m) * l) / k_m) * ((l + l) / (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e-6:
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
	else:
		tmp = ((math.cos(k_m) * l) / k_m) * ((l + l) / (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(Float64(l + l) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * 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 <= 1.7e-6)
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	else
		tmp = ((cos(k_m) * l) / k_m) * ((l + l) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * 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, 1.7e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e-6

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. 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. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 64.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 64.0

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. *-commutative N/A

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

   8. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 74.3%

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

   if 1.70000000000000003e-6 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 83.0%

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

   10. Applied rewrites 83.9%

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

Alternative 2: 86.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(t \cdot k\_m\right)\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-6)
   (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m)))
   (/
    (* (* (* (cos k_m) l) l) 2.0)
    (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* t k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * (t * k_m)) * k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.7d-6) then
        tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
    else
        tmp = (((cos(k_m) * l) * l) * 2.0d0) / (((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * (t * k_m)) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((Math.cos(k_m) * l) * l) * 2.0) / (((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * (t * k_m)) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e-6:
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
	else:
		tmp = (((math.cos(k_m) * l) * l) * 2.0) / (((0.5 - (math.cos((k_m + k_m)) * 0.5)) * (t * k_m)) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(t * k_m)) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e-6)
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	else
		tmp = (((cos(k_m) * l) * l) * 2.0) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * (t * k_m)) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e-6

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. 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. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 64.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 64.0

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. *-commutative N/A

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

   8. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 74.3%

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

   if 1.70000000000000003e-6 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

Alternative 3: 86.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\_m\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-6)
   (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m)))
   (/
    (* (* (* l l) (cos k_m)) 2.0)
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((l * l) * cos(k_m)) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.7d-6) then
        tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
    else
        tmp = (((l * l) * cos(k_m)) * 2.0d0) / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((l * l) * Math.cos(k_m)) * 2.0) / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e-6:
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
	else:
		tmp = (((l * l) * math.cos(k_m)) * 2.0) / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) * cos(k_m)) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e-6)
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	else
		tmp = (((l * l) * cos(k_m)) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e-6

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. 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. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 64.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 64.0

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. *-commutative N/A

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

   8. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 74.3%

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

   if 1.70000000000000003e-6 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 70.8

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

   8. Applied rewrites 70.8%

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

Alternative 4: 86.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k\_m \cdot \ell\right) \cdot \left(\ell + \ell\right)}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-6)
   (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m)))
   (/
    (* (* (cos k_m) l) (+ l l))
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = ((cos(k_m) * l) * (l + l)) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.7d-6) then
        tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
    else
        tmp = ((cos(k_m) * l) * (l + l)) / ((((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-6) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = ((Math.cos(k_m) * l) * (l + l)) / ((((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e-6:
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
	else:
		tmp = ((math.cos(k_m) * l) * (l + l)) / ((((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-6)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) * Float64(l + l)) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e-6)
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	else
		tmp = ((cos(k_m) * l) * (l + l)) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e-6], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e-6

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. 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. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 64.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 64.0

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. *-commutative N/A

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

   8. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 74.3%

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

   if 1.70000000000000003e-6 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 83.0%

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

   10. Applied rewrites 70.8%

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

Alternative 5: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 60000000000:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 \cdot \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{2}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 60000000000.0)
   (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m)))
   (*
    (* (/ (* 1.0 l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t)) (/ l k_m))
    (/ 2.0 k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 60000000000.0) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((1.0 * l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (2.0 / k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 60000000000.0d0) then
        tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
    else
        tmp = (((1.0d0 * l) / ((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * t)) * (l / k_m)) * (2.0d0 / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 60000000000.0) {
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	} else {
		tmp = (((1.0 * l) / ((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (2.0 / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 60000000000.0:
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
	else:
		tmp = (((1.0 * l) / ((0.5 - (math.cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (2.0 / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 60000000000.0)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t)) * Float64(l / k_m)) * Float64(2.0 / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 60000000000.0)
		tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
	else
		tmp = (((1.0 * l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * t)) * (l / k_m)) * (2.0 / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 60000000000.0], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e10

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. 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. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 64.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lower-+.f64 64.0

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. pow2 N/A

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

      24. *-commutative N/A

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

   8. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. lift-/.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 74.3%

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

   if 6e10 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 65.7%

       \[\leadsto \left(\frac{1 \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell + \ell}{k\_m \cdot k\_m}}{t} \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ (+ l l) (* k_m k_m)) t) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l + l) / Float64(k_m * k_m)) / t) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l + l) / (k_m * k_m)) / t) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k$95$m * 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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 64.0

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 64.0

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

   21. lift-*.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. pow2 N/A

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

   24. *-commutative N/A

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

8. Applied rewrites 64.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. frac-times N/A

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

   11. pow2 N/A

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

   12. associate-/l/ N/A

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

   13. lift-/.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. associate-*r/ N/A

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

10. Applied rewrites 74.3%

    \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]
11. Add Preprocessing

Alternative 7: 74.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell + \ell}{k\_m}}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)} \end{array} \]

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(Float64(l + l) / k_m) / k_m) * Float64(l / Float64(k_m * Float64(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[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f64 72.8

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

8. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 74.1

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

10. Applied rewrites 74.1%

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

Alternative 8: 72.8% accurate, 5.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* 2.0 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 ((2.0 * 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 = ((2.0d0 * 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 ((2.0 * l) / (k_m * k_m)) * (l / (k_m * (k_m * t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 * 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(2.0 * l) / Float64(k_m * k_m)) * Float64(l / Float64(k_m * Float64(k_m * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 * 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[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * t), $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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 72.8

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

8. Applied rewrites 72.8%

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

Alternative 9: 72.8% 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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 72.8

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

8. Applied rewrites 72.8%

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

Alternative 10: 71.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell + \ell}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (+ l l) (* (* (* k_m k_m) t) k_m)) (/ l k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l + l) / (((k_m * k_m) * t) * k_m)) * (l / k_m);
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l + l) / (((k_m * k_m) * t) * k_m)) * (l / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l + l) / (((k_m * k_m) * t) * k_m)) * (l / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) / (((k_m * k_m) * t) * k_m)) * (l / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l + l) / Float64(Float64(Float64(k_m * k_m) * t) * k_m)) * Float64(l / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l + l) / (((k_m * k_m) * t) * k_m)) * (l / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 64.0

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 64.0

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

   21. lift-*.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. pow2 N/A

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

   24. *-commutative N/A

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

8. Applied rewrites 64.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-/.f64 71.8

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

10. Applied rewrites 71.8%

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

Alternative 11: 70.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (+ l l) (/ l (* (* (* (* k_m k_m) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l + l) * (l / ((((k_m * k_m) * t) * 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 / ((((k_m * k_m) * t) * 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 / ((((k_m * k_m) * t) * k_m) * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l + l) * (l / ((((k_m * k_m) * t) * 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(Float64(k_m * k_m) * t) * k_m) * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l + l) * (l / ((((k_m * k_m) * t) * 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[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
5. 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. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 72.8

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

6. Applied rewrites 72.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 64.0

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 64.0

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

   21. lift-*.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. pow2 N/A

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

   24. *-commutative N/A

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \]

   3. associate-/l* N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   5. lower-/.f64 70.5

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   8. associate-*r* N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \]

   12. pow2 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left({k}^{2} \cdot t\right) \cdot k\right) \cdot k} \]

   13. lower-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left({k}^{2} \cdot t\right) \cdot k\right) \cdot k} \]

   14. pow2 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 70.5

       \[\leadsto \left(\ell + \ell\right) \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k} \]

10. Applied rewrites 70.5%

    \[\leadsto \left(\ell + \ell\right) \cdot \color{blue}{\frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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))))